This document outlines a talk on using grossone in optimization. It discusses single and multi-objective linear programming and nonlinear optimization. It covers linear programming and the simplex method, including preliminary results, basic feasible solutions, associated bases, and convergence. It also discusses the lexicographic rule and recent results.
This document outlines a talk on the use of 1 in mathematical programming and summarizes several topics to be covered, including degeneracy and the simplex method, nonlinear optimization, equality constraints, inequality constraints, and data envelopment analysis. It provides details on linear programming and the simplex method, including preliminary results, basic feasible solutions, the single iteration process, and the lexicographic rule for selecting leaving variables. The document contains mathematical notation and definitions to explain these concepts.
This document outlines a talk on nonlinear programming and grossone theory and algorithms. It discusses equality constraints, inequality constraints, quadratic problems, and algorithms. For equality constraints, it presents the Lagrangian function and Karush-Kuhn-Tucker (KKT) first-order optimality conditions. It then discusses penalty functions and the sequential penalty method. Two examples applying the theory to problems with equality constraints are provided. Inequality constraints and first-order optimality conditions for problems with equality and inequality constraints are also covered.
The document discusses various topics related to analytics including:
1. It defines analytics as transforming data into insights for better decision making and describes the Deming cycle of plan, do, check, act.
2. It provides definitions and descriptions of different types of innovation - product, process, marketing, and organizational innovation.
3. It discusses how analytics can drive innovation and describes descriptive, predictive, and prescriptive analytics categories and common analytics tools.
4. Supply chain management and inventory optimization are provided as examples of analytics applications.
The document discusses power production and storage in microgrids. It presents a case study of optimizing the Leaf Community microgrid in Italy, which contains a photovoltaic plant, hydroelectric plant, battery storage, and loads from an office building and industrial facility. The goal is to minimize energy costs by determining the optimal strategy for buying and selling power to the grid and charging/discharging the battery storage. The optimization problem is formulated as a mixed-integer linear program to minimize costs while meeting loads based on forecasts of renewable production and demand over multiple days. The results show that renewable energy is used first to meet loads and the battery charges from low-cost power and discharges during high-cost periods.
This document outlines a talk on the use of 1 in mathematical programming and summarizes several topics to be covered, including degeneracy and the simplex method, nonlinear optimization, equality constraints, inequality constraints, and data envelopment analysis. It provides details on linear programming and the simplex method, including preliminary results, basic feasible solutions, the single iteration process, and the lexicographic rule for selecting leaving variables. The document contains mathematical notation and definitions to explain these concepts.
This document outlines a talk on nonlinear programming and grossone theory and algorithms. It discusses equality constraints, inequality constraints, quadratic problems, and algorithms. For equality constraints, it presents the Lagrangian function and Karush-Kuhn-Tucker (KKT) first-order optimality conditions. It then discusses penalty functions and the sequential penalty method. Two examples applying the theory to problems with equality constraints are provided. Inequality constraints and first-order optimality conditions for problems with equality and inequality constraints are also covered.
The document discusses various topics related to analytics including:
1. It defines analytics as transforming data into insights for better decision making and describes the Deming cycle of plan, do, check, act.
2. It provides definitions and descriptions of different types of innovation - product, process, marketing, and organizational innovation.
3. It discusses how analytics can drive innovation and describes descriptive, predictive, and prescriptive analytics categories and common analytics tools.
4. Supply chain management and inventory optimization are provided as examples of analytics applications.
The document discusses power production and storage in microgrids. It presents a case study of optimizing the Leaf Community microgrid in Italy, which contains a photovoltaic plant, hydroelectric plant, battery storage, and loads from an office building and industrial facility. The goal is to minimize energy costs by determining the optimal strategy for buying and selling power to the grid and charging/discharging the battery storage. The optimization problem is formulated as a mixed-integer linear program to minimize costs while meeting loads based on forecasts of renewable production and demand over multiple days. The results show that renewable energy is used first to meet loads and the battery charges from low-cost power and discharges during high-cost periods.
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
BREEDING METHODS FOR DISEASE RESISTANCE.pptxRASHMI M G
Plant breeding for disease resistance is a strategy to reduce crop losses caused by disease. Plants have an innate immune system that allows them to recognize pathogens and provide resistance. However, breeding for long-lasting resistance often involves combining multiple resistance genes
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
Creative operations teams expect increased AI use in 2024. Currently, over half of tasks are not AI-enabled, but this is expected to decrease in the coming year. ChatGPT is the most popular AI tool currently. Business leaders are more actively exploring AI benefits than individual contributors. Most respondents do not believe AI will impact workforce size in 2024. However, some inhibitions still exist around AI accuracy and lack of understanding. Creatives primarily want to use AI to save time on mundane tasks and boost productivity.
Organizational culture includes values, norms, systems, symbols, language, assumptions, beliefs, and habits that influence employee behaviors and how people interpret those behaviors. It is important because culture can help or hinder a company's success. Some key aspects of Netflix's culture that help it achieve results include hiring smartly so every position has stars, focusing on attitude over just aptitude, and having a strict policy against peacocks, whiners, and jerks.
PEPSICO Presentation to CAGNY Conference Feb 2024Neil Kimberley
PepsiCo provided a safe harbor statement noting that any forward-looking statements are based on currently available information and are subject to risks and uncertainties. It also provided information on non-GAAP measures and directing readers to its website for disclosure and reconciliation. The document then discussed PepsiCo's business overview, including that it is a global beverage and convenient food company with iconic brands, $91 billion in net revenue in 2023, and nearly $14 billion in core operating profit. It operates through a divisional structure with a focus on local consumers.
Content Methodology: A Best Practices Report (Webinar)contently
This document provides an overview of content methodology best practices. It defines content methodology as establishing objectives, KPIs, and a culture of continuous learning and iteration. An effective methodology focuses on connecting with audiences, creating optimal content, and optimizing processes. It also discusses why a methodology is needed due to the competitive landscape, proliferation of channels, and opportunities for improvement. Components of an effective methodology include defining objectives and KPIs, audience analysis, identifying opportunities, and evaluating resources. The document concludes with recommendations around creating a content plan, testing and optimizing content over 90 days.
How to Prepare For a Successful Job Search for 2024Albert Qian
The document provides guidance on preparing a job search for 2024. It discusses the state of the job market, focusing on growth in AI and healthcare but also continued layoffs. It recommends figuring out what you want to do by researching interests and skills, then conducting informational interviews. The job search should involve building a personal brand on LinkedIn, actively applying to jobs, tailoring resumes and interviews, maintaining job hunting as a habit, and continuing self-improvement. Once hired, the document advises setting new goals and keeping skills and networking active in case of future opportunities.
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
The document provides career advice for getting into the tech field, including:
- Doing projects and internships in college to build a portfolio.
- Learning about different roles and technologies through industry research.
- Contributing to open source projects to build experience and network.
- Developing a personal brand through a website and social media presence.
- Networking through events, communities, and finding a mentor.
- Practicing interviews through mock interviews and whiteboarding coding questions.
Google's Just Not That Into You: Understanding Core Updates & Search IntentLily Ray
1. Core updates from Google periodically change how its algorithms assess and rank websites and pages. This can impact rankings through shifts in user intent, site quality issues being caught up to, world events influencing queries, and overhauls to search like the E-A-T framework.
2. There are many possible user intents beyond just transactional, navigational and informational. Identifying intent shifts is important during core updates. Sites may need to optimize for new intents through different content types and sections.
3. Responding effectively to core updates requires analyzing "before and after" data to understand changes, identifying new intents or page types, and ensuring content matches appropriate intents across video, images, knowledge graphs and more.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Beginners Guide to TikTok for Search - Rachel Pearson - We are Tilt __ Bright...
Main
1. LION11 1 / 40
The use of grossone in optimization: a survey and some
recent results
R. De Leone
School of Science and Technology
Universit`a di Camerino
June 2017
2. Outline of the talk
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
LION11 2 / 40
Single and Multi Objective Linear Programming
Nonlinear Optimization
Some recent results
3. Single and Multi Objective
Linear Programming
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 3 / 40
4. Linear Programming and the Simplex Method
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 4 / 40
min
x
cT x
subject to Ax = b
x ≥ 0
The simplex method proposed by George Dantzig in 1947
■ starts at a corner point (a Basic Feasible Solution, BFS)
■ verifies if the current point is optimal
■ if not, moves along an edge to a new corner point
until the optimal corner point is identified or it discovers that the problem
has no solution.
5. Preliminary results and notations
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 5 / 40
Let
X = {x ∈ IRn
: Ax = b, x ≥ 0}
where A ∈ IRm×n, b ∈ IRm, m ≤ n.
A point ¯x ∈ X is a Basic Feasible Solution (BFS) iff the columns of A
corresponding to positive components of ¯x are linearly independent.
6. Preliminary results and notations
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 5 / 40
Let
X = {x ∈ IRn
: Ax = b, x ≥ 0}
where A ∈ IRm×n, b ∈ IRm, m ≤ n.
A point ¯x ∈ X is a Basic Feasible Solution (BFS) iff the columns of A
corresponding to positive components of ¯x are linearly independent.
Let ¯x be a BFS and define ¯I = I(¯x) := {j : ¯xj > 0} then
rank(A.¯I) = |¯I|. Note: |¯I| ≤ m
7. Preliminary results and notations
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 5 / 40
Let
X = {x ∈ IRn
: Ax = b, x ≥ 0}
where A ∈ IRm×n, b ∈ IRm, m ≤ n.
A point ¯x ∈ X is a Basic Feasible Solution (BFS) iff the columns of A
corresponding to positive components of ¯x are linearly independent.
Let ¯x be a BFS and define ¯I = I(¯x) := {j : ¯xj > 0} then
rank(A.¯I) = |¯I|. Note: |¯I| ≤ m
Vertex Point, Extreme Points and Basic Feasible Solution Point coin-
cide
BFS ≡ Vertex ≡ Extreme Point
8. BFS and associated basis
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 6 / 40
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B) = 0
N = {1, . . . , n} − B
Let ¯x be a BFS. .
9. BFS and associated basis
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 6 / 40
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B) = 0
N = {1, . . . , n} − B
Let ¯x be a BFS. .
If |{j : ¯xj > 0}| = m the BFS is said to be non–degenerate and
there is only a single base B := {j : ¯xj > 0} associated to ¯x
Non-degenerate BFS
10. BFS and associated basis
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 6 / 40
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B) = 0
N = {1, . . . , n} − B
Let ¯x be a BFS. .
If |{j : ¯xj > 0}| < m the BFS is said to be degenerate and
there are more than one base B1, B2, . . . , Bl associated to ¯x with
{j : ¯xj > 0} ⊆ Bi
Degenerate BFS
11. BFS and associated basis
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 6 / 40
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B) = 0
N = {1, . . . , n} − B
Let ¯x be a BFS. Let B a base associated to ¯x.
Then
¯xN = 0, ¯xB = A−1
.B b ≥ 0
12. BFS and associated basis
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 6 / 40
Assume now
rank(A) = m ≤ n
A base B is a subset of m linearly independent columns of A.
B ⊆ {1, . . . , n} , det(A.B) = 0
N = {1, . . . , n} − B
Let ¯x be a BFS. Let B a base associated to ¯x.
13. Convergence of the Simplex Method
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 7 / 40
Convergence of the simplex method is ensured if all basis visited by the
method are nondegenerate
14. Convergence of the Simplex Method
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 7 / 40
Convergence of the simplex method is ensured if all basis visited by the
method are nondegenerate
In presence of degenerate BFS, the Simplex method may not terminate
(cycling)
15. Convergence of the Simplex Method
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 7 / 40
Convergence of the simplex method is ensured if all basis visited by the
method are nondegenerate
In presence of degenerate BFS, the Simplex method may not terminate
(cycling)
⇓
Hence, specific anti-cycling procedures must be implemented (Bland’s
rule, lexicographic order)
16. Lexicographic Rule
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 8 / 40
At each iteration of the simplex method we choose the leaving variable
using the lexicographic rule
17. Lexicographic Rule
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 8 / 40
Let B0 be the initial base and N0 = {1, . . . , n} − B0.
We can always assume, after columns reordering, that A has the form
A = A.Bo
... A.No
18. Lexicographic Rule
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 8 / 40
Let
¯ρ = min
i: ¯Aijr >0
(A.−1
B b)i
¯Aijr
if such minimum value is reached in only one index this is the leaving
variable.
OTHERWISE
19. Lexicographic Rule
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 8 / 40
Among the indices i for which
min
i: ¯Aijr >0
(A.−1
B b)i
¯Aijr
= ¯ρ
we choose the index for which
min
i: ¯Aijr >0
(A.−1
B A.Bo)i1
¯Aijr
If the minimum is reached by only one index this is the leaving variable.
OTHERWISE
20. Lexicographic Rule
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 8 / 40
Among the indices reaching the minimum value, choose the index for which
min
i: ¯Aijr >0
(A.−1
B A.Bo)i2
¯Aijr
Proceed in the same way.
This procedure will terminate providing a single index since the rows of the
matrix (A.−1
B A.Bo) are linearly independent.
21. Lexicographic rule and RHS perturbation
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 9 / 40
The procedure outlined in the previous slides is equivalent to perturb each
component of the RHS vector b by a very small quantity.
If this perturbation is small enough, the new linear programming problem is
nondegerate and the simplex method produces exactly the same pivot
sequence as the lexicographic pivot rule
However, is very difficult to determine how small this perturbation must be.
More often a symbolic perturbation is used (with higher computational
costs)
22. Lexicographic rule and RHS perturbation and ①
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 10 / 40
Replace bi with bi with
bi +
j∈Bo
Aij①−j
.
23. Lexicographic rule and RHS perturbation and ①
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 10 / 40
Replace bi with bi with
bi +
j∈Bo
Aij①−j
.
Let
e =
①−1
①−2
...
①−m
and
b = A.−1
B (b + A.Boe) = A.−1
B b + A.−1
B A.Boe.
24. Lexicographic rule and RHS perturbation and ①
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 10 / 40
Replace bi with bi with
bi +
j∈Bo
Aij①−j
.
Therefore b = (A.−1
B b)i +
m
k=1
(A.−1
B A.Bo)ik
①−k
and
min
i: ¯Aijr >0
(A.−1
B b)i +
m
k=1
(A.−1
B A.Bo)ik
①−k
¯Aijr
=
min
i: ¯Aijr >0
(A.−1
B b)i
¯Aijr
+
(A.−1
B A.Bo)i1
¯Aijr
①−1
+ . . . +
(A.−1
B A.Bo)im
¯Aijr
①−m
25. Lexicographic multi-objective Linear Programming
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 11 / 40
lexmax
x
c1T
x, c2T
x, . . . , crT x
subject to Ax = b
x ≥ 0
The set
S := {Ax = b, x ≥ 0}
is bounded and non-empty.
26. Lexicographic multi-objective Linear Programming
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 11 / 40
lexmax
x
c1T
x, c2T
x, . . . , crT x
subject to Ax = b
x ≥ 0
Preemptive Scheme
Starts considering the first objective function alone:
max
x
c1x
subject to Ax = b
x ≥ 0
Let x∗1 be an optimal solution and β1 = c1T
x∗1.
27. Lexicographic multi-objective Linear Programming
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 11 / 40
lexmax
x
c1T
x, c2T
x, . . . , crT x
subject to Ax = b
x ≥ 0
Preemptive Scheme
Then solve
max
x
c2T
x
subject to Ax = b
c1T
x = c1T
x∗1
x ≥ 0
28. Lexicographic multi-objective Linear Programming
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 11 / 40
lexmax
x
c1T
x, c2T
x, . . . , crT x
subject to Ax = b
x ≥ 0
Preemptive Scheme
Repeat above schema until either the last problem is solved or an unique
solution has been determined.
29. Lexicographic multi-objective Linear Programming
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 11 / 40
lexmax
x
c1T
x, c2T
x, . . . , crT x
subject to Ax = b
x ≥ 0
Non–Preemptive Scheme
30. Lexicographic multi-objective Linear Programming
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 11 / 40
lexmax
x
c1T
x, c2T
x, . . . , crT x
subject to Ax = b
x ≥ 0
Non–Preemptive Scheme
There always exists a finite scalar MIR such that the solution of the above
problem can be obtained by solving the one single-objective LP problem
max
x
˜cT x
subject to Ax = b
x ≥ 0
where ˜c =
r
i=1
M−i+1
ci
.
31. Non–Preemptive grossone-based scheme
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 12 / 40
Solve the LP
max
x
˜cT x
subject to Ax = b
x ≥ 0
where
˜c =
r
i=1
①−i+1
ci
Note that
˜cT
x = c1T
x ①0
+ c2T
x ①−1
+ . . . crT
x ①r−1
32. Non–Preemptive grossone-based scheme
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 12 / 40
Solve the LP
max
x
˜cT x
subject to Ax = b
x ≥ 0
where
˜c =
r
i=1
①−i+1
ci
Note that
˜cT
x = c1T
x ①0
+ c2T
x ①−1
+ . . . crT
x ①r−1
The main advantage of this scheme is that it does not require the
specification of a real scalar value M
33. Non–Preemptive grossone-based scheme
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 12 / 40
Solve the LP
max
x
˜cT x
subject to Ax = b
x ≥ 0
where
˜c =
r
i=1
①−i+1
ci
Note that
˜cT
x = c1T
x ①0
+ c2T
x ①−1
+ . . . crT
x ①r−1
M. Cococcioni, M. Pappalardo, Y.D. Sergeyev
34. Theoretical results
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 13 / 40
max
x
˜cT x
subject to Ax = b
x ≥ 0
35. Theoretical results
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 13 / 40
max
x
˜cT x
subject to Ax = b
x ≥ 0
If the LP above has solution, there is always a solution that a vertex.
All optimal solutions of the lexicographic problem are feasible for the above
problem and have the objective value.
Any optimal solutions of the lexicographic problem is optimal for the above
problem, and viceversa.
36. Theoretical results
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 13 / 40
max
x
˜cT x
subject to Ax = b
x ≥ 0
The dual problem is
min
π
bT π
subject to AT π ≤ ˜c
If ¯x is feasible for the primal problem and ¯π feasible for the dual problem
˜cT
¯x ≤ bT
¯π
37. Theoretical results
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 13 / 40
max
x
˜cT x
subject to Ax = b
x ≥ 0
The dual problem is
min
π
bT π
subject to AT π ≤ ˜c
If x∗ is feasible for the primal problem and π∗ feasible for the dual problem
and
˜cT
x∗
= bT
π∗
x∗ is primal optimal and π∗ dual optimal.
38. The gross-simplex Algorithm
Outline of the talk
Single and Multi
Objective Linear
Programming
Linear Programming
and the Simplex
Method
Preliminary results
and notations
BFS and associated
basis
Convergence of the
Simplex Method
Lexicographic Rule
Lexicographic rule and
RHS perturbation
Lexicographic rule and
RHS perturbation and
①
Lexicographic
multi-objective Linear
Programming
Non–Preemptive
grossone-based
scheme
Theoretical results
The gross-simplex
Algorithm
Nonlinear Optimization
Some recent resultsLION11 14 / 40
Main issues:
1) Solve
AT
.Bπ = ˜cB
Use LU decomposition of A.B. Note: no divisions by gross-number are
required.
2) Calculate reduced cost vector
¯˜cN = ˜cN − AT
.N π
Also in this case only multiplications and additions of gross-numbers are
required.
¯˜cN =
7.331 ①−1
+ 0.331 ①−2
4 0 − 3.331 ①−1
− 0.33 ①−2
¯˜cN =
3.67 ①−1
0.17 ①−2
4 ①0
+ 0.33 ①−1
− 0.17 ①−2
39. Nonlinear Optimization
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 15 / 40
40. The case of Equality Constraints
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 16 / 40
min
x
f(x)
subject to h(x) = 0
where f : IRn → IR and h : IRn → IRk
L(x, π) := f(x) +
k
j=1
πjhj(x) = f(x) + πT
h(x)
41. Penalty Functions
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 17 / 40
A penalty function P : IRn → IR satisfies the following condition
P(x)
= 0 if x belongs to the feasible region
> 0 otherwise
42. Penalty Functions
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 17 / 40
A penalty function P : IRn → IR satisfies the following condition
P(x)
= 0 if x belongs to the feasible region
> 0 otherwise
P(x) =
k
j=1
|hj(x)|
P(x) =
k
j=1
h2
j (x)
43. Exactness of a Penalty Function
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 18 / 40
The optimal solution of the constrained problem
min
x
f(x)
subject to h(x) = 0
can be obtained by solving the following unconstrained minimization
problem
min f(x) +
1
σ
P(x)
for sufficiently small but fixed σ > 0.
44. Exactness of a Penalty Function
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 18 / 40
The optimal solution of the constrained problem
min
x
f(x)
subject to h(x) = 0
can be obtained by solving the following unconstrained minimization
problem
min f(x) +
1
σ
P(x)
for sufficiently small but fixed σ > 0.
P(x) =
k
j=1
|hj(x)|
45. Exactness of a Penalty Function
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 18 / 40
The optimal solution of the constrained problem
min
x
f(x)
subject to h(x) = 0
can be obtained by solving the following unconstrained minimization
problem
min f(x) +
1
σ
P(x)
for sufficiently small but fixed σ > 0.
P(x) =
k
j=1
|hj(x)|
Non–smooth function!
46. Introducing ①
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 19 / 40
Let
P(x) =
k
j=1
h2
j (x)
Solve
min f(x) + ①P(x) =: φ (x, ①)
47. Convergence Results
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 20 / 40
min
x
f(x)
subject to h(x) = 0
(1)
48. Convergence Results
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 20 / 40
min
x
f(x)
subject to h(x) = 0
(1)
min
x
f(x) +
1
2
① h(x) 2
(2)
49. Convergence Results
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 20 / 40
min
x
f(x)
subject to h(x) = 0
(1)
min
x
f(x) +
1
2
① h(x) 2
(2)
Let
x∗
= x∗0
+ ①−1
x∗1
+ ①−2
x∗2
+ . . .
be a stationary point for (2) and assume that the LICQ condition holds at
x∗0
then
50. Convergence Results
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 20 / 40
min
x
f(x)
subject to h(x) = 0
(1)
min
x
f(x) +
1
2
① h(x) 2
(2)
Let
x∗
= x∗0
+ ①−1
x∗1
+ ①−2
x∗2
+ . . .
be a stationary point for (2) and assume that the LICQ condition holds at
x∗0
then
the pair x∗0, π∗ = h(1)(x∗) is a KKT point of (1).
51. Example 1
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 21 / 40
min
x
1
2x2
1 + 1
6 x2
2
subject to x1 + x2 = 1
The pair (x∗, π∗) with x∗ =
1
4
3
4
, π∗ = −1
4 is a KKT point.
52. Example 1
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 21 / 40
min
x
1
2x2
1 + 1
6 x2
2
subject to x1 + x2 = 1
The pair (x∗, π∗) with x∗ =
1
4
3
4
, π∗ = −1
4 is a KKT point.
f(x) + ①P(x) =
1
2
x2
1 +
1
6
x2
2 +
1
2
①(1 − x1 − x2)2
53. Example 1
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 21 / 40
f(x) + ①P(x) =
1
2
x2
1 +
1
6
x2
2 +
1
2
①(1 − x1 − x2)2
First Order Optimality Condition
x1 + ①(x1 + x2 − 1) = 0
1
3 x2 + ①(x1 + x2 − 1) = 0
54. Example 1
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 21 / 40
f(x) + ①P(x) =
1
2
x2
1 +
1
6
x2
2 +
1
2
①(1 − x1 − x2)2
x∗
1 =
1①
1 + 4①
, x∗
2 =
3①
1 + 4①
55. Example 1
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 21 / 40
f(x) + ①P(x) =
1
2
x2
1 +
1
6
x2
2 +
1
2
①(1 − x1 − x2)2
x∗
1 =
1①
1 + 4①
, x∗
2 =
3①
1 + 4①
x∗
1 =
1
4
− ①−1
(
1
16
−
1
64
①−1
. . .)
x∗
2 =
3
4
− ①−1
(
3
16
−
3
64
①−1
. . .)
56. Example 1
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 21 / 40
f(x) + ①P(x) =
1
2
x2
1 +
1
6
x2
2 +
1
2
①(1 − x1 − x2)2
x∗
1 =
1①
1 + 4①
, x∗
2 =
3①
1 + 4①
x∗
1 + x∗
2 − 1 =
1
4
−
1
16
①−1
+
1
64
①−2
. . .
+
3
4
−
3
16
①−1
+
3
64
①−2
. . . − 1
= −
4
16
①−1
−
3
16
①−1
+
4
64
①−2
. . .
and h(1)(x∗) = −1
4 = π∗
57. Example 2
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 22 / 40
min x1 + x2
subject to x2
1 + x2
2 − 2 = 0
L(x, π) = x1 + x2 + π x2
1 + x2
2 − 2
The optimal solution is x∗ =
−1
−1
and the pair x∗, π∗ = 1
2 satisfies
the KKT conditions.
58. Example 2
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 22 / 40
φ (x, ①) = x1 + x2 +
①
2
x2
1 + x2
2 − 2
2
First–Order Optimality Conditions
x1 + 2①x1 x2
1 + x2
2 − 2
2
= 0
x2 + 2①x2 x2
1 + x2
2 − 2
2
= 0
The solution is given by
x1 = −1 − ①−1 1
8 + ①−2
C
x2 = −1 − ①−1 1
8 + ①−2
C
59. Example 2
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 22 / 40
Moreover
x2
1 + x2
2 − 2 = 1 +
1
64
①−2
+ ①−4
C2 1
4
①−1
− 2①−2
−
1
4
①−3
C +
1 +
1
64
①−2
+ ①−4
C2 1
4
①−1
− 2①−2
−
1
4
①−3
C
=
1
2
①−1
+
1
32
− 4C ①−2
+ −
1
2
C ①−3
+ −2C2
60. Inequality Constraints
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 23 / 40
min
x
f(x)
subject to g(x) ≤ 0
h(x) = 0
where f : IRn → IR, g : IRn → IRm h : IRn → IRk.
L(x, π, µ) := f(x) +
m
i=1
µigi(x) +
k
j=1
πjhj(x)
= f(x) + µT
g(x) + πT
h(x)
61. Modified LICQ condition
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 24 / 40
Let x0 ∈ IRn. The Modified LICQ (MLICQ) condition is said to hold at x0 if
the vectors
∇gi(x0
), i : gi(x0
) ≥ 0, ∇hj(x0
), j = 1, . . . , k
are linearly independent.
62. Convergence Results
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 25 / 40
min
x
f(x)
subject to g(x) ≤ 0
h(x) = 0
min
x
f(x) +
①
2
max{0, gi(x)} 2
+
①
2
h(x) 2
x∗
= x∗0
+ ①−1
x∗1
+ ①−2
x∗2
+ . . .
⇓ (MLICQ)
63. Convergence Results
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 25 / 40
min
x
f(x)
subject to g(x) ≤ 0
h(x) = 0
min
x
f(x) +
①
2
max{0, gi(x)} 2
+
①
2
h(x) 2
x∗
= x∗0
+ ①−1
x∗1
+ ①−2
x∗2
+ . . .
⇓ (MLICQ)
x∗0
, µ∗
= g(1)
(x∗
), π∗
= h(1)
(x∗
)
64. The importance of CQs
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 26 / 40
min x1 + x2
subject to x2
1 + x2
2 − 2
2
= 0
L(x, π) = x1 + x2 + π x2
1 + x2
2 − 2
2
The optimal solution is x∗ =
−1
−1
65. The importance of CQs
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 26 / 40
φ (x, ①) = x1 + x2 +
①
2
x2
1 + x2
2 − 2
4
First–Order Optimality Conditions
1 + 4①x1 x2
1 + x2
2 − 2
3
= 0
1 + 4①x2 x2
1 + x2
2 − 2
3
= 0
66. The importance of CQs
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 26 / 40
Let the solution of the above system be
x∗
1 = x∗
2 = A + B①−1
+ C①−2
with A, B, and C ∈ IR. Now
4①x∗
1 = 4A① + 4B + 4C①−1
and
1 + 4①x∗
1 (x∗
1)2
+ (x∗
2)2
− 2
3
= 1 + 4A① + 4B + 4C①−1
2A2 − 2 + 2AB①−1
+ D①−2
3
.
67. The importance of CQs
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 26 / 40
1 + 4①x∗
1 (x∗
1)2
+ (x∗
2)2
− 2
3
= 1 + 4A① + 4B + 4C①−1
2A2
− 2 + 2AB①−1
+ D①−2
3
.
68. The importance of CQs
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 26 / 40
1 + 4①x∗
1 (x∗
1)2
+ (x∗
2)2
− 2
3
= 1 + 4A① + 4B + 4C①−1
2A2
− 2 + 2AB①−1
+ D①−2
3
.
If 2A2 − 2 = 0 there is still a term of the order ① unless A = 0.
69. The importance of CQs
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 26 / 40
1 + 4①x∗
1 (x∗
1)2
+ (x∗
2)2
− 2
3
= 1 + 4A① + 4B + 4C①−1
2A2
− 2 + 2AB①−1
+ D①−2
3
.
If 2A2 − 2 = 0 there is still a term of the order ① unless A = 0.
If 2A2 − 2 = 0 a term ①−1
can be factored out
1 + 4A① + 4B + 4C①−1
①−3
+2AB + D①−1
3
and the finite term cannot be equal to 0.
70. The importance of CQs
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 26 / 40
1 + 4①x∗
1 (x∗
1)2
+ (x∗
2)2
− 2
3
= 1 + 4A① + 4B + 4C①−1
2A2
− 2 + 2AB①−1
+ D①−2
3
.
If 2A2 − 2 = 0 there is still a term of the order ① unless A = 0.
If 2A2 − 2 = 0 a term ①−1
can be factored out
1 + 4A① + 4B + 4C①−1
①−3
+2AB + D①−1
3
and the finite term cannot be equal to 0.
When Constraint Qualification conditions do not hold, the solution
of ∇F(x) = 0 does not provide a KKT pair for the constrained
problem.
71. Conjugate Gradient Method
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 27 / 40
Data: Set k = 0, y0 = 0, r0 = b − Ay0.
If r0 = 0, then STOP. Else, set p0 = r0.
Step k: Compute αk = rT
k pk/pT
k Apk,
yk+1 = yk + αkpk,
rk+1 = rk − αkApk.
If rk+1 = 0, then STOP.
Else, set βk =
−rT
k+1Apk
pT
k Apk
=
rk+1
2
|rk
2
, and
pk+1 = rk+1 + βkpk,
k = k + 1.
Go to Step k.
72. pT
k Apk
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 28 / 40
When the matrix A is positive definite then
λm(A) pk
2
≤ pT
k Apk
and pT
k Apk is bounded from below.
If the matrix A is not positive definite, then such a bound does not hold,
being potentially pT
k Apk = 0,.
73. pT
k Apk
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 28 / 40
When the matrix A is positive definite then
λm(A) pk
2
≤ pT
k Apk
and pT
k Apk is bounded from below.
If the matrix A is not positive definite, then such a bound does not hold,
being potentially pT
k Apk = 0,.
R. De Leone, G. Fasano, Y.D. Sergeyev
Use
pT
k Apk = s①
where s = O(①−1
) if the Step k is a non-degenerate CG step, and
s = O(①−2
) if the Step k is a degenerate CG step.
74. Variable Metric Method for convex nonsmooth optimization
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 29 / 40
xk+1
= xk
− αk Bk
−1
ξk
a
where ξk
a current aggregate subgradient, and the positive definite
variable-metric n × n matrix, approximation of the Hessian matrix.
Then
Bk+1
= Bk
+ ∆k
and
Bk+1
δk
≈ γk
with γk = gk+1 − gk (subgradients) and δk = xk+1 − xk and diagonal.
The focus on the updating technique of matrix Bk
75. Matrix Updating scheme
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 30 / 40
min
b
Bδk − γk
subject to Bii ≥ ǫ
Bij = 0, i = j
76. Matrix Updating scheme
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 30 / 40
min
b
Bδk − γk
subject to Bii ≥ ǫ
Bij = 0, i = j
Bk+1
ii = max ǫ,
γk
i
δk
1
77. Matrix Updating scheme
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 30 / 40
min
b
Bδk − γk
subject to Bii ≥ ǫ
Bij = 0, i = j
M. Gaudioso, G. Giallombardo, M. Mukhametzhanov
¯γk
i =
γk
i if |γk
i | > ǫ
①−1
otherwise
¯δk
i =
δk
i if |δk
i | > ǫ
①−1
otherwise
78. Matrix Updating scheme
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
The case of Equality
Constraints
Penalty Functions
Exactness of a Penalty
Function
Introducing ①
Convergence Results
Example 1
Example 2
Inequality Constraints
Modified LICQ
condition
Convergence Results
The importance of
CQs
Conjugate Gradient
Method
pT
k Apk
Variable Metric
Method for convex
nonsmooth
optimization
Matrix Updating
scheme
LION11 30 / 40
min
b
Bδk − γk
subject to Bii ≥ ǫ
Bij = 0, i = j
bk
i =
①−1
if 0 <
¯γk
i
¯δk
i
≤ ǫ
¯γk
i
¯δk
i
otherwise
Bk+1
ii = max ①−1
, bk
i
79. Some recent results
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 31 / 40
80. Quadratic Problems
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 32 / 40
min
x
1
2xT Mx
subject to Ax = b
x ≥ 0
81. Quadratic Problems
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 32 / 40
min
x
1
2xT Mx
subject to Ax = b
x ≥ 0
KKT conditions
Mx + q − AT
u − v = 0
Ax − b = 0
x ≥ 0, v ≥ 0, xT
v = 0
82. Quadratic Problems
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 32 / 40
min
x
1
2xT Mx
subject to Ax = b
x ≥ 0
min
1
2
xT
Mx +
①
2
Ax − b 2
2 +
①
2
max{0, −x} 2
2 =: F(x)
83. Quadratic Problems
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 32 / 40
min
x
1
2xT Mx
subject to Ax = b
x ≥ 0
min
1
2
xT
Mx +
①
2
Ax − b 2
2 +
①
2
max{0, −x} 2
2 =: F(x)
∇F(x) = Mx + q + ①AT
(Ax − b) − ① max{0, −x}
84. Quadratic Problems
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 32 / 40
min
x
1
2xT Mx
subject to Ax = b
x ≥ 0
min
1
2
xT
Mx +
①
2
Ax − b 2
2 +
①
2
max{0, −x} 2
2 =: F(x)
∇F(x) = Mx + q + ①AT
(Ax − b) − ① max{0, −x}
x = x(0)
+ ①−1
x(1)
+ ①−2
x(2)
+ . . .
b = b(0)
+ ①−1
b(1)
+ ①−2
b(2)
+ . . .
A ∈ IRm×n
rank(A) = m
85. ∇F(x) = 0
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 33 / 40
0 = Mx + q + ①AT A x(0) + ①−1
x(1) + ①−2
x(2) + . . .
−b(0) − ①−1
b(1) − ①−2
b(2) + . . .
+① max 0, −x(0) − ①−1
x(1) − ①−2
x(2) + . . .
86. ∇F(x) = 0
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 33 / 40
0 = Mx + q + ①AT A x(0) + ①−1
x(1) + ①−2
x(2) + . . .
−b(0) − ①−1
b(1) − ①−2
b(2) + . . .
+① max 0, −x(0) − ①−1
x(1) − ①−2
x(2) + . . .
Looking at the ① terms
Ax(0)
− b(0)
= 0
max 0, −x(0)
= 0 and hence x(0)
≥ 0
87. ∇F(x) = 0
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 33 / 40
0 = Mx + q + ①AT A x(0) + ①−1
x(1) + ①−2
x(2) + . . .
−b(0) − ①−1
b(1) − ①−2
b(2) + . . .
+① max 0, −x(0) − ①−1
x(1) − ①−2
x(2) + . . .
Looking at the ①0
terms
Mx(0)
+ q + AT
Ax(1)
− b(1)
− v = 0
where
vj = max 0, −x
(1)
j
only for the indices j for which x
(0)
j = 0, otherwise vj = 0
88. ∇F(x) = 0
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 33 / 40
0 = Mx + q + ①AT A x(0) + ①−1
x(1) + ①−2
x(2) + . . .
−b(0) − ①−1
b(1) − ①−2
b(2) + . . .
+① max 0, −x(0) − ①−1
x(1) − ①−2
x(2) + . . .
Set
u = Ax(1)
− b(1)
vj =
0 if x
(0)
j = 0
max 0, −x
(1)
j otherwise
Then
Mx(0)
+ q + AT
u − v = 0
v ≥ 0, vT
x0
= 0
89. A Generic Algorithm
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 34 / 40
min
x
f(x)
f(x) = ①f(1)
(x) + f(0)
(x) + ①−1
f(−1)
(x) + . . .
∇f(x) = ①∇f(1)
(x) + ∇f(0)
(x) + ①−1
∇f(−1)
(x) + . . .
90. A Generic Algorithm
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 34 / 40
min
x
f(x)
At iteration k
91. A Generic Algorithm
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 34 / 40
min
x
f(x)
At iteration k
If
∇f(1)
(xk
) = 0 and ∇f(0)
(xk
) = 0
STOP
92. A Generic Algorithm
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 34 / 40
min
x
f(x)
At iteration k
otherwise find xk+1 such that
93. A Generic Algorithm
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 34 / 40
min
x
f(x)
At iteration k
otherwise find xk+1 such that
If ∇f(1)(xk) = 0
f(1)
(xk+1
) ≤ f(1)
(xk
) + σ ∇f(1)
(xk
)
f(0)
(xk+1
) ≤ max
0≤j≤lk
f(0)
(xk−j
) + σ ∇f(0)
(xk
)
94. A Generic Algorithm
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 34 / 40
min
x
f(x)
At iteration k
otherwise find xk+1 such that
If ∇f(1)(xk) = 0
f(0)
(xk+1
) ≤ f(0)
(xk
) + σ ∇f(0)
(xk
)
f(1)
(xk+1
) ≤ max
0≤j≤mk
f(1)
(xk−j
)
95. A Generic Algorithm
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 34 / 40
min
x
f(x)
m0 = 0, mk+1 ≤ max {mk + 1, M}
l0 = 0, kk+1 ≤ max {lk + 1, L}
σ(.) is a forcing function.
Non–monotone optimization technique, Zhang-Hager, Grippo-
Lampariello-Lucidi, Dai
96. Convergence
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 35 / 40
Case 1: ∃¯k such that ∇f(1)(xk) = 0, k ≥ ¯k
Then
f(1)
(xk+1
) ≤ max
0≤j≤mk
f(1)
(xk−j
), k ≥ ¯k
and hence
max
0≤i≤M
f(1)
(x
¯k+Ml+i
) ≤ max
0≤i≤M
f(1)
(x
¯k+M(l−1)+i
)
and
f(0)
(xk+1
) ≤ f(0)
(xk
) + σ ∇f(0)
(xk
) , k ≥ ¯k
Assuming that the level sets for f(1)(x0) and f(0)(x0) are compact sets,
then the sequence has at least one accumulation point x∗ and any
accumulation point satisfies ∇f(1)(x∗) = 0 and ∇f(0)(x∗) = 0
97. Convergence
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 35 / 40
Case 2: ∃ a subsequence jk such that ∇f(1)(xjk ) = 0
Then
f(1)
(xjk+1
) ≤ f(1)
(xjk
) + +σ ∇f(1)
(xjk
)
Again
max
0≤i≤M
f(1)
(xjk+Mt+i
) ≤ max
0≤i≤M
f(1)
(xjk+M(t−1)+i
)+σ ∇f(1)
(xjk
)
and hence ∇f(1)(xjk ) → 0.
98. Convergence
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 35 / 40
Case 2: ∃ a subsequence jk such that ∇f(1)(xjk ) = 0
Then
f(1)
(xjk+1
) ≤ f(1)
(xjk
) + +σ ∇f(1)
(xjk
)
Again
max
0≤i≤M
f(1)
(xjk+Mt+i
) ≤ max
0≤i≤M
f(1)
(xjk+M(t−1)+i
)+σ ∇f(1)
(xjk
)
and hence ∇f(1)(xjk ) → 0. Moreover,
max
0≤i≤L
f(0)
(xjk+Lt+i
) ≤ max
0≤i≤L
f(0)
(xjk+L(t−1)+i
)+σ ∇f(0)
(xjk
)
and hence ∇f(0)(xjk ) → 0.
99. Gradient Method
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 36 / 40
At iterations k calculate ∇f(xk).
If ∇f(1)(xk) = 0
xk+1
= min
α≥0,β≥0
f xk
− α∇f(1)
(xk
) − β∇f(0)
(xk
)
If ∇f(1)(xk) = 0
xk+1
= min
α≥0
f(0)
xk
− α∇f(0)
(xk
)
100. Example A
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 37 / 40
min
x
1
2x2
1 + 1
6 x2
2
subject to x1 + x2 − 1 = 0
f(x) =
1
2
x2
1 +
1
6
x2
2 +
1
2
①(1 − x1 − x2)2
x0
=
4
1
→ x1
=
0.31
0.69
→ x2
=
−0.1
0.39
→ x3
=
0.26
0.74
→
x4
=
−0.12
0.38
→ x5
=
0.25
0.75
101. Example B
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 38 / 40
min
x
x1 + x2
subject to x1
1 + x2
2 − 2 = 0
f(x) =
1
2
x2
1 +
1
6
x2
2 +
1
2
①(1 − x1 − x2)2
x0
=
0.25
0.75
→ x1
=
−1.22
−0.72
→ x2
=
−7.39
−6.89
→ x3
=
1.04
0.95
x4
=
−7.10
−7.19
→ x5
=
−1
−1
102. Conclusions (?)
Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 39 / 40
■ The use of ① is extremely beneficial in various aspects in Linear and
Nonlinear Optimization
■ Difficult problems in NLP can be approached in a simpler way using ①
■ A new convergence theory for standard algorithms (gradient, Newton’s,
Quasi-Newton) needs to be developed in theis new framework
103. Outline of the talk
Single and Multi
Objective Linear
Programming
Nonlinear Optimization
Some recent results
Quadratic Problems
∇F (x) = 0
A Generic Algorithm
Convergence
Gradient Method
Example A
Example B
Conclusions (?)
LION11 40 / 40
Thanks for your attention