AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 1.
More info at http://summerschool.ssa.org.ua
Quadratic Programming : KKT conditions with inequality constraintsMrinmoy Majumder
In the case of Quadratic Programming for optimization, the objective function is a quadratic function. One of the techniques for solving quadratic optimization problems is KKT Conditions which is explained with an example in this tutorial.
The document introduces dynamic programming as a technique for making optimal decisions over multiple time periods. It discusses how dynamic programming breaks large problems into smaller subproblems and solves each in order, working backwards from the last period. The document provides an example of using dynamic programming to find the shortest route between two cities by breaking the problem into stages and working backwards from the final destination.
The document discusses the theory of NP-completeness. It begins by defining the complexity classes P, NP, NP-hard, and NP-complete. It then explains the concepts of reduction and how none of the NP-complete problems can be solved in polynomial time deterministically. The document provides examples of NP-complete problems like satisfiability (SAT), vertex cover, and the traveling salesman problem. It shows how nondeterministic algorithms can solve these problems and how they can be transformed into SAT instances. Finally, it proves that SAT is the first NP-complete problem by showing it is in NP and NP-hard.
This document outlines an introduction to convex optimization. It begins with an introduction stating that convex optimization problems can be solved efficiently to find the global optimum. It then provides an outline covering convex sets, convex functions, convex optimization problems, and references. The body of the document defines convex sets as sets where a line segment between any two points lies entirely within the set. It also provides examples of convex sets including norm balls and intersections of convex sets. It defines convex functions as functions where the graph lies below any line segment between two points, and provides conditions for checking convexity using derivatives. Finally, it discusses convex optimization problems and solving them efficiently.
The document discusses optimization and gradient descent algorithms. Optimization aims to select the best solution given some problem, like maximizing GPA by choosing study hours. Gradient descent is a method for finding the optimal parameters that minimize a cost function. It works by iteratively updating the parameters in the opposite direction of the gradient of the cost function, which points in the direction of greatest increase. The process repeats until convergence. Issues include potential local minimums and slow convergence.
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
Quadratic Programming : KKT conditions with inequality constraintsMrinmoy Majumder
In the case of Quadratic Programming for optimization, the objective function is a quadratic function. One of the techniques for solving quadratic optimization problems is KKT Conditions which is explained with an example in this tutorial.
The document introduces dynamic programming as a technique for making optimal decisions over multiple time periods. It discusses how dynamic programming breaks large problems into smaller subproblems and solves each in order, working backwards from the last period. The document provides an example of using dynamic programming to find the shortest route between two cities by breaking the problem into stages and working backwards from the final destination.
The document discusses the theory of NP-completeness. It begins by defining the complexity classes P, NP, NP-hard, and NP-complete. It then explains the concepts of reduction and how none of the NP-complete problems can be solved in polynomial time deterministically. The document provides examples of NP-complete problems like satisfiability (SAT), vertex cover, and the traveling salesman problem. It shows how nondeterministic algorithms can solve these problems and how they can be transformed into SAT instances. Finally, it proves that SAT is the first NP-complete problem by showing it is in NP and NP-hard.
This document outlines an introduction to convex optimization. It begins with an introduction stating that convex optimization problems can be solved efficiently to find the global optimum. It then provides an outline covering convex sets, convex functions, convex optimization problems, and references. The body of the document defines convex sets as sets where a line segment between any two points lies entirely within the set. It also provides examples of convex sets including norm balls and intersections of convex sets. It defines convex functions as functions where the graph lies below any line segment between two points, and provides conditions for checking convexity using derivatives. Finally, it discusses convex optimization problems and solving them efficiently.
The document discusses optimization and gradient descent algorithms. Optimization aims to select the best solution given some problem, like maximizing GPA by choosing study hours. Gradient descent is a method for finding the optimal parameters that minimize a cost function. It works by iteratively updating the parameters in the opposite direction of the gradient of the cost function, which points in the direction of greatest increase. The process repeats until convergence. Issues include potential local minimums and slow convergence.
Many Decision Problems in business and social systems can be modeled using mathematical optimization, which seeks to maximize or minimize some objective which is a function of the decisions.
Stochastic Optimization Problems are mathematical programs where some of the data incorporated into the objective or constraints are Uncertain.
whereas, Deterministic Optimization Problems are formulated with known parameters.
The document discusses gradient descent methods for unconstrained convex optimization problems. It introduces gradient descent as an iterative method to find the minimum of a differentiable function by taking steps proportional to the negative gradient. It describes the basic gradient descent update rule and discusses convergence conditions such as Lipschitz continuity, strong convexity, and condition number. It also covers techniques like exact line search, backtracking line search, coordinate descent, and steepest descent methods.
This document provides an introduction to XGBoost, including:
1. XGBoost is an important machine learning library that is commonly used by winners of Kaggle competitions.
2. A quick example is shown using XGBoost to predict diabetes based on patient data, achieving good results with only 20 lines of simple code.
3. XGBoost works by creating an ensemble of decision trees through boosting, and focuses on explaining concepts at a high level rather than detailed algorithms.
This document discusses reinforcement learning. It defines reinforcement learning as a learning method where an agent learns how to behave via interactions with an environment. The agent receives rewards or penalties based on its actions but is not told which actions are correct. Several reinforcement learning concepts and algorithms are covered, including model-based vs model-free approaches, passive vs active learning, temporal difference learning, adaptive dynamic programming, and exploration-exploitation tradeoffs. Generalization methods like function approximation and genetic algorithms are also briefly mentioned.
This document discusses probability and Bayes' theorem. It provides examples of basic probability concepts like the probability of a coin toss. It then defines conditional probability as the probability of an event given another event. Bayes' theorem is introduced as a way to revise a probability based on new information. An example problem demonstrates how to calculate the probability of rain given a weather forecast using Bayes' theorem.
This document discusses optimization problems and their solutions. It begins by defining optimization problems as seeking to maximize or minimize a quantity given certain limits or constraints. Both deterministic and stochastic models are discussed. Examples of discrete optimization problems include the traveling salesman and shortest path problems. Solution methods mentioned include integer programming, network algorithms, dynamic programming, and approximation algorithms. The document then focuses on convex optimization problems, which can be solved efficiently. It discusses using tools like CVX for solving convex programs and the duality between primal and dual problems. Finally, it presents the collaborative resource allocation algorithm for solving non-convex optimization problems in a suboptimal way.
This document provides a summary of Markov chains. It begins by defining stochastic processes and Markov chains. A Markov chain is a stochastic process where the probability of the next state depends only on the current state, not on the sequence of events that preceded it. The document discusses n-step transition probabilities, classification of states, and steady-state probabilities. It provides examples of Markov chains for cola purchases and camera store inventory to illustrate the concepts.
The document discusses various path planning techniques for mobile robots to navigate between a starting point and destination while avoiding collisions. It describes methods like visibility graphs, roadmaps, cell decomposition, and potential fields. It also covers implementing techniques like breadth-first search on visibility graphs and optimizing robot trajectories using factors like travel time, distance and sensor information.
Stochastic gradient descent and its tuningArsalan Qadri
This paper talks about optimization algorithms used for big data applications. We start with explaining the gradient descent algorithms and its limitations. Later we delve into the stochastic gradient descent algorithms and explore methods to improve it it by adjusting learning rates.
This document discusses constraint satisfaction problems (CSPs) and techniques for solving them. It begins by defining CSPs as problems with variables, domains of possible values, and constraints limiting assignments. Backtracking search and heuristics like minimum remaining values are described as standard approaches. Constraint propagation techniques like forward checking and arc consistency are explained, which aim to detect inconsistencies earlier. The 4-queens problem is provided as an example CSP.
This presentation contains an introduction to reinforcement learning, comparison with others learning ways, introduction to Q-Learning and some applications of reinforcement learning in video games.
This document summarizes key concepts in unconstrained optimization of functions with two variables, including:
1) Critical points are found by taking the partial derivatives and setting them equal to zero, generalizing the first derivative test for single-variable functions.
2) The Hessian matrix generalizes the second derivative, with its entries being the partial derivatives evaluated at a critical point.
3) The second derivative test classifies critical points as local maxima, minima or saddle points based on the signs of the Hessian matrix's eigenvalues.
4) Taylor polynomial approximations in two variables involve partial derivatives up to second order, analogous to single-variable Taylor series.
5) An example classifies the critical points
Methods of Optimization in Machine LearningKnoldus Inc.
In this session we will discuss about various methods to optimise a machine learning model and, how we can adjust the hyper-parameters to minimise the cost function.
Particle swarm optimization (PSO) is an evolutionary computation technique for optimizing problems. It initializes a population of random solutions and searches for optima by updating generations. Each potential solution, called a particle, tracks its best solution and the overall best solution to change its velocity and position in search of better solutions. The algorithm involves initializing particles with random positions and velocities, then updating velocities and positions iteratively based on the particles' local best solution and the global best solution until termination criteria are met. PSO has advantages of being simple, quick, and effective at locating good solutions.
The document discusses various neural network learning rules:
1. Error correction learning rule (delta rule) adapts weights based on the error between the actual and desired output.
2. Memory-based learning stores all training examples and classifies new inputs based on similarity to nearby examples (e.g. k-nearest neighbors).
3. Hebbian learning increases weights of simultaneously active neuron connections and decreases others, allowing patterns to emerge from correlations in inputs over time.
4. Competitive learning (winner-take-all) adapts the weights of the neuron most active for a given input, allowing unsupervised clustering of similar inputs across neurons.
The Big M Method is a variant of the simplex method for solving linear programming problems. It introduces artificial variables and a large number M to convert inequalities into equalities. The transformed problem is then solved using the simplex method, eliminating artificial variables until an optimal solution is found. However, the method has drawbacks in determining a sufficiently large M value and not knowing feasibility until optimality is reached. It is inferior to the two-phase method and not used in commercial solvers.
The document discusses inference rules for quantifiers in first-order logic. It describes the rules of universal instantiation and existential instantiation. Universal instantiation allows inferring sentences by substituting terms for variables, while existential instantiation replaces a variable with a new constant symbol. The document also introduces unification, which finds substitutions to make logical expressions identical. Generalized modus ponens is presented as a rule that lifts modus ponens to first-order logic by using unification to substitute variables.
This presentation discuses the following topics:
What is A-Star (A*) Algorithm in Artificial Intelligence?
A* Algorithm Steps
Why is A* Search Algorithm Preferred?
A* and Its Basic Concepts
What is a Heuristic Function?
Admissibility of the Heuristic Function
Consistency of the Heuristic Function
Probability
Random variables and Probability Distributions
The Normal Probability Distributions and Related Distributions
Sampling Distributions for Samples from a Normal Population
Classical Statistical Inferences
Properties of Estimators
Testing of Hypotheses
Relationship between Confidence Interval Procedures and Tests of Hypotheses.
This document provides a concise probability cheatsheet compiled by William Chen and others. It covers key probability concepts like counting rules, sampling tables, definitions of probability, independence, unions and intersections, joint/marginal/conditional probabilities, Bayes' rule, random variables and their distributions, expected value, variance, indicators, moment generating functions, and independence of random variables. The cheatsheet is licensed under CC BY-NC-SA 4.0 and the last updated date is March 20, 2015.
The document discusses gradient descent methods for unconstrained convex optimization problems. It introduces gradient descent as an iterative method to find the minimum of a differentiable function by taking steps proportional to the negative gradient. It describes the basic gradient descent update rule and discusses convergence conditions such as Lipschitz continuity, strong convexity, and condition number. It also covers techniques like exact line search, backtracking line search, coordinate descent, and steepest descent methods.
This document provides an introduction to XGBoost, including:
1. XGBoost is an important machine learning library that is commonly used by winners of Kaggle competitions.
2. A quick example is shown using XGBoost to predict diabetes based on patient data, achieving good results with only 20 lines of simple code.
3. XGBoost works by creating an ensemble of decision trees through boosting, and focuses on explaining concepts at a high level rather than detailed algorithms.
This document discusses reinforcement learning. It defines reinforcement learning as a learning method where an agent learns how to behave via interactions with an environment. The agent receives rewards or penalties based on its actions but is not told which actions are correct. Several reinforcement learning concepts and algorithms are covered, including model-based vs model-free approaches, passive vs active learning, temporal difference learning, adaptive dynamic programming, and exploration-exploitation tradeoffs. Generalization methods like function approximation and genetic algorithms are also briefly mentioned.
This document discusses probability and Bayes' theorem. It provides examples of basic probability concepts like the probability of a coin toss. It then defines conditional probability as the probability of an event given another event. Bayes' theorem is introduced as a way to revise a probability based on new information. An example problem demonstrates how to calculate the probability of rain given a weather forecast using Bayes' theorem.
This document discusses optimization problems and their solutions. It begins by defining optimization problems as seeking to maximize or minimize a quantity given certain limits or constraints. Both deterministic and stochastic models are discussed. Examples of discrete optimization problems include the traveling salesman and shortest path problems. Solution methods mentioned include integer programming, network algorithms, dynamic programming, and approximation algorithms. The document then focuses on convex optimization problems, which can be solved efficiently. It discusses using tools like CVX for solving convex programs and the duality between primal and dual problems. Finally, it presents the collaborative resource allocation algorithm for solving non-convex optimization problems in a suboptimal way.
This document provides a summary of Markov chains. It begins by defining stochastic processes and Markov chains. A Markov chain is a stochastic process where the probability of the next state depends only on the current state, not on the sequence of events that preceded it. The document discusses n-step transition probabilities, classification of states, and steady-state probabilities. It provides examples of Markov chains for cola purchases and camera store inventory to illustrate the concepts.
The document discusses various path planning techniques for mobile robots to navigate between a starting point and destination while avoiding collisions. It describes methods like visibility graphs, roadmaps, cell decomposition, and potential fields. It also covers implementing techniques like breadth-first search on visibility graphs and optimizing robot trajectories using factors like travel time, distance and sensor information.
Stochastic gradient descent and its tuningArsalan Qadri
This paper talks about optimization algorithms used for big data applications. We start with explaining the gradient descent algorithms and its limitations. Later we delve into the stochastic gradient descent algorithms and explore methods to improve it it by adjusting learning rates.
This document discusses constraint satisfaction problems (CSPs) and techniques for solving them. It begins by defining CSPs as problems with variables, domains of possible values, and constraints limiting assignments. Backtracking search and heuristics like minimum remaining values are described as standard approaches. Constraint propagation techniques like forward checking and arc consistency are explained, which aim to detect inconsistencies earlier. The 4-queens problem is provided as an example CSP.
This presentation contains an introduction to reinforcement learning, comparison with others learning ways, introduction to Q-Learning and some applications of reinforcement learning in video games.
This document summarizes key concepts in unconstrained optimization of functions with two variables, including:
1) Critical points are found by taking the partial derivatives and setting them equal to zero, generalizing the first derivative test for single-variable functions.
2) The Hessian matrix generalizes the second derivative, with its entries being the partial derivatives evaluated at a critical point.
3) The second derivative test classifies critical points as local maxima, minima or saddle points based on the signs of the Hessian matrix's eigenvalues.
4) Taylor polynomial approximations in two variables involve partial derivatives up to second order, analogous to single-variable Taylor series.
5) An example classifies the critical points
Methods of Optimization in Machine LearningKnoldus Inc.
In this session we will discuss about various methods to optimise a machine learning model and, how we can adjust the hyper-parameters to minimise the cost function.
Particle swarm optimization (PSO) is an evolutionary computation technique for optimizing problems. It initializes a population of random solutions and searches for optima by updating generations. Each potential solution, called a particle, tracks its best solution and the overall best solution to change its velocity and position in search of better solutions. The algorithm involves initializing particles with random positions and velocities, then updating velocities and positions iteratively based on the particles' local best solution and the global best solution until termination criteria are met. PSO has advantages of being simple, quick, and effective at locating good solutions.
The document discusses various neural network learning rules:
1. Error correction learning rule (delta rule) adapts weights based on the error between the actual and desired output.
2. Memory-based learning stores all training examples and classifies new inputs based on similarity to nearby examples (e.g. k-nearest neighbors).
3. Hebbian learning increases weights of simultaneously active neuron connections and decreases others, allowing patterns to emerge from correlations in inputs over time.
4. Competitive learning (winner-take-all) adapts the weights of the neuron most active for a given input, allowing unsupervised clustering of similar inputs across neurons.
The Big M Method is a variant of the simplex method for solving linear programming problems. It introduces artificial variables and a large number M to convert inequalities into equalities. The transformed problem is then solved using the simplex method, eliminating artificial variables until an optimal solution is found. However, the method has drawbacks in determining a sufficiently large M value and not knowing feasibility until optimality is reached. It is inferior to the two-phase method and not used in commercial solvers.
The document discusses inference rules for quantifiers in first-order logic. It describes the rules of universal instantiation and existential instantiation. Universal instantiation allows inferring sentences by substituting terms for variables, while existential instantiation replaces a variable with a new constant symbol. The document also introduces unification, which finds substitutions to make logical expressions identical. Generalized modus ponens is presented as a rule that lifts modus ponens to first-order logic by using unification to substitute variables.
This presentation discuses the following topics:
What is A-Star (A*) Algorithm in Artificial Intelligence?
A* Algorithm Steps
Why is A* Search Algorithm Preferred?
A* and Its Basic Concepts
What is a Heuristic Function?
Admissibility of the Heuristic Function
Consistency of the Heuristic Function
Probability
Random variables and Probability Distributions
The Normal Probability Distributions and Related Distributions
Sampling Distributions for Samples from a Normal Population
Classical Statistical Inferences
Properties of Estimators
Testing of Hypotheses
Relationship between Confidence Interval Procedures and Tests of Hypotheses.
This document provides a concise probability cheatsheet compiled by William Chen and others. It covers key probability concepts like counting rules, sampling tables, definitions of probability, independence, unions and intersections, joint/marginal/conditional probabilities, Bayes' rule, random variables and their distributions, expected value, variance, indicators, moment generating functions, and independence of random variables. The cheatsheet is licensed under CC BY-NC-SA 4.0 and the last updated date is March 20, 2015.
Monte Carlo methods use random sampling to solve quantitative problems. They were first used by Stanislaw Ulam and Nicholas Metropolis to solve non-random problems by transforming them into random forms. Monte Carlo simulations play a major role in experimental physics by designing experiments, evaluating potential outputs and risks, and validating results. Random numbers are generated using pseudorandom number generators or by transforming uniform random variables using probability distribution functions. The accuracy of Monte Carlo simulations improves as the number of samples increases, with the standard error declining proportionally with the square root of the number of samples.
Noise is unwanted sound considered unpleasant, loud, or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound.
Maximum likelihood estimation of regularisation parameters in inverse problem...Valentin De Bortoli
This document discusses an empirical Bayesian approach for estimating regularization parameters in inverse problems using maximum likelihood estimation. It proposes the Stochastic Optimization with Unadjusted Langevin (SOUL) algorithm, which uses Markov chain sampling to approximate gradients in a stochastic projected gradient descent scheme for optimizing the regularization parameter. The algorithm is shown to converge to the maximum likelihood estimate under certain conditions on the log-likelihood and prior distributions.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, moments, and more. The cheatsheet is regularly updated with comments and suggestions submitted through a GitHub repository.
1. The document covers probability axioms and rules including the additive rule, conditional probability, independence, and Bayes' rule. It also defines discrete and continuous random variables and their probability distributions.
2. Important discrete distributions discussed include the Bernoulli distribution for a binary outcome experiment and the binomial distribution for repeated Bernoulli trials.
3. Techniques for counting permutations, combinations, and sequences of events are presented to handle probability problems involving counting.
Accounting for uncertainty is a crucial component in decision making (e.g., classification) because of ambiguity in our measurements.
Probability theory is the proper mechanism for accounting for uncertainty.
This document provides an introduction to radial basis function (RBF) interpolation of scattered data. It discusses how RBFs choose basis functions centered at data points to guarantee a well-posed interpolation problem. Common RBF kernels include the multiquadric, inverse multiquadric, and Gaussian functions. While RBF interpolation is guaranteed to have a unique solution, it can still be ill-conditioned depending on the shape parameter choice. Considerations for using RBFs include that the interpolation matrix is dense, requiring optimization of the shape parameter, and interpolation error increases near boundaries.
random variables-descriptive and contincuousar9530
The document discusses discrete and continuous random variables and their probability mass functions (pmf), probability density functions (pdf), and cumulative distribution functions (cdf). It provides examples of discrete and continuous random variables. It defines pmf as P(X=x) and pdf as f(x)=P[(x-a/2)≤X≤(x+a/2)]/a for discrete and continuous variables respectively. The cdf is defined as F(x)=P(X≤x). It also discusses mathematical expectation (mean) as E(X)=ΣxP(x) for discrete variables and ∫xf(x)dx for continuous variables.
Basics of probability in statistical simulation and stochastic programmingSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 2.
More info at http://summerschool.ssa.org.ua
The document discusses cumulative distribution functions (CDFs) and probability density functions (PDFs) for continuous random variables. It provides definitions and properties of CDFs and PDFs. For CDFs, it describes how they give the probability that a random variable is less than or equal to a value. For PDFs, it explains how they provide the probability of a random variable taking on a particular value. The document also gives examples of CDFs and PDFs for exponential and uniform random variables.
- The document proposes a new class of coherent risk measures called spectral measures of risk. These measures are generated by expanding on expected shortfall measures using a risk aversion function φ.
- The risk aversion function φ assigns weights to different confidence levels of portfolio losses, allowing for a more flexible representation of subjective risk preferences than expected shortfall alone.
- φ must be positive, decreasing, and normalized to define a coherent spectral risk measure. This provides an intuitive interpretation of coherence as assigning larger weights to worse outcomes. φ allows investors to express their individual risk aversion profile.
Accelerating Metropolis Hastings with Lightweight Inference CompilationFeynman Liang
This document summarizes research on accelerating Metropolis-Hastings sampling with lightweight inference compilation. It discusses background on probabilistic programming languages and Bayesian inference techniques like variational inference and sequential importance sampling. It introduces the concept of inference compilation, where a neural network is trained to construct proposals for MCMC that better match the posterior. The paper proposes a lightweight approach to inference compilation for imperative probabilistic programs that trains proposals conditioned on execution prefixes to address issues with sequential importance sampling.
Statistical Inference Part II: Types of Sampling DistributionDexlab Analytics
This is an in-depth analysis of the way different types of sampling distribution works focusing on their specific functions and interrelations as part of the discussion on the theory of sampling.
This document outlines the key concepts that will be covered in Lecture 2 on Bayesian modeling. It introduces the likelihood function and how it can be used to determine the most likely parameter values given observed data. It provides examples of applying Bayesian modeling to proportions, normal distributions, linear regression with one predictor, and linear regression with multiple predictors. The lecture aims to give students a basic understanding of how Bayesian analysis works and prepare them for fitting linear mixed models.
This presentation guide you through Basic Probability Theory and Statistics, those are Random Experiment, Sample Space, Random Variables, Probability, Conditional Probability, Variance, Probability Distribution, Joint Probability Distribution, Conditional Probability Distribution (CPD) and Factor.
For more topics stay tuned with Learnbay.
This document provides a probability cheatsheet compiled by William Chen and Joe Blitzstein with contributions from others. It is licensed under CC BY-NC-SA 4.0 and contains information on topics like counting rules, probability definitions, random variables, expectations, independence, and more. The cheatsheet is designed to summarize essential concepts in probability.
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 3.
More info at http://summerschool.ssa.org.ua
This document defines key concepts related to probability distributions and random variables. It explains that a random variable can take on a set of possible values with different probabilities, and these probabilities are defined by a probability function. Probability functions for discrete random variables are called probability mass functions, while those for continuous random variables are called probability density functions. Both have cumulative distribution functions that give the probability that the random variable is less than or equal to a given value. Expected value and variance are used to characterize probability distributions. Examples are provided of common discrete and continuous distributions and how to calculate probabilities and expected values.
Similar to Statement of stochastic programming problems (20)
This document discusses student organizations and the university system in Germany. It provides an overview of the different types of higher education institutions in Germany, including universities, universities of applied sciences, and arts universities. It describes the degree system including bachelor's, master's, and Ph.D. programs. It also outlines the systems of student participation at universities, using the examples of Leipzig and Hanover. Student councils, departments, and faculty student organizations are discussed.
The document discusses grand challenges in energy and perspectives on moving towards more sustainable systems. It notes that while global energy demand and CO2 emissions rebounded in 2010 after the economic downturn, urgent changes are still needed. It explores perspectives on changing direction, including overcoming barriers like technologies, economies, management, and mindsets. The document advocates a systems approach and backcasting from desirable futures to identify pathways for transitioning between states.
Engineering can play an important role in sustainable development by focusing on meeting human needs over wants and prioritizing projects that serve the most vulnerable populations. Engineers should consider how their work impacts sustainability, affordability, and accessibility. A socially sustainable product is manufactured sustainably and also improves people's lives. Engineers are not neutral and should strive to serve societal needs rather than just generate profits. They can help redefine commerce and an engineering culture focused on meeting needs sustainably through services rather than creating unnecessary products and infrastructure.
Consensus and interaction on a long term strategy for sustainable developmentSSA KPI
The document discusses the need for a long-term vision for sustainable development to address major challenges like climate change, resource depletion, and inequity. A long-term perspective is required because these problems will take consistent action over many years to solve. However, short-term solutions may counteract long-term goals if not guided by an overall strategic vision. Developing a widely accepted long-term sustainable development vision requires input from many stakeholders to find balanced solutions and avoid dead ends. Strategic decisions with long-lasting technological and social consequences need a vision that can adapt to changing conditions over time.
Competences in sustainability in engineering educationSSA KPI
The document discusses competencies in sustainability for engineering education. It defines competencies and lists taxonomies that classify competencies into categories like knowledge, skills, attitudes, and ethics. Engineering graduates are expected to have competencies like critical thinking, systemic thinking, and interdisciplinarity. Analysis of competency frameworks from different universities found that competencies are introduced at varying levels, from basic knowledge to complex problem solving and valuing sustainability challenges. The document also outlines the University of Polytechnic Catalonia's framework for its generic sustainability competency.
The document discusses concepts related to sustainability including carrying capacity, ecological footprint, and the IPAT equation. It provides data on historical and projected world population growth. Examples are given showing the ecological footprint of different countries and how it is calculated based on factors like energy use, agriculture, transportation, housing, goods and services. The human development index is also introduced as a broader measure than GDP for assessing well-being. Graphs illustrate the relationship between increasing HDI, ecological footprint, and the goal of transitioning to sustainable development.
From Huygens odd sympathy to the energy Huygens' extraction from the sea wavesSSA KPI
Huygens observed that two pendulum clocks suspended near each other would synchronize their swings to be 180 degrees out of phase. He conducted experiments that showed the synchronization was caused by small movements transmitted through their common frame. While this discovery did not help solve the longitude problem as intended, it sparked further investigations into coupled oscillators and synchronization phenomena.
1) The document discusses whether dice rolls and other mechanical randomizers can truly produce random outcomes from a dynamics perspective.
2) It analyzes the equations of motion for different dice shapes and coin tossing, showing that outcomes are theoretically predictable if initial conditions can be reproduced precisely.
3) However, in reality small uncertainties in initial conditions mean mechanical randomizers can approximate random processes, even if they are deterministic based on their underlying dynamics.
This document discusses the concept of energy security costs. It defines energy security costs as externalities associated with short-term macroeconomic adjustments to changes in energy prices and long-term impacts of monopoly or monopsony power in energy markets. The document provides references on calculating health and environmental impacts of electricity generation and assessing costs and benefits of oil imports. It also outlines a proposed 4-hour course on basic concepts, examples, and a case study analyzing energy security costs for Ukraine based on impacts of increasing natural gas import prices.
Naturally Occurring Radioactivity (NOR) in natural and anthropic environmentsSSA KPI
This document provides an overview of naturally occurring radioactivity (NOR) and naturally occurring radioactive materials (NORM) with a focus on their relevance to the oil and gas industry. It discusses the main radionuclides of interest, including radium-226, radium-228, uranium, radon-222, and lead-210. It also summarizes the origins of NORM in the oil and gas industry and the types of radiation emitted by NORM.
Advanced energy technology for sustainable development. Part 5SSA KPI
All energy technologies involve risks that must be carefully evaluated and minimized to ensure sustainable development. No technology is perfectly safe, so ongoing analysis of benefits, risks and impacts is needed. Public understanding and acceptance of risks is also important.
Advanced energy technology for sustainable development. Part 4SSA KPI
The document discusses the impacts and benefits of energy technology research, using fusion research as a case study. It outlines four pathways through which energy research can impact economies and societies: 1) direct economic effects, 2) impacts on local communities, 3) impacts on industrial technology capabilities, and 4) long-term impacts on energy markets and technologies. It then analyzes the direct and indirect economic impacts of fusion research investments and the technical spin-offs that fusion research has produced. Finally, it evaluates the potential future role of fusion electricity in global energy markets under environmental constraints.
Advanced energy technology for sustainable development. Part 3SSA KPI
This document discusses using fusion energy for sustainable development through biomass conversion. It proposes a system where fusion energy is used to provide heat for gasifying biomass into synthetic fuels like methane and diesel. Experiments show biomass can be over 95% converted to hydrogen, carbon monoxide and methane gases using nickel catalysts at temperatures of 600-1000 degrees Celsius. A conceptual biomass reactor is presented that could process 6 million tons of biomass per year, consisting of 70% cellulose and 30% lignin, into synthetic fuels to serve as carbon-neutral transportation fuels. Fusion energy could provide the high heat needed for the gasification and synthesis processes.
Advanced energy technology for sustainable development. Part 2SSA KPI
The document summarizes fusion energy technology and its potential for sustainable development. Fusion occurs at extremely high temperatures and is the process that powers the Sun and stars. Researchers are working to develop fusion energy on Earth using hydrogen isotopes as fuel. Key challenges include confining the hot plasma long enough at high density for fusion reactions to produce net energy gain. Progress is being made towards achieving the conditions needed for a sustainable fusion reaction as defined by Lawson's criteria.
Advanced energy technology for sustainable development. Part 1SSA KPI
1. The document discusses the concept of sustainability and sustainable systems. It provides an example of a closed ecosystem with algae, water fleas, and fish, where energy and material balances must be maintained for long-term stability.
2. Key requirements for a sustainable system include energy balance between inputs and outputs, recycling of materials or wastes, and mechanisms to control population relationships and prevent overconsumption of resources.
3. Historically, the environment was seen as external and unchanging, but it is now recognized that the environment co-evolves interactively with the living creatures within it.
This document discusses the use of fluorescent proteins in current biological research. It begins with an overview of the development of optical microscopy and fluorescence techniques. It then focuses on the green fluorescent protein (GFP) and how it has been used as a molecular tag to study protein expression and interactions in living cells through techniques like gene delivery, transfection, viral infection, FRET, and optogenetics. The document concludes that fluorescent proteins have revolutionized cell biology by enabling the real-time visualization and control of molecular pathways and signaling processes in living systems.
Neurotransmitter systems of the brain and their functionsSSA KPI
1. Neurotransmitters are chemical substances released at synapses that transmit signals between neurons. The main neurotransmitters in the brain are acetylcholine, serotonin, dopamine, norepinephrine, glutamate, GABA, and endorphins.
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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Pride Month Slides 2024 David Douglas School District
Statement of stochastic programming problems
1. Lecture 1
Introduction. Statement of stochastic
programming problems
Leonidas Sakalauskas
Institute of Mathematics and Informatics
Vilnius, Lithuania <sakal@ktl.mii.lt>
EURO Working Group on Continuous Optimization
2. Content
Introduction
Example
Basics of Probability
Unconstrained Stochastic Optimization
Nonlinear Stochastic Programming
Two-stage linear Programming
Multi-Stage Linear Programming
3. Introduction
o Many decision problems in business and social systems
are modeled using mathematical programs, which seek to
maximize or minimize some objective, which is a function
of the decisions to be done.
oDecisions are represented by variables, which may be,
for example, nonnegative or integer. Objectives and
constraints are functions of the variables, and problem
data.
oThe feasible decisions are constrained according to limits in
resources, minimum requirements, etc.
oExamples of problem data include unit costs, production
rates, sales, or capacities.
4. Introduction
Stochastic programming is a framework for
modelling optimization problems that involve
uncertainty.
Whereas deterministic optimization problems are
formulated with known parameters, real world
problems almost invariably include some unknown
and uncertain parameters.
Stochastic programming models take advantage of
the fact that probability distributions governing
the data are known or can be estimated.
5. Introduction
The goal here is to find some policy that is
feasible for all (or almost all) the possible data
change scenarios and maximizes (or minimizes)
the probability of some event or expectation of
some function depending on the decisions and
the random variables.
This course is aimed to give the knowledge
about the statement and solving of stochastic
linear and nonlinear programs
The issues are also emphasized on continuous
optimization and applicability of programs
7. Introduction
Sources:
www.stoprog.org
J. Birge & F. Louvaux (1997) Introduction to
Stochastic Programming. Springer
L.Sakalauskas (2006)Towards Implementable
Nonlinear Stochastic Programming. Lecture
Notes in Economics and Mathematical
Systems, vol. 581, pp. 257-279
8. Introduction
An First Example
Farmer Fred can plant his land with either
corn, wheat, or beans.
For simplicity, assume that the season will
either be wet or dry – nothing in between.
If it is wet, corn is the most profitable
If it is dry, wheat is the most profitable.
9. Profit
All Corn All Wheat All Beans
Wet 100 70 80
Dry -10 40 35
Assume the probability of a wet season is p,
the expected profit of planting the different crops:
Corn: -10 + 110p
Wheat: 40 + 30p
Beans: 35 + 45p
10. What is the answer ?
Suppose p = 0.5, can anyone suggest a
planting plan?
Plant 1/2 corn, 1/2 wheat ?
Expected Profit:
0.5 (-10 + 110(0.5)) + 0.5 (40 +
30(0.5))= 50
Is this optimal?
11. !!!
Suppose p = 0.5, can anyone suggest a
planting plan?
Plant all beans!
Expected Profit: 35 + 45(0.5) = 57.5!
The expected profit in behaving optimally is
15% better than in behaving reasonably !
12. What Did We Learn ?
Averaging Solutions Doesn’t Work!
You can’t replace random parameters by
their mean value and solve the problem.
The best decision for today, when faced
with a number of different outcomes for the
future, is in general not equal to the
“average” of the decisions that would be
best for each specific future outcome.
13. Statement of stochastic programs
Mathematical Programming.
The general form of a mathematical program is
minimize f(x1, x2,..., xn) - objective function
subject to g1(x1, x2,..., xn) ≤ 0
.. - constraints
gm(x1, x2,..., xn) ≤ 0
where the vector
x=(x1, x2,..., xn) ϵ X,
supposes the decisions should be done, X is a set that be,
e.g., all nonnegative real numbers.
For example, xi can represent amount of production of the
ith from n products.
14. Statement of stochastic programs
Stochastic programming
is like mathematical (deterministic) programming but with
“random” parameters. Denote E as symbol of expectation and
Prob as symbol of probability.
Thus, now the objective (or constraint) function becomes by
mathematical expectation of some random function :
F(x)=Ef(x, ζ),
or probability of some event A(x):
F(x)=Prob(ζ ϵ A(x))
x=(x1, x2,..., xn) is a vector of a decision variable, ζ is a vector of
random variables, defining the uncertainty (scenarios, outcome
of some experiment).
15. Statement of stochastic programs
It makes sense to do just a bit of review of
probability.
ζ ϵ Ω is “outcome” of a random experiment,
called by an elementary event.
The set of all possible outcomes is Ω.
The outcomes can be combined into subsets
A Ω of ζ (called by events).
16. Random variable
Random variable ζ is described by
1) Set of support Ω=SUPP(ζ)
2) Probability measure
Probability measure is defined by the
cumulative distribution function:
F ( x) Pr ob ( X x) Pr ob ( X 1 x1 ,..., X n xn )
18. Continuous r.v.
Continuous random variable (or random
vector) are defined by probability density
function:
n
p( z ) :
Thus, in an uni-variate case:
x
F ( x) p( z )dz
19. Continuous r.v.
If the probability measure is absolutely
continuous, the expected value of
random function f (x, ) is integral:
F ( x) Ef ( x, ) f ( x, z ) p( z )dz
20. Continuous r.v.
The probability of some event (set of scenarios)
A is defined by the integral, too:
Pr ob ( A) Eh( ) p ( z )dz
z A
where
1, A
h( )
0, A
is the characteristic-function of set A.
21. What did we learn ?
Remark. Since any nonnegative function
n
p:
that
p ( z )dz 1
is the density function of certain random
variable (or vector) some multivariate
integrals can be changed by expectation of
some random variable (or vector).
22. Discrete r. v.
Discrete r.v. ζ is described by mass
probabilities of all elementary events:
z1 , z 2 ,..., z K
p1 , p2 ,..., pK ,
that
p1 p2 ... pK 1
23. Discrete r. v.
If probability measure is discrete, the expected
value of random function is the sum or series:
K
Ef ( X ) f ( zi ) pi
i 1
24. Singular random variable
Singular r.v. probabilistic measure is
concentrated on the set having the
zero Borel measure (say, the Cantor
set).
25. Statement of stochastic programs
Unconstrained continuous (nonlinear)
stochastic programming problem:
F ( x) Ef x, f ( x, z ) p( z )dz min
x X.
It is easy to extend this statement to discrete
model of uncertainty and constrained
optimization
26. Statement of stochastic programs
Constrained continuous (nonlinear )stochastic
programming problem is
F0 ( x) Ef 0 x, n
f 0 ( x, z ) p( z )dz min
R
F1 ( x) Ef1 x, n
f1 ( x, z ) p( z )dz 0,
R
x X.
If the constraint function is the probability of some
event depending on the decision variable, the problem
becomes by chance-constrained stochastic
programming problem
27. Statement of stochastic programs
Note, the expectation can enter the objective
function by nonlinear way, i.e.
F ( x) Ef x, min
x X.
Programs with functions of such kind are often
considered in statistics: Bayesian analysis, likelihood
estimation, etc., that are solved by Monte-Carlo
Markov Chain (MCMC) approach.
28. Statement of stochastic programs
The stochastic two-stage programming.
The most widely applied and studied stochastic
programming models are two-stage linear programs.
Here the decision maker takes some action in the first
stage, after which a random event occurs affecting the
outcome of the first-stage decision.
A recourse decision can then be made in the second
stage that compensates for any bad or undesired effects
that might have been experienced as a result of the
first-stage decision.
29. Statement of stochastic programs
The stochastic two-stage programming.
The optimal policy from such a model is a single
first-stage policy and a collection of recourse
decisions (a decision rule) defining which
second-stage action should be taken in
response to each random outcome.
30. Statement of stochastic programs
two-stage stochastic linear programming
The two-stage stochastic linear programming (SLP) problem
with recourse is formulated as
F ( x) c x E min y q y min
W y T x h, y Rm ,
Ax b, x X,
assume vectors q, h and matrices W, T be random in general.
31. Statement of stochastic programs
multi-stage stochastic linear programming
F ( x) c x E min y1 q1 y1 E (min y2 (q2 y2 ) ...) min
W1 y1 T1 x h1, W2 y2 T2 y1 h2 , ... ,
y1 R m1 , y2 R m2 , ... ,
Ax b, x X,
32. Statement of stochastic programs
An First Example
Thus, Farmer Tedd have to solve the
optimization problem that to make the best
decision:
F ( x1 , x2 , x3 ) x1 (100 p 10 (1 p))
x2 (70 p 40 (1 p))
x3 (80 p 35 (1 p)) max
subject to x1 0, x2 0, x3 0, x1 x2 x3 1.
33. Wrap-up and conclusions
Stochastic programming problems are
formulated as mathematical programming
tasks with the objective and constraints
defined as expectations of some random
functions or probabilities of some sets of
scenarios
Expectations are defined by multivariate
integrals (scenarios distributed
continuously) or finite series (scenarios
distributed discretely).