Primal Dual
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The document discusses primal-dual algorithms as an approximation technique for optimization problems. It provides an overview and introduction to primal-dual algorithms. As an example, it describes how the minimum weighted vertex cover problem can be solved using a primal-dual strategy. Specifically, it formulates the problem as an integer linear program and linear program. Rather than fully solving the linear program, the primal-dual algorithm maintains feasible integer and dual solutions and iteratively derives more feasible solutions until the integer solution is feasible.
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Keep Smiling & Mailing
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Keep Smiling & Mailing
With Warm Regards,
P.Gopi,
Inspection Engr,
CHEM P&S - QA&I,
L&T, E&C Div, Powai
Mumbai - 400072.
Extn - 4279 / 3049
Phone: +9122 6705 4279
Fax: +9122 6705 1383 / 1890
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aaoczc2252
Keep Smiling & Mailing
With Warm Regards,
P.Gopi,
Inspection Engr,
CHEM P&S - QA&I,
L&T, E&C Div, Powai
Mumbai - 400072.
Extn - 4279 / 3049
Phone: +9122 6705 4279
Fax: +9122 6705 1383 / 1890
Keep Smiling & Mailing
With Warm Regards,
P.Gopi,
Inspection Engr,
CHEM P&S - QA&I,
L&T, E&C Div, Powai
Mumbai - 400072.
Extn - 4279 / 3049
Phone: +9122 6705 4279
Fax: +9122 6705 1383 / 1890
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This document discusses P, NP and NP-complete problems. It begins by introducing tractable and intractable problems, and defines problems that can be solved in polynomial time as tractable, while problems that cannot are intractable. It then discusses the classes P and NP, with P containing problems that can be solved deterministically in polynomial time, and NP containing problems that can be solved non-deterministically in polynomial time. The document concludes by defining NP-complete problems as those in NP that are as hard as any other problem in the class, in that any NP problem can be reduced to an NP-complete problem in polynomial time.
UNIT -IV DAA.pdf
This document discusses NP-completeness and NP-hard problems. It begins by reviewing P vs NP problem classes and introduces the concepts of NP-hard and NP-complete problems. It provides examples of well-known NP-complete problems and discusses techniques for proving that a problem is NP-complete, such as reducing it to the SAT, CLIQUE, or vertex cover problems. The document aims to define key complexity classes like P, NP, NP-hard and NP-complete and explain the relationships between them.
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C&s sparse june_2010
Sparse & Redundant Representation Modeling of Images: Theory and Applications
by Michael Elad,the coolest ppt I have ever seen.
Dynamic programming
computer science, mathematics, management science, economics and bioinformatics, dynamic programming is a method for solving a complex problem
A Decomposition Technique For Solving Integer Programming Problems
This document summarizes a research paper that develops an algorithm for solving large-scale integer programming problems using Dantzig-Wolfe decomposition and column generation. The algorithm is tested on capital budgeting and scheduling problems. Numerical examples are provided to demonstrate the method. Key aspects of the algorithm include generating columns iteratively to solve a pricing problem at each iteration, and using computer software to code the algorithm and output results.
A New Lagrangian Relaxation Approach To The Generalized Assignment Problem
This summary provides the key details about a new Lagrangian relaxation approach to solving the generalized assignment problem (GAP) in 3 sentences:
The approach reformulates GAP into an equivalent problem by introducing auxiliary variables and coupling constraints, then relaxes the coupling constraints. This results in subproblems where the constraint structures of GAP are both active, providing stronger lower bounds than traditional approaches. The method was tested on small problems and shown to generate upper bounds more easily than traditional Lagrangian relaxation approaches for GAP.
Np
This document discusses NP-completeness and computational complexity. It begins by introducing the concept of problems that cannot be solved by algorithms, even with unlimited time, such as the halting problem. It then discusses NP-complete problems, which are problems whose complexity status is unknown, and may or may not be solvable in polynomial time. The document defines key complexity classes like P, NP, NP-complete and NP-hard, and explains the concept of reducing one problem to another to prove complexity.
7f44bdd880a385b7c1338293ea4183f930ea
This document summarizes a research paper that presents a new algorithm for solving the 0-1 Knapsack Problem. The algorithm combines previous approaches that generated valid inequalities and used surrogate relaxation. It is able to solve classical test instances with up to 10,000 variables in under 0.2 seconds. The key aspects of the new algorithm include:
1) Using a new initial "core" problem that is likely to produce a well-filled knapsack, rather than just items near the break item ratio.
2) Generating cardinality constraints if the number of states grows too large, relaxing them with the original weight constraint to obtain a good upper bound.
3) Attempting to improve the lower
Probabilistic Abductive Logic Programming using Possible Worlds
Reasoning in very complex contexts often requires purely deductive reasoning to be supported by a variety of techniques that can cope with incomplete data. Abductive inference allows to guess information that has not been explicitly observed. Since there are many explanations for such guesses, there is the need for assigning a probability to each one. This work exploits logical abduction to produce multiple explanations consistent with a given background knowledge and defines a strategy to prioritize them using their chance of being true. Another novelty is the introduction of probabilistic integrity constraints rather than hard ones. Then we propose a strategy that learns model and parameters from data and exploits our Probabilistic Abductive Proof Procedure to classify never-seen instances. This approach has been tested on some standard datasets showing that it improves accuracy in presence of corruptions and missing data.
NON LINEAR PROGRAMMING
This document discusses nonlinear programming (NLP) problems. NLP problems involve objective functions and/or constraints that contain nonlinear terms, making them more difficult to solve than linear programs. While exact solutions cannot always be found, algorithms can typically find approximate solutions within an acceptable error range of the optimum. However, for some NLP problems there is no reliable way to find the global maximum, as algorithms may stop at a local maximum instead. The document describes different types of NLP problems and techniques for solving them, including using Excel Solver with multiple starting values to attempt finding the global rather than just local optima.
Deep VI with_beta_likelihood
I am using DL & Actor critic tools for solving Variational inference problem. The intriguing part from my hand is that the likelihood has a Beta distribution.Thus we handle both VI issues and a non common distributions
Introduction to Max-SAT and Max-SAT Evaluation
This document provides an introduction to Max-SAT and Max-SAT evaluation. It discusses SAT and related problems like Max-SAT and pseudo-boolean optimization. The author shares their experience submitting their solver "toysat" to the Max-SAT evaluation in 2013. For Max-SAT 2014, the author plans to submit improved versions of SCIP, FibreSCIP, and toysat. The document concludes by discussing interactions between AI/CP and OR communities in developing solvers.
Analysis and Design of Algorithms notes
This document provides an introduction to algorithms and their analysis. It defines what an algorithm is and discusses different aspects of analyzing algorithm performance, including time complexity, space complexity, asymptotic analysis using Big O, Big Theta, and Big Omega notations. It also covers greedy algorithms, their characteristics, and examples like the knapsack problem. Greedy algorithms make locally optimal choices at each step without reconsidering prior decisions. Not all problems can be solved greedily, and the document discusses when greedy algorithms can and cannot be applied.
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A Decomposition Technique For Solving Integer Programming Problems
A Decomposition Technique For Solving Integer Programming Problems
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
ここ3000字までしか入らないけどタイトルの方がたくさん文字入ると思います。
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Mind map of terminologies used in context of Generative AI
Mind map of common terms used in context of Generative AI.
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みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
みなさんこんにちはこれ何文字まで入るの？40文字以下不可とか本当に意味わからないけどこれ限界文字数書いてないからマジでやばい文字数いけるんじゃないの？えこ...
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Primal Dual
- 1. Carlo Lombardi, June 2008 Theoretical Computer Science Primal-Dual Algorithms A brief survey of Primal-Dual Algorithms as an approximation technique for optimization problems Scribe: Carlo Lombardi [email_address]
- 4. Carlo Lombardi, June 2008 Theoretical Computer Science Minimum Weighted Vertex Cover Vertex Cover Problem “ Each edge is covered by at least one node” + Weighted Verteces “ Each vertex has a weight” + Minimization of total weight “ Minimize the total weight” = Minimum Weighted Vertex Cover (WVC)
- 5. Carlo Lombardi, June 2008 Theoretical Computer Science WVC: ILP and LP formulation We formulate the WVC as an Integer Linear Program (ILP) defining a variable x i for each vertex ( x i =1 if vertex i belongs to the cover, 0 otherwise). ILP FORMULATION LP FORMULATION by relaxing integrality constraints
- 6. Carlo Lombardi, June 2008 Theoretical Computer Science WVC: Rounding the LP solution Primal-Dual Method We need to solve LP formulation …it can be expensive for problems having many constraints!!! Can we do something clever?
- 7. Carlo Lombardi, June 2008 Theoretical Computer Science A different approach to LP relaxations: Primal-Dual strategy Main idea: !!! Don’t solve LP totally !!! Obtain a feasible integral solution to the LP ( Primal) from scratch using a related LP ( Dual ) to guide your decision . LP Primal LP Dual Good approximated solution “ Solve me” “ I’ll be your guide”
- 8. Carlo Lombardi, June 2008 Theoretical Computer Science P-D strategy: Background theoretic properties (1/2) PRIMAL DUAL (Weak Duality) For any feasible Primal-Dual solution pair (x,y) : = if (x,y) is optimal (Strong Duality) If either the Primal or Dual have bounded optimal solution, the both of them do. Moreover, their objective functions values are qual. That is: (Complementary Slackness) Let (x,y) be a solutions to a primal-dual pair of LPs with bounded optima. Then x and y are both optimal iff all of the following hold
- 9. Carlo Lombardi, June 2008 Theoretical Computer Science P-D strategy: Background theoretic properties (2/2) (Weak Duality) For any feasible Primal-Dual solution pair (x,y) : The dual solution is a lover bound for primal solution = if (x,y) is optimal (Strong Duality) If either the Primal or Dual have bounded optimal solution, the both of them do. Moreover, their objective functions values are qual. That is: At the optimum the evaluation of solutions coincides (Complementary Slackness) Let (x,y) be a solutions to a primal-dual pair of LPs with bounded optima. Then x and y are both optimal iff all of the following hold Only If a dual constraints is tight the corresponding primal variables can be greater than 0 (it can participate to the primal solution)
- 10. Carlo Lombardi, June 2008 Theoretical Computer Science Primal-Dual strategy
- 12. Carlo Lombardi, June 2008 Theoretical Computer Science Analysis of Program 2.7 (Weak Duality) Note that for every it holds: (1) The o.f. is infact From the (1) Because we are considering all vertices in V Each edge in E is taken two times