Linear Programming and Extensions

Linear Programming and Extensions. Instructor: Prof. Prabha Sharma, Department of Mathematics and Statistics, IIT Kanpur. The objective of this course is to introduce those real life problems which can be formulated as Linear Programming Problems (LPP). The course will be taught as a first course in optimization, hence all the concepts will be properly motivated and explained with examples. Topics covered in this course include Simplex algorithm; Duality theory and its ramifications; Basic ideas of the ellipsoid algorithm and Karmarkar's algorithm; Special cases of LPP; and Dynamic programming and PERT/CPM algorithms. (from

Lecture 40 - Interior Point Methods

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Lecture 01 - Introduction to Linear Programming Problems
Lecture 02 - Vector Space, Linear Independence and Dependence, Basis
Lecture 03 - Moving from One Basic Feasible Solution to Another, Optimality Criteria
Lecture 04 - Basic Feasible Solutions, Existence and Derivation
Lecture 05 - Convex Sets, Dimension of a Polyhedron Faces, Example of a Polytope
Lecture 06 - Direction of a Polyhedron, Correspondence between BFS and Extreme Points
Lecture 07 - Representation Theorem, LPP Solution is a BFS
Lecture 08 - Development of the Simplex Algorithm, Unboundedness, Simplex Tableau
Lecture 09 - Simplex Tableau and Algorithm, Cycling, Bland's Anti-Cycling Rules
Lecture 10 - Big-M Method, Graphical Solutions, Adjacent Extreme PTS and Adjacent BFS
Lecture 11 - Progress of Simplex Algorithm on a Polytope, Bounded Variables LPP
Lecture 12 - LPP Bounded Variable, Revised Simplex Algorithm, Duality Theory, Weak Duality Theorem
Lecture 13 - Weak Duality Theorem, Economic Interpretation of Dual Variables, Fundamental Theorem of Duality
Lecture 14 - Examples of Writing the Dual Complementary Slackness Theorem
Lecture 15 - Complementary Slackness Conditions, Dual Simplex Algorithm
Lecture 16 - Primal-Dual Algorithm
Lecture 17 - Starting Dual Feasible Solutions, Shortest Path Problem
Lecture 18 - Shortest Path Problem, Primal-Dual Method
Lecture 19 - Shortest Path Problem-Complexity, Interpretation of Dual Variables, Changes in the Cost Vector
Lecture 20 - Post-Optimality Analysis, Changes in B, Adding a New Constraint
Lecture 21 - Parametric LPP-Right Hand Side Vector
Lecture 22 - Parametric Cost Vector LPP
Lecture 23 - Parametric Cost Vector LPP, Introduction to Min-Cost Flow Problem
Lecture 24 - Min-Cost Flow Problem, Transportation Problem
Lecture 25 - Transportation Problem Degeneracy, Cycling
Lecture 26 - Sensitivity Analysis
Lecture 27 - Sensitivity Analysis (cont.)
Lecture 28 - Bounded Variable Transportation Problem, Min-Cost Flow Problem
Lecture 29 - Min-Cost Flow Problem
Lecture 30 - Starting Feasible Solution, Lexicographic Method for Preventing Cycling, Strongly Feasible Solution
Lecture 31 - Shortest Path Problem, Shortest Path between Any Two Nodes, Detection of Negative Cycles
Lecture 32 - Min-Cost-Flow Sensitivity Analysis, Shortest Path Problem Sensitivity Analysis
Lecture 33 - Min-Cost Flow Changes in ARC Capacities, Max-Flow Problem
Lecture 34 - Min-Cut Max-Flow Theorem, Labelling Algorithm
Lecture 35 - Max-Flow-Critical Capacity of an ARC, Starting Solution for Min-Cost Flow Problem
Lecture 36 - Improved Max-Flow Algorithm
Lecture 37 - Critical Path Method (CPM)
Lecture 38 - Programme Evaluation and Review Technique (PERT)
Lecture 39 - Simplex Algorithm is not Polynomial Time - An example
Lecture 40 - Interior Point Methods