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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 11: The Simplex Algorithm Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay We saw last time that to solve the linear programming problem, it was enough to nd an extreme point of maximum cost. The so called simplex algorithm looks at extreme points in a certain order. The algorithm consists of the following two steps. 1. Start at an extreme point. 2. Move to a neighbouring extreme point of greater cost if one exists and repeat this step. If no such neighbour exists then exit with this point as the optimum point. First some questions. 1. How do we identify the rst extreme point? 2. What do we mean by neighbouring extreme points? How do we nd them? 3. Is the algorithm correct? That is, when it terminates do we have an optimum? 4. How many iterations does this take before stopping in the worst case? Answering some of these questions will be our next task. We will leave the rst as an exercise. We begin with the second. 1 The notion of directions Let us consider an example in two dimensions. Example 1 Consider two half-planes in 2D given by 3 x + 4 y 12 7 x + 3 y 6 We have to nd out their point of intersection and the direction of the vectors v 1 and v 2 along the lines. The coordinates of the point of intersection ( x; y ) is the solution to: 7 3 3 4 x y = 6 12 Also the direction of potential neighbours can be determined to be the vectors 4 3 and 3 7 , respec- tively. If we write v 1 and v 2 as columns of a matrix B , 3 7 4 3 then we observe that AB is a diagonal matrix with negative entries on the diagonal. Consider a corner p of the cube. We see that its neighbours are the three corners you get by walking along the three edges which meet at p . Note that each edge is the intersection of two of the three planes intersecting at p . Let the bottom face of the cube be labelled 1 ; 2 ; 3 ; 4 in the clockwise direction and the...
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This note was uploaded on 01/04/2010 for the course CSE CS435 taught by Professor Profsundar during the Summer '09 term at IIT Bombay.
- Summer '09
- Computer Science