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Unformatted text preview: MS&E111 Lecture 5 Introduction to Optimization April 19, 2006 Prof. Amin Saberi Scribed by: Yusuf Ozuysal 1 Simplex Algorithm In this lecture, we will be going through the steps of simplex algorithm for solving an LP in the standard form. In the last lecture, we gave the following rough description of the algorithm: a) Start from any basic feasible solution x . b) Repeat until x is optimal- If there exists a basic feasible solution y adjacent to x such that c T y ≥ c T x , then x ← y. Otherwise, x is optimal. Now let us focus on the first step and talk about how we can find a basic feasible solution: 1.1 Initiating the Simplex Algorithm Consider an LP in standard form minimize z = c T x subject to Ax = b x ≥ Reminder: A feasible solution in which at most m variables are nonzero, is called a basic feasible solution . One might try to find a basic feasible solution by selecting m linearly independent columns from the constraints and solving for the corresponding variables and setting the rest of the variables to 0. For example, for the linear programming below: minimize z = 2 x 1 + 3 x 2 + x 3 subject to x 1 + x 2- x 3 = 3- x 1 + 2 x 2- 2 x 3 =- 3 x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 2 MS&E 111: Introduction to Optimization - Spring 06 Here let us select x 1 and x 2 as the basic variables and set x 3 to 0. We will get the independent equations 1 1- 1 2 x 1 x 2 = 3- 3 which gives the basic feasible solution as x 1 = 3 , x 2 = 0 , x 3 = 0. However, we were lucky that both x 1 and x 2 turned out to be non-negative. In general, it is not easy to find a set of m linearly independent columns which give us a basic solution that is non-negative...
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This note was uploaded on 09/23/2009 for the course ENGR 62 taught by Professor Unknown during the Spring '06 term at Stanford.
- Spring '06