Lecture4 - MS&E111 Introduction to Optimization Prof Amin...

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MS&E111 Lecture 4 Introduction to Optimization April 17, 2006 Prof. Amin Saberi Scribed by: Yusuf Ozuysal 1 Solving a linear program In this lecture, we will talk about the simplex algorithm and how it finds the optimum solution for a linear program. First, we will show how to transform all linear programs to a generic form known as the standard form. 1.1 Standard form A linear program is in standard form if: All its variables are required to be non-negative All of its constraints (except the non-negativity constraints) are in equality form The constants on the right hand side in all constraints are non-negative Every linear program can be written in the standard format. For example, consider the linear program: maximize z = 3 x 1 + 2 x 2 - x 3 + x 4 subject to x 1 + 2 x 2 + x 3 - x 4 5 - 2 x 1 - 4 x 2 + x 3 + x 4 ≤ - 1 x 1 0 , x 2 0 We multiply any constraints in which the right hand side is non-negative by - 1 maximize z = 3 x 1 + 2 x 2 - x 3 + x 4 subject to x 1 + 2 x 2 + x 3 - x 4 5 2 x 1 + 4 x 2 - x 3 - x 4 1 x 1 0 , x 2 0
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