LP via Complementary pivoting

LP via Complementary pivoting - Complementary pivoting Carl...

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Complementary pivoting Carl Lemke created a sensation in the with a computational scheme that can be thought of as “re-wiring” the ideas in George Dantzig’s simplex method. Lemke’s scheme is called “complementary pivoting.” It solves linear programs, quadratic programs (whatever they are) and bi-matrix games. It is easiest to introduce in the context of a linear program and, in particular, a linear program that has been written in this format: LP. max c x , subject to the constraints (1) A x b , x 0 . The dual to this linear program is: DUAL . min y b , subject to the constraints (2) y A c , y 0 . Solutions to (1) and (2) are known to be optimal if and only if they satisfy the condition that is known as complementary slackness . To describe complementary slackness compactly, we insert slack variables in A x ≤ b , surplus variables in y A c and then multiply the latter by minus 1. This places (1) and (2) in the equivalent form (1) A x + s = b , x 0 ,
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LP via Complementary pivoting - Complementary pivoting Carl...

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