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Unformatted text preview: EE236A (Fall 2007-08) Lecture 15 Self-dual formulations initialization and infeasibility detection skew-symmetric LPs homogeneous self-dual formulation self-dual formulation 151 Solution of an LP given a pair of primal and dual LPs minimize c T x subject to Ax + s = b s maximize- b T z subject to A T z + c = 0 z , classify problem as solvable, primal infeasible, or dual infeasible if solvable, find optimal x , s , z Ax + s = b, A T z + c = 0 , c T x + b T z = 0 , s , z if primal infeasible, find certificate z : A T z = 0 , z , b T z < if dual infeasible, find certificate x : Ax , c T x < Self-dual formulations 152 Methods for initialization and infeasibility detection phase I phase II minimize t subject to Ax b + t 1 , t disadvantage: phase I is as expensive as phase II big M method minimize c T x + Mt subject to Ax b + t 1 , t for some large M disadvantage: large M causes numerical problems infeasible-start methods (lecture 14) disadvantage: do not return certificate of (primal or dual) infeasibility self-dual embeddings: this lecture Self-dual formulations 153 Self-dual LP primal LP (variables u , v ) minimize f T u + g T v subject to Cu + Dv f- D T u + Ev = g u with C =- C T , E =- E T dual LP (variables u , v ) maximize- f T u- g T v subject to C u + D v f- D T u + E v = g u primal LP = dual LP we assume the problem is feasible: hence p = d =- p = 0 Self-dual formulations 154 Optimality conditions for self-dual LP...
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This note was uploaded on 09/26/2009 for the course CAAM 236 taught by Professor Dr.vandenber during the Spring '07 term at Monmouth IL.
- Spring '07