pdlb1 - ORIE 3300/5300 Prof. Bland Fall 2011 Motivating the...

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ORIE 3300/5300 Fall 2011 Prof. Bland Motivating the Primal-Dual Log Barrier Method These notes are based on lectures by Professor Michael Todd. Consider a canonical pair of primal and dual l.p. problems: max cx min yb ( P ) s.t. Ax b ( D ) s.t. yA c x 0 y 0 , where x = ( x 1 ,...,x n ) T denotes the vector of primal decision variables and y = ( y 1 ,...,y m ) denotes the vector of dual decision variables. Add slack variables s = ( s 1 ,...,s m ) T in (P) and surplus variables t = ( t 1 ,...,t n ) in (D). Now consider the optimality conditions based on part (2) of the Complementary Slackness Theorem: x is opt for ( P ) and y opt for ( D ) if s = b - Ax 0 t = yA - c 0 ( PF ) x 0 ( DF ) y 0 and t j x j = 0 j = 1 ,...,n ( CS ) y i s i = 0 i = 1 ,...,m Rather than maintaining ( PF ) and ( CS ) and working toward ( DF ), as in the simplex method, or maintaining ( DF ) and ( CS ) and working toward ( PF ), as in the dual simplex method, we will consider maintaining ( PF
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This note was uploaded on 03/08/2012 for the course ORIE 3300 at Cornell.

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pdlb1 - ORIE 3300/5300 Prof. Bland Fall 2011 Motivating the...

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