Lecture24

# Lecture24 - Advanced Mathematical Programming IE417 Lecture...

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Advanced Mathematical Programming IE417 Lecture 24 Dr. Ted Ralphs

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IE417 Lecture 24 1 Reading for This Lecture Sections 11.2-11.2
IE417 Lecture 24 2 The Linear Complementarity Problem Given M R p × p and q R p , the linear complementarity problem is to ﬁnd w,z R p such that Iw - Mz = q w,z 0 w j z j = 0 j This importance of this problem, for our purposes is in solving quadratic programming problems.

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IE417 Lecture 24 3 Terminology A cone spanned by p vectors, one from each pair e j , - m j is called a complementary cone . Note that we are trying to show that q belongs to at least one complementary cone. Vectors w,z satisfying w j z j = 0 j are called complementary . A solution to the given system is called a complementary feasible solution .
IE417 Lecture 24 4 Solving the LCP We can formulate the LCP as an optimization problem: min X [ y j w j + (1 - y j ) z j ] s.t. Iw - Mz = q w,z 0 y ∈ { 0 , 1 } The optimal solution is zero if and only if the solution is complementary. The variable y j indicates which of w j and z j is nonzero.

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IE417 Lecture 24 5 Another Approach A solution is called a complementary basic feasible solution
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Lecture24 - Advanced Mathematical Programming IE417 Lecture...

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