Abstract
In this paper, we study the sparsest solutions of linear complementarity problems (LCPs), which study has many applications, such as bimatrix games and portfolio selections. Mathematically, the underlying model is NP-hard in general. By transforming the complementarity constraints into a fixed point equation with projection type, we propose an l 1 regularization projection minimization model for relaxation. Through developing a thresholding representation of solutions for a key subproblem of this regularization model, we design a shrinkage-thresholding projection (STP) algorithm to solve this model and also analyze convergence of STP algorithm. Numerical results demonstrate that the STP method can efficiently solve this regularized model and get a sparsest solution of LCP with high quality.MSC:90C33, 90C26, 90C90.
Highlights
Given a matrix M ∈ Rn×n and a vector q ∈ Rn, the linear complementarity problem, denoted by linear complementarity problems (LCPs)(q, M), is to find a vector x ∈ Rn such that x ≥, Mx + q ≥, xT (Mx + q) = .The set of solutions to this problem is denoted by SOL(q, M)
In Section, we develop a shrinkage-thresholding representation theory for the subproblem of ( ) and propose a shrinkage-thresholding projection (STP) algorithm for ( )
4 Numerical experiments we present some numerical experiments to demonstrate the effectiveness of our STP algorithm
Summary
The set of solutions to this problem is denoted by SOL(q, M). We try to find the sparsest solution of the LCP, which has the smallest number of nonzero entries. In this paper we consider applying the l norm to find the sparsest solution of LCPs, and we obtain the following minimization problem to approximate ( ): min x∈Rn x s.t. x = x – F(x) +, ( )
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