A half thresholding projection algorithm for sparse solutions of LCPs

In this paper, we aim to find sparse solutions of the linear complementarity problems (LCPs), which has many applications such as bimatrix games and portfolio selection. Mathematically, the underlying model is NP-hard in general. Thus, an $$\ell _{1/2}$$ℓ1/2 regularized projection minimization model is proposed for relaxation. A half thresholding projection (HTP) algorithm is then designed for this regularization model, and the convergence of HTP algorithm is studied. Numerical results demonstrate that the HTP algorithm can effectively solve this regularization model and output very sparse solutions of LCPs with high quality.

[1]  P. Toint.  Global Convergence of a a of Trust-Region Methods for Nonconvex Minimization in Hilbert Space , 1988 .

[2]  Duan Li,et al.  Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach , 2014, INFORMS J. Comput..

[3]  David J. Field,et al.  Emergence of simple-cell receptive field properties by learning a sparse code for natural images , 1996, Nature.

[4]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[5]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[6]  Michael C. Ferris,et al.  Complementarity: Applications, Algorithms and Extensions , 2010 .

[7]  Masao Fukushima,et al.  An Implementable Active-Set Algorithm for Computing a B-Stationary Point of a Mathematical Program with Linear Complementarity Constraints , 2002, SIAM J. Optim..

[8]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[9]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[10]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[11]  Masao Fukushima,et al.  Some Feasibility Issues in Mathematical Programs with Equilibrium Constraints , 1998, SIAM J. Optim..

[12]  Gui-Hua Lin,et al.  New reformulations for stochastic nonlinear complementarity problems , 2006, Optim. Methods Softw..

[13]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[14]  Zongben Xu,et al.  $L_{1/2}$ Regularization: A Thresholding Representation Theory and a Fast Solver , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[15]  Duan Li,et al.  Optimal Cardinality Constrained Portfolio Selection , 2013, Oper. Res..

[16]  Emmanuel J. Candès,et al.  Highly Robust Error Correction byConvex Programming , 2006, IEEE Transactions on Information Theory.

[17]  D. L. Donoho,et al.  Compressed sensing , 2006, IEEE Trans. Inf. Theory.

[18]  Chao Zhang,et al.  Minimal Zero Norm Solutions of Linear Complementarity Problems , 2014, J. Optim. Theory Appl..