Global Optimization Method for Solving Mathematical Programs with Linear Complementarity Constraints

We propose a method for finding a global optimal solution of programs with linear complementarity constraints. This problem arises for instance in bilevel programming. The main idea of the method is to generate a sequence of points either ending at a global optimal solution within a finite number of iterations or converging to a global optimal solution. The construction of such sequence is based on branch-and-bound techniques, which have been used successfully in global optimization. Results on a numerical test of the algorithm are reported.

[1]  Panos M. Pardalos,et al.  Introduction to Global Optimization , 2000, Introduction to Global Optimization.

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

[3]  J. Bard,et al.  Nondifferentiable and Two-Level Mathematical Programming , 1996 .

[4]  Nguyen V. Thoai,et al.  Convergent Algorithms for Minimizing a Concave Function , 1980, Math. Oper. Res..

[5]  Nguyen V. Thoai,et al.  Global optimization method for solving the minimum maximal flow problem , 2003, Optim. Methods Softw..

[6]  Jianzhong Zhang,et al.  A New Extreme Point Algorithm and Its Application in PSQP Algorithms for Solving Mathematical Programs with Linear Complementarity Constraints , 2001, J. Glob. Optim..

[7]  Nimrod Megiddo,et al.  A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems , 1991, Lecture Notes in Computer Science.

[8]  Masao Fukushima,et al.  A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints , 1998, Comput. Optim. Appl..

[9]  Jonathan F. Bard,et al.  An explicit solution to the multi-level programming problem , 1982, Comput. Oper. Res..

[10]  Zhi-Quan Luo,et al.  Exact penalization and stationarity conditions of mathematical programs with equilibrium constraints , 1996, Math. Program..

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

[12]  Lonnie Hamm,et al.  GLOBAL OPTIMIZATION METHODS , 2002 .

[13]  Jonathan F. BARD,et al.  Convex two-level optimization , 1988, Math. Program..

[14]  Horst Reiner,et al.  Introduction to Global Optimization. Second Edition , 2000 .