A global QP-free algorithm for mathematical programs with complementarity constraints
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(2020) 2020:213
RESEARCH
Open Access
A global QP-free algorithm for mathematical programs with complementarity constraints Jian Ling Li1* and Qi Zhang1 *
Correspondence: [email protected] 1 College of Mathematics and Information Science, Guangxi University, Nanning, China
Abstract In this paper, a primal–dual interior point QP-free algorithm for mathematical programs with complementarity constraints is presented. Firstly, based on Fischer–Burmeister function and smoothing techniques, the investigated problem is approximated by a smooth nonlinear constrained optimization problem. Secondly, combining with an effective penalty function technique and working set, a QP-free algorithm is proposed to solve the smooth constrained optimization problem. At each iteration, only two reduced linear equations with the same coefficient matrix are solved to obtain the search direction. Under some mild conditions, the proposed algorithm possesses global convergence. Finally, some numerical results are reported. Keywords: Complementarity constraints; Working set; QP-free algorithm; Global convergence
1 Introduction In this paper, we discuss the following mathematical programming problem with complementarity constraints (MPCC for short): min s.t.
f (x, y) g(x, y) ≤ 0,
(1)
0 ≤ F(x, y) ⊥ y ≥ 0, where f : Rn+m → R, g = (g1 , . . . , gmg )T : Rn+m → Rmg , F = (F1 , . . . , Fm )T : Rn+m → Rm are continuously differentiable functions. “F(x, y) ⊥ y” means that the vectors F(x, y) and y are perpendicular to each other. MPCC (1) has a broad of applications in real world, such as engineering design, traffic transportation, game theory and so on. A detailed overview of MPCC applications can be found in [1] and the monographs [2–4]. Since MPCC (1) is a nonconvex optimization problem and the standard Mangasarian– Fromovitz constraint qualification (MFCQ) is violated at any feasible point, the welldeveloped algorithms for the standard nonlinear programs (for example, [5–18]) typically have severe difficulties if they are directly used to solve the MPCC (1). Hence, MPCCtailed algorithms are desired. © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Li and Zhang Journal of Inequalities and Applications
(2020) 2020:213
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