An Inexact Solution Approach for the Mathematical Program with Complementarity Constraints

We present a new smoothing method based on a logarithm-exponential function for mathematical program with complementarity constraints (MPCC). Different from the existing smoothing methods available in the literature, we construct an approximate smooth problem of MPCC by partly smoothing the complementarity constraints. With this new method, it is proved that the Mangasarian-Fromovitz constraint qualification holds for the approximate smooth problem. Convergence of the approximate solution sequence, generated by solving a series of smooth perturbed subproblems, is investigated. Under the weaker constraint qualification MPCC-Cone-Continuity Property, it is proved that any accumulation point of the approximate solution sequence is a M-stationary point of the original MPCC. Preliminary numerical results indicate that the developed algorithm based on the partly smoothing method is efficient, particularly in comparison with the other similar ones.

Download Full-text

A Regularization SAA Scheme for a Stochastic Mathematical Program with Complementarity Constraints

Journal of Applied Mathematics ◽

10.1155/2014/321010 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12

Author(s):

Yu-xin Li ◽

Jie Zhang ◽

Zun-quan Xia

Keyword(s):

Recent Literature ◽

Mathematical Program ◽

Uncertain Data ◽

Almost Sure Convergence ◽

Sample Average Approximation ◽

Accumulation Point ◽

Complementarity Constraints ◽

Convergence Properties ◽

Sample Average ◽

Average Approximation

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC-MFCQ, which is weaker than MPCC-LICQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed.

Download Full-text

Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints

Journal of Industrial & Management Optimization ◽

10.3934/jimo.2020050 ◽

2017 ◽

Vol 13 (5) ◽

pp. 0-0

Author(s):

Li Chu ◽

◽

Bo Wang ◽

Jie Zhang ◽

Hong-Wei Zhang ◽

...

Keyword(s):

Convergence Analysis ◽

Mathematical Program ◽

Second Order ◽

Complementarity Constraints ◽

Second Order Cone

Download Full-text

Numerical Studies on Reformulation Techniques for Continuous Network Design with Asymmetric User Equilibria

International Journal of Operations Research and Information Systems ◽

10.4018/joris.2010101304 ◽

2010 ◽

Vol 1 (1) ◽

pp. 52-72 ◽

Cited By ~ 7

Author(s):

Michael Ferris ◽

Henry X. Liu

Keyword(s):

Nonlinear Programming ◽

Network Design ◽

Mathematical Program ◽

Network Design Problem ◽

Complementarity Constraints ◽

Nonlinear Programming Problem ◽

Design Problems ◽

Nonlinear Complementarity ◽

Numerical Studies ◽

Constraints Model

In this article, we aim to find the most effective reformulation techniques to solve the MPCC (mathematical program with complementarity constraints) model that we proposed recently for continuous network design problems under asymmetric user equilibria. The MPCC model is based on a link-node nonlinear complementarity formulation for asymmetric user equilibria. By applying various reformulation techniques for the lower level nonlinear complementarity, the original bilevel formulation can be converted to a single level nonlinear programming problem. We show that certain reformulations are more effective than others to solve the proposed MPCC model. Recommendations are thus provided on how to choose a reformulation of the continuous network design problem that can be solved effectively and/or efficiently.

Download Full-text