Mathematical Programs with Second-Order Cone Complementarity Constraints: Strong Stationarity and Approximation Method

2019 ◽  
Vol 181 (2) ◽  
pp. 521-540
Author(s):  
Xide Zhu ◽  
Jin Zhang ◽  
Jinchuan Zhou ◽  
Xinmin Yang
Keyword(s):  
Approximation Method ◽  
Second Order ◽  
Complementarity Constraints ◽  
Mathematical Programs ◽  
Strong Stationarity ◽  
Second Order Cone

CONVERGENCE ANALYSIS OF A REGULARIZED SAMPLE AVERAGE APPROXIMATION METHOD FOR STOCHASTIC MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS

2011 ◽  
Vol 28 (06) ◽  
pp. 755-771 ◽  
Author(s):  
YONGCHAO LIU ◽  
GUI-HUA LIN
Keyword(s):  
Convergence Analysis ◽  
Approximation Method ◽  
Regularization Parameter ◽  
Sample Average Approximation ◽  
Stationary Points ◽  
Complementarity Constraints ◽  
One Stage ◽  
Mathematical Programs ◽  
Sample Average ◽  
Average Approximation

Regularization method proposed by Scholtes (2011) has been a recognized approach for deterministic mathematical programs with complementarity constraints (MPCC). Meng and Xu (2006) applied the approach coupled with Monte Carlo techniques to solve a class of one stage stochastic MPCC and presented some promising numerical results. However, Meng and Xu have not presented any convergence analysis of the regularized sample approximation method. In this paper, we fill out this gap. Specifically, we consider a general class of one stage stochastic mathematical programs with complementarity constraint where the objective and constraint functions are expected values of random functions. We carry out extensive convergence analysis of the regularized sample average approximation problems including the convergence of statistical estimators of optimal solutions, C-stationary points, M-stationary points and B-stationary points as sample size increases and the regularization parameter tends to zero.

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