scholarly journals Expected Residual Minimization Method for a Class of Stochastic Quasivariational Inequality Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Hui-Qiang Ma ◽  
Nan-Jing Huang
Keyword(s):  
Sample Average Approximation ◽  
Gap Function ◽  
Stationary Points ◽  
Quasivariational Inequality ◽  
Minimization Method ◽  
Limiting Behavior ◽  
Regularized Gap Function ◽  
Expected Residual Minimization ◽  
Sample Average ◽  
Residual Minimization

We consider the expected residual minimization method for a class of stochastic quasivariational inequality problems (SQVIP). The regularized gap function for quasivariational inequality problem (QVIP) is in general not differentiable. We first show that the regularized gap function is differentiable and convex for a class of QVIPs under some suitable conditions. Then, we reformulate SQVIP as a deterministic minimization problem that minimizes the expected residual of the regularized gap function and solve it by sample average approximation (SAA) method. Finally, we investigate the limiting behavior of the optimal solutions and stationary points.

scholarly journals A New Gap Function for Vector Variational Inequalities with an Application

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hui-qiang Ma ◽  
Nan-jing Huang ◽  
Meng Wu ◽  
Donal O'Regan
Keyword(s):  
Variational Inequality ◽  
Optimization Problem ◽  
Dimensional Space ◽  
Sample Average Approximation ◽  
Gap Function ◽  
Vector Variational Inequality ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Sample Average ◽  
Stochastic Vector

We consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the vector variational inequality. As an application, we use the new gap function to reformulate a stochastic vector variational inequality as a deterministic optimization problem. We solve this optimization problem by employing the sample average approximation method. The convergence of optimal solutions of the approximation problems is also investigated.

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.

Sign in / Sign up

Export Citation Format

Share Document