Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach

Liping Pang; Fanyun Meng; Jinhe Wang

doi:10.3934/jimo.2018116

Article Contents

2019, Volume 15, Issue 4: 1653-1675. Doi: 10.3934/jimo.2018116

This issue Previous Article Coordinating a multi-echelon supply chain under production disruption and price-sensitive stochastic demand Next Article Recovering optimal solutions via SOC-SDP relaxation of trust region subproblem with nonintersecting linear constraints

Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2.
School of Information and Control Engineering, Qingdao University of Technology, Qingdao 266033, China

^* Corresponding author: Fanyun Meng
^* Corresponding author: Fanyun Meng

Received: January 31, 2017

Revised: April 30, 2018

Early access: August 2018

Published: July 2019

The research is partially supported by Huzhou science and technology plan on No.2016GY03, the Natural Science Foundation of China, Grant 11601061.

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Abstract

We consider the sample average approximation method for a stochastic multiobjective programming problem constrained by parametric variational inequalities. The first order necessary conditions for the original problem and its sample average approximation (SAA) problem are established under constraint qualifications. By graphical convergence of set-valued mappings, the stationary points of the SAA problem converge to the stationary points of the true problem when the sample size tends to infinity. Under proper assumptions, the convergence of optimal solutions of SAA problems is also obtained. The numerical experiments on stochastic multiobjective optimization problems with variational inequalities are given to illustrate the efficiency of SAA estimators.

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Mathematics Subject Classification: Primary: 90C30, 90C29; Secondary: 65K10.

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Figure 1. Convergence of SAA optimal values

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Figure 2. Convergence of SAA optimal solutions

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Figure 3. The boundary of set $\mathbb{E}[\phi(C,\cdot)]$

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Figure 4. Convergence of SAA optimal values

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Figure 5. Convergence of SAA optimal solutions

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