STABILITY ANALYSIS OF STOCHASTIC MULTIOBJECTIVE OPTIMIZATION PROBLEMS WITH COMPLEMENTARITY CONSTRAINTS

We study the stability of the optimal solutions and the stationary points of the stochastic multiobjective optimization problems with complementarity constraints (SMOPCC) when the underlying probability measure varies in some metric probability space. We show under some moderate conditions that the optimal solutions and the optimal value function are continuous with respect to probability measure. Based on some new results on stochastic generalized equations, we also show that the set-valued mapping of M-stationary points is upper semi-continuous with respect to the variation of the probability measure. A particular focus is given to the empirical probability measure approximation which is also known as the sample average approximation (SAA). We present that the stationary points of SAA problems converge to their true counterparts with probability one (w.p.1.) at exponential rate as the sample size increases.