Smoothing sample average approximation method for solving stochastic second-order-cone complementarity problems

Meiju Luo,Yan Zhang

doi:10.1186/s13660-018-1674-2

Abstract

In this paper, we consider stochastic second-order-cone complementarity problems (SSOCCP). We first use the so-called second-order-cone complementarity function to present an expected residual minimization (ERM) model for giving reasonable solutions of SSOCCP. Then, we introduce a smoothing function, by which we obtain a smoothing approximate ERM model. We further show that the global solution sequence and weak stationary point sequence of this smoothing approximate ERM model converge to the global solution and the weak stationary point of the original ERM model as the smoothing parameter tends to zero respectively. Moreover, since the ERM formulation contains an expectation, we employ a sample average approximate method for solving the smoothing ERM model. As the convergence analysis, we first show that the global optimal solution of this smoothing sample average approximate problem converges to the global optimal solution of the ERM problem with probability one. Subsequently, we consider the weak stationary points’ convergence results of this smoothing sample average approximate problem of ERM model. Finally, some numerical examples are given to explain that the proposed methods are feasible.

Highlights

If F(x, y, z) = f (x) – y with a continuously differentiable function f : Rn → Rn and K = Kn, we can rewrite second-order-cone complementarity problems (SOCCP) as follows: Find (x, y) ∈ Rn × Rn such that x ∈ Kn, y ∈ Kn, x, y = 0, y = f (x)
The second-order-cone (SOC) in Rn (n ≥ 1) is defined as follows: Kn = (x1, x2) ∈ R × Rn–1 | x2 ≤ x1, where · denotes the Euclidean norm
Motivated by the work of Chen and Fukushima [4] for the special case of stochastic second-order-cone complementarity problems (SSOCCP), we propose the following deterministic formulation for SSOCCP, called the expected residual minimization (ERM) problem below, in which we try to find a vector (x, y, z) ∈ Rn × Rn × Rl that minimizes an expected residual for φ0 and F(x, y, z, ξ ), that is, min θ (x, y, z) := E F(x, y, z, ξ ) 2 + φ0(x, y) 2

Summary

Introduction

If F(x, y, z) = f (x) – y with a continuously differentiable function f : Rn → Rn and K = Kn, we can rewrite SOCCP as follows: Find (x, y) ∈ Rn × Rn such that x ∈ Kn, y ∈ Kn, x, y = 0, y = f (x). The convergence analysis of global solution and weak stationary point of this approximation problem are established in Sect.

Results

Conclusion