Stochastic absolute value equations

TL;DR

This paper introduces stochastic absolute value equations, explores their expected value formulations, establishes existence conditions, and proposes a smoothing gradient method for solving the associated minimization problem, supported by numerical results.

Contribution

It presents a novel class of stochastic absolute value equations, develops solution methods, and provides theoretical and numerical analysis for these equations.

Findings

Existence conditions for solutions are established.

A smoothing gradient method is proposed for solving the equations.

Numerical results demonstrate the effectiveness of the proposed approach.

Abstract

We propose a new kind of stochastic absolute value equations involving absolute values of variables. By utilizing an equivalence relation to stochastic bilinear program, we investigate the expected value formulation for the proposed stochastic absolute value equations. We also consider the expected residual minimization formulation for the proposed stochastic absolute value equations. Under mild assumptions, we give the existence conditions for the solution of the stochastic absolute value equations. The solution of the stochastic absolute value equations can be gotten by solving the discrete minimization problem. And we also propose a smoothing gradient method to solve the discrete minimization problem. Finally, the numerical results and some discussions are given.