Chance constrained optimization of elliptic PDE systems with a smoothing convex approximation^*

¹ Department of Process Optimization, Institute for Automation and Systems Engineering, Technische Universität Ilmenau, P.O. Box 10 05 65, 98684 Ilmenau, Germany.
² Department of Mathematical Methods of Operations Research, Institute of Mathematics, Technische Universität Ilmenau, P.O. Box 10 05 65, 98684 Ilmenau, Germany.

^** Corresponding author: abebe.geletu@tu-ilmenau.de

Received: 25 February 2018
Accepted: 27 November 2019

Abstract

In this paper, we consider chance constrained optimization of elliptic partial differential equation (CCPDE) systems with random parameters and constrained state variables. We demonstrate that, under standard assumptions, CCPDE is a convex optimization problem. Since chance constrained optimization problems are generally nonsmooth and difficult to solve directly, we propose a smoothing inner-outer approximation method to generate a sequence of smooth approximate problems for the CCPDE. Thus, the optimal solution of the convex CCPDE is approximable through optimal solutions of the inner-outer approximation problems. A numerical example demonstrates the viability of the proposed approach.