A tractable approximation of non-convex chance constrained optimization with non-Gaussian uncertainties

Abstract

Chance constrained optimization problems in engineering applications possess highly nonlinear process models and non-convex structures. As a result, solving a nonlinear non-convex chance constrained optimization (CCOPT) problem remains as a challenging task. The major difficulty lies in the evaluation of probability values and gradients of inequality constraints which are nonlinear functions of stochastic variables. This article proposes a novel analytic approximation to improve the tractability of smooth non-convex chance constraints. The approximation uses a smooth parametric function to define a sequence of smooth nonlinear programs (NLPs). The sequence of optimal solutions of these NLPs remains always feasible and converges to the solution set of the CCOPT problem. Furthermore, Karush-Kuhn-Tucker (KKT) points of the approximating problems converge to a subset of KKT points of the CCOPT problem. Another feature of this approach is that it can handle uncertainties with both Gaussian and/or non-Gaussian distributions.

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

“…The viability of the proposed approach is demonstrated through a numerical example. This work extends our recent research findings in [26,27] with respect to finite dimensional chance constrained optimization problem to infinite dimensional CCPDE problems. We point out to our readers that recent numerical considerations in [21,31] w.r.t.…”

“…In addition, Assumption A6 along with properties P1 and P4 of the approximation function ψ(τ, u, x) guarantees that, for α < 1, the feasible set M(τ ) is non-empty for sufficiently small τ ∈ (0, 1) and M(τ ) converges to P, for τ 0. This is already shown in [27,Theorem 4.2] (and also in [26, Theorem 6]) for finite dimensional chance constrained optimization problems. Lemma 4.5.…”

“…In case of convexity, the norm convergence of the approximate solutions to a solution of CCPDE red can be proved. For this purpose, we employ and extend our recent result from [26,27], where inner-outer approximation methods, for finite-dimensional chance constrained optimization problems, were proposed. In the work [26,27] convexity issues were not considered.…”

“…For this purpose, we employ and extend our recent result from [26,27], where inner-outer approximation methods, for finite-dimensional chance constrained optimization problems, were proposed. In the work [26,27] convexity issues were not considered. In the following, the properties of the underlying parametric functions are briefly summarized, and their convexity properties are studied.…”

“…The properties (P1)-(P5) were introduced in [26,27] in the context of finite dimensional chance constrained optimization problems to characterize admissible families of parametric functions. A special type of parametric function was also introduced in [26,27] for the purpose of inner-outer approximations of the feasible set corresponding to lower-upper approximations of the function p(…”