Deriving robust counterparts of nonlinear uncertain inequalities

Abstract

In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.

Robust Optimization for the Pooling Problem

“…A benefit of safety factors is that they do not increase the complexity of the problem at all. A disadvantage is, however, that one generally has to rely on experience when choosing a suitable value for s. Note that Problem (19) is equivalent to a robust version of Problem (7) with uncertain bounds P U j,k and P L j,k and a box uncertainty set, e.g. :…”

“…For s = 1, Problem (19) is equivalent to the nominal problem and for s → ∞ the optimal solution is y l,j = z i,j = 0 for all (l, j), (i, j), i.e., the safest solution is producing nothing. To compare the safety factor approach with the robust optimization approach, we find the smallest safety factor s min which gives a feasible solution for the robust problem with a given uncertainty set U p .…”

“…(18) separation problem with solution from Eqn. (y l,j , z i,j , q i,l ) ← Eqn (19) safety factor problem 12:C i,k ← arg max Eqn (18) separation problem 13: if (y l,j , z i,j , q i,l ) is a feasible solution to Eqn.…”

Sensor Network Deployment under Distance Uncertainty with Robust Optimization

“…This approach, however, tends to find solutions that are too conservative to provide much more optimality. Later, Ben-Tal et al [21][22][23] carry out further research on the robust optimization theory and have made significant progress in robust convex optimization. However, as the resulting robust counterparts involve nonlinear problems, such methods cannot be applied to discrete optimization.…”

Nonlinear robust optimization for process design

“…Recently, robust counterparts of nonlinear uncertain inequalities have been derived based on convex analysis including support function, conjugate functions, Fenchel duality and conic duality, and in this work, both simple and complex types of uncertainty set have been studied. 11 In this article, a novel nonlinear robust optimization framework is proposed to address general nonlinear problems under uncertainty. The proposed method is based on linearization with respect to uncertain parameter around multiple realizations and an iterative algorithm.…”