Robust Optimization for Models with Uncertain Second-Order Cone and Semidefinite Programming Constraints

Abstract

In this paper, we consider uncertain second-order cone (SOC) and semidefinite programming (SDP) constraints with polyhedral uncertainty, which are in general computationally intractable. We propose to reformulate an uncertain SOC or SDP constraint as a set of adjustable robust linear optimization constraints with an ellipsoidal or semidefinite representable uncertainty set, respectively. The resulting adjustable problem can then (approximately) be solved by using adjustable robust linear optimization techniques. For example, we show that if linear decision rules are used, then the final robust counterpart consists of SOC or SDP constraints, respectively, which have the same computational complexity as the nominal version of the original constraints. We propose an efficient method to obtain good lower bounds. Moreover, we extend our approach to other classes of robust optimization problems, such as nonlinear problems that contain wait-and-see variables, linear problems that contain bilinear uncertainty, and general conic constraints. Numerically, we apply our approach to reformulate the problem on finding the minimum volume circumscribing ellipsoid of a polytope and solve the resulting reformulation with linear and quadratic decision rules as well as Fourier-Motzkin elimination. We demonstrate the effectiveness and efficiency of the proposed approach by comparing it with the state-of-the-art copositive approach. Moreover, we apply the proposed approach to a robust regression problem and a robust sensor network problem and use linear decision rules to solve the resulting adjustable robust linear optimization problems, which solve the problem to (near) optimality. Summary of Contribution: Computing robust solutions for nonlinear optimization problems with uncertain second-order cone and semidefinite programming constraints are of much interest in real-life applications, yet they are in general computationally intractable. This paper proposes a computationally tractable approximation for such problems. Extensive computational experiments on (i) computing the minimum volume circumscribing ellipsoid of a polytope, (ii) robust regressions, and (iii) robust sensor networks are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

A Robust Optimization Approach to Deep Learning

“…Since obtaining exact robust counterparts in these cases is generally infeasible, safe approximations are considered instead (Bertsimas et al, 2020). For instance, Zhen et al (2017) develop safe approximations for the specific cases of second order cone and semidefinite programming constraints with polyhedral uncertainty. These techniques are generalized in Roos et al (2020), where the authors convert the robust counterpart to an adjustable robust optimization problem that produces a safe approximation for any problem that is convex in the optimization variables as well as in the the uncertain parameters.…”

Optimization problems in graphs with locational uncertainty

“…Actually, for Euclidean metric spaces based on the vector space R , ∈ Z + , d(u i , u j ) = u i − u j 2 is convex in u i and u j . Function u i − u j 2 is closely related to the second-order cone (SOC) constraints considered by Zhen et al (2021) for robust problems with polyhedral uncertainty sets. Zhen et al (2021) et al (2017), who rely on computational geometry techniques to provide constant-factor approximation algorithms in the special case where F contains all Hamiltonian cycles of G.…”

“…To summarize, we see that while Zhen et al (2021) provide valuable tools for addressing problems defined in Euclidean metric spaces considering uncertainty polytopes, their approaches cannot be used for graph-induced metric spaces, such as those mentioned in…”

“…The second one is composed of strategic facility location instances. The former application relies on two-dimensional Euclidean metric spaces so we can further include the affine decision rule reformulation from Zhen et al (2021) to the comparison.…”

“…In Section 7, we present our numerical experiments. Several appendices are annexed to the paper, containing most of the proofs, details on fixed-parameter tractable algorithms (FPT ), and the affine decision rules reformulation from Zhen et al (2021).…”