Scenario approach for minmax optimization with emphasis on the nonconvex case: positive results and caveats

TL;DR

This paper investigates the scenario approach for robust minmax optimization, focusing on high-dimensional, nonconvex cases, highlighting limitations and providing finite sample guarantees.

Contribution

It extends the analysis of the scenario approach to nonconvex problems, identifying obstructions to consistency and developing finite sample probabilistic guarantees.

Findings

Obstruction to consistency in noncompact decision sets.

Finite sample probabilistic guarantees for nonconvex problems.

Impact of high-dimensional concentration phenomena.

Abstract

We treat the so-called scenario approach, a popular probabilistic approximation method for robust minmax optimization problems via independent and indentically distributed (i.i.d) sampling from the uncertainty set, from various perspectives. The scenario approach is well-studied in the important case of convex robust optimization problems, and here we examine how the phenomenon of concentration of measures affects the i.i.d sampling aspect of the scenario approach in high dimensions and its relation with the optimal values. Moreover, we perform a detailed study of both the asymptotic behaviour (consistency) and finite time behaviour of the scenario approach in the more general setting of nonconvex minmax optimization problems. In the direction of the asymptotic behaviour of the scenario approach, we present an obstruction to consistency that arises when the decision set is noncompact. In the direction of finite sample guarantees, we establish a general methodology for extracting `probably approximately correct' type estimates for the finite sample behaviour of the scenario approach for a large class of nonconvex problems.