Probabilistic guarantees on the objective value for the scenario approach via sensitivity analysis

TL;DR

This paper introduces a new method to derive probabilistic bounds on the objective value in scenario-based robust convex optimization, improving existing results through sensitivity analysis and complexity considerations.

Contribution

It proposes a novel approach using max-min reformulation and sensitivity analysis to obtain tighter probabilistic bounds on objective values in scenario optimization.

Findings

New bounds outperform existing literature

Explicit bounds derived under regularity conditions

Numerical example demonstrates improved results

Abstract

This paper is concerned with objective value performance of the scenario approach for robust convex optimization. A novel method is proposed to derive probabilistic bounds for the objective value from scenario programs with a finite number of samples. This method relies on a max-min reformulation and the concept of complexity of robust optimization problems. With additional continuity and regularity conditions, via sensitivity analysis, we also provide explicit bounds which outperform an existing result in the literature. To illustrate the improvements of our results, we also provide a numerical example.