On Objective Function Value Performance of the Scenario Approach Under Regularity Conditions

We study the objective function 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 on the concept of complexity of robust optimization problems. With additional continuity and regularity conditions, via sensitivity analysis, we also provide explicit bounds which outperform the previously existing bounds. To illustrate our contribution, we also provide numerical examples. Finally, we apply our method to a planar antenna array synthesis problem, where we investigate the overfitting issue based on the derived probabilistic objective value bounds.