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

TL;DR

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, and an obstruction to consistency that arises when the decision set is noncompact is presented.

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.

Figures (4)

• Figure 1: Variation of the error in scenario approximations of the problem \ref{['eq:num exp']} with respect to the dimension ($n$) of the uncertainty set and number ($m$) of i.i.d samples drawn from the uncertainty set. The i.i.d samples were drawn according to the uniform distribution on $\Theta = \left[-1, 1 \right]^{n}$. The numbers reported here correspond to the average error of the scenario approximations over 25 sets of samples of length $m$ from $\Theta = \left[-1, 1 \right]^{n}$.
• Figure 2: Variation of the error in scenario approximations of the problem \ref{['eq:num exp']} with respect to the number ($m$) of i.i.d samples drawn from the uncertainty set when the dimension ($n$) of the uncertainty set is fixed at 20. The i.i.d samples were drawn according to the uniform distribution on $\Theta = \left[-1,1 \right]^{n}$. The numbers reported here correspond to the average error of the scenario approximations over 25 sets of samples of length $m$ from $\Theta = \left[-1, 1\right]^{n}$.
• Figure 3: Variation of the error in scenario approximations of the problem \ref{['eq:num exp 2']} with respect to the dimension ($n$) of the uncertainty set and number ($m$) of i.i.d samples drawn from the uncertainty set. The i.i.d samples were drawn according to the Gaussian distribution with mean 0 and variance $I_n$ on $\Theta = \mathbb{R}^{n}$. The numbers reported here correspond to the average error of the scenario approximations over 25 sets of samples of length $m$ from $\Theta = \mathbb{R}^{n}$.
• Figure 4: Variation of the error in scenario approximations of the problem \ref{['eq:num exp 2']} with respect to the number ($m$) of i.i.d samples drawn from the uncertainty set when the dimension ($n$) of the uncertainty set is fixed at 20. The i.i.d samples were drawn according to the Gaussian distribution with mean 0 and variance $I_n$ on $\Theta = \mathbb{R}^{n}$. The numbers reported here correspond to the average error of the scenario approximations over 25 sets of samples of length $m$ from $\Theta = \mathbb{R}^{n}$.

Theorems & Definitions (36)

• Definition 1.1: ref:Par-67
• Lemma 1.2
• Lemma 1.4
• Theorem 1.5: ref:Ram-18
• Theorem 1.6: ref:EsfSutLyg-15
• Example 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Remark 2.4