Decision-dependent distributionally robust standard quadratic optimization with Wasserstein ambiguity

TL;DR

This paper studies a distributionally robust approach to the standard quadratic optimization problem under Wasserstein ambiguity, providing a deterministic reformulation and out-of-sample performance guarantees.

Contribution

It introduces a Wasserstein-based distributionally robust formulation for StQP and shows its equivalence to a modified deterministic problem, extending recent research.

Findings

Equivalent deterministic reformulation of robust StQP

Out-of-sample performance guarantees established

Experimental validation of the approach

Abstract

The standard quadratic optimization problem (StQP) consists of minimizing a quadratic form over the standard simplex. Without assuming convexity or concavity of the quadratic form, the StQP is NP-hard. This problem has many interesting applications ranging from portfolio optimization to machine learning. Sometimes, the data matrix is uncertain but some information about its distribution can be inferred, e.g. a distance to a reference distribution (typically, the empirical distribution after sampling). In distributionally robust optimization, the goal is to hedge against the worst case of all possible distributions in an ambiguity set, defined by above mentioned distance. In this paper we will focus on distributionally robust StQPs under Wasserstein distance, and show equivalence to an accordingly modified deterministic instance of an StQP. This blends well into recent findings for other approaches of StQPs under uncertainty. We will also address out-of-sample performance guarantees. Carefully designed experiments shall complement and illustrate the approach.