Decision-dependent distributionally robust standard quadratic optimization with Wasserstein ambiguity

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.

💡 Research Summary

This paper investigates the standard quadratic optimization problem (StQP), which seeks to minimize a quadratic form (x^{\top}Qx) over the standard simplex (\Delta), under distributional uncertainty of the data matrix (Q). While StQP is already NP‑hard in the deterministic setting, real‑world applications often involve noisy or partially observed matrices, motivating a robust treatment. The authors adopt a distributionally robust optimization (DRO) framework based on Wasserstein ambiguity sets.

First, they define a Wasserstein ball (B_{\theta,p}(\hat P_N)={P\in\mathcal P(\mathbb R^m):W_p(P,\hat P_N)\le\theta}) centered at the empirical distribution (\hat P_N) built from (N) samples of the unknown true distribution. They prove that the set of first moments of all distributions inside this ball coincides with a Euclidean ball centered at the empirical mean, which allows the inner supremum over distributions to be replaced by a simple norm constraint on the mean.

Because the StQP objective is linear in the uncertain matrix (Q) (i.e., (x^{\top}Qx) is affine in (Q)), the worst‑case expectation over the Wasserstein ball can be expressed analytically using push‑forward measures and dual norms. This yields a deterministic reformulation:

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