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Uncertain standard quadratic optimization under distributional assumptions: a chance-constrained epigraphic approach

Immanuel M. Bomze, Daniel de Vicente

TL;DR

The paper addresses uncertain StQPs where the data matrix $\widetilde{\mathsf Q}$ is random with a known distribution. It introduces a chance-constrained epigraphic formulation (CCEStQP) based on Value-at-Risk and shows that, under a location/scale distribution for $x^T \widetilde{\mathsf Q} x$, the problem reduces to a deterministic StQP with $\overline{\mathsf Q} = M + F^{-1}(\alpha) S$. In the GOE perturbation special case, this yields $\overline{\mathsf Q} = Q^{nom} + \sqrt{2}\beta\, \Phi^{-1}(\alpha) I$, linking to a robust StQP with Frobenius-ball radius $\rho = \sqrt{2}\beta\,\Phi^{-1}(\alpha)$ and a PSD condition $\alpha \ge \Phi(|\lambda_{\min}(Q^{nom})|/(\sqrt{2}\beta))$. Numerical experiments show the CCEStQP can be less conservative than robust approaches at moderate confidence levels, providing a practical framework for stochastic StQPs in portfolio optimization and clustering. The work highlights a principled way to integrate distributional information into StQP handling when the objective is quadratic and the feasible set is the simplex.

Abstract

The standard quadratic optimization problem (StQP) consists of minimizing a quadratic form over the standard simplex. Without convexity or concavity of the quadratic form, the StQP is NP-hard. This problem has many relevant real-life applications ranging portfolio optimization to pairwise clustering and replicator dynamics. Sometimes, the data matrix is uncertain. We investigate models where the distribution of the data matrix is known but where both the StQP after realization of the data matrix and the here-and-now problem are indefinite. We test the performance of a chance-constrained epigraphic StQP to the uncertain StQP.

Uncertain standard quadratic optimization under distributional assumptions: a chance-constrained epigraphic approach

TL;DR

The paper addresses uncertain StQPs where the data matrix is random with a known distribution. It introduces a chance-constrained epigraphic formulation (CCEStQP) based on Value-at-Risk and shows that, under a location/scale distribution for , the problem reduces to a deterministic StQP with . In the GOE perturbation special case, this yields , linking to a robust StQP with Frobenius-ball radius and a PSD condition . Numerical experiments show the CCEStQP can be less conservative than robust approaches at moderate confidence levels, providing a practical framework for stochastic StQPs in portfolio optimization and clustering. The work highlights a principled way to integrate distributional information into StQP handling when the objective is quadratic and the feasible set is the simplex.

Abstract

The standard quadratic optimization problem (StQP) consists of minimizing a quadratic form over the standard simplex. Without convexity or concavity of the quadratic form, the StQP is NP-hard. This problem has many relevant real-life applications ranging portfolio optimization to pairwise clustering and replicator dynamics. Sometimes, the data matrix is uncertain. We investigate models where the distribution of the data matrix is known but where both the StQP after realization of the data matrix and the here-and-now problem are indefinite. We test the performance of a chance-constrained epigraphic StQP to the uncertain StQP.

Paper Structure

This paper contains 14 sections, 5 theorems, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Chance-constrained epigraphic models of random StQPs
  3. Definition and Value-at-Risk
  4. Distributional models for random (indefinite) symmetric matrices
  5. A distributional assumption and main results
  6. A reasonable distributional assumption on random StQPs
  7. Deterministic reformulation
  8. An important special case: GOE perturbation model
  9. Numerical experiments
  10. Nominal instance generation
  11. Random data generation
  12. Comparing solutions to the nominal with solutions to random instances
  13. Comparing conservativity
  14. Conclusion

Key Result

Proposition 4

For the GOE perturbation model def_Q_normal and the shifted Wishart model def_Q_wishart there are two symmetric $n\times n$ matrices ${\mathsf M}$ and ${\mathsf S}$, the latter satisfying ${\mathbf x}^\top{\mathsf S}{\mathbf x}>0$ for all ${\mathbf x}\in \Delta$, and a continuous cumulative distribu where are location and scale parameters, respectively.

Figures (4)

  • Figure 1: $|\ell^{\rm (nom)}_{\rm cce,\alpha} - \ell^{\rm (nom)}|$
  • Figure 2: $|\ell^{\rm (emp)}_{\rm cce,\alpha} - \ell^{\rm (emp)}|$
  • Figure 3: $|\ell^{\rm (nom)}_{\rm cce,\alpha} - \ell^{\rm (nom)}|$ vs $|\ell^{\rm (nom)}_{\rm rob } - \ell^{\rm (nom)}|$
  • Figure 4: $|\ell^{\rm (emp)}_{\rm cce,\alpha}-\ell^{\rm (emp)}|$ vs $|\ell^{\rm (emp)}_{\rm rob}-\ell^{\rm (emp)}|$

Theorems & Definitions (13)

  • Definition 1: Chance-Constrained Epigraphic Standard Quadratic Optimization Problem (CCEStQP)
  • Definition 2
  • Definition 3
  • Proposition 4
  • proof
  • Remark 6
  • Theorem 7
  • proof
  • Corollary 8
  • proof
  • ...and 3 more