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On the Semidefinite Representability of Continuous Quadratic Submodular Minimization With Applications to Moment Problems

Samuel Burer, Karthik Natarajan

TL;DR

This work addresses the problem of minimizing a continuous quadratic submodular function over the box $[0,1]^n$ by establishing a tight semidefinite relaxation that exactly matches the original optimum, enabling polynomial-time SDP solvability. The main technical contribution is a tight SDP reformulation for quadratic continuous submodular minimization (QSMB) that remains exact under general submodular data, with a proof by induction leveraging a technical lemma. The authors then apply this to two moment problems: (i) a novel distributionally robust optimization ambiguity set on $[0,1]^n$ with fixed means, fixed diagonal second moments, and lower bounds on cross moments, and (ii) tight covariance bounds for bounded variables given means and variances, including a closed-form solution in dimension $2$. Numerical results illustrate tighter bounds relative to existing approaches on Laplacian-energy problems, underscoring the practical impact for robust optimization and covariance estimation. Overall, the paper extends tractability from convex minimization to continuous submodular minimization via exact SDP relaxations and opens new application avenues in moment problems and robust optimization.

Abstract

We show that continuous quadratic submodular minimization with bounds is solvable in polynomial time using semidefinite programming, and we apply this result to two moment problems arising in distributionally robust optimization and the computation of covariance bounds. Accordingly, this research advances the ongoing study of continuous submodular minimization and opens new application areas therein.

On the Semidefinite Representability of Continuous Quadratic Submodular Minimization With Applications to Moment Problems

TL;DR

This work addresses the problem of minimizing a continuous quadratic submodular function over the box by establishing a tight semidefinite relaxation that exactly matches the original optimum, enabling polynomial-time SDP solvability. The main technical contribution is a tight SDP reformulation for quadratic continuous submodular minimization (QSMB) that remains exact under general submodular data, with a proof by induction leveraging a technical lemma. The authors then apply this to two moment problems: (i) a novel distributionally robust optimization ambiguity set on with fixed means, fixed diagonal second moments, and lower bounds on cross moments, and (ii) tight covariance bounds for bounded variables given means and variances, including a closed-form solution in dimension . Numerical results illustrate tighter bounds relative to existing approaches on Laplacian-energy problems, underscoring the practical impact for robust optimization and covariance estimation. Overall, the paper extends tractability from convex minimization to continuous submodular minimization via exact SDP relaxations and opens new application avenues in moment problems and robust optimization.

Abstract

We show that continuous quadratic submodular minimization with bounds is solvable in polynomial time using semidefinite programming, and we apply this result to two moment problems arising in distributionally robust optimization and the computation of covariance bounds. Accordingly, this research advances the ongoing study of continuous submodular minimization and opens new application areas therein.

Paper Structure

This paper contains 16 sections, 11 theorems, 83 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions and Structure of the Paper
  3. Notations
  4. A Tight Semidefinite Relaxation
  5. Existing results
  6. Our result
  7. Three examples
  8. Distributionally Robust Optimization
  9. A new moment ambiguity set
  10. Semidefinite representation
  11. Comparison with an existing moment ambiguity set
  12. Covariance Bounds
  13. Closed-form solution in dimension 2
  14. Numerical Results
  15. Technical Lemma
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose $(Q,c,\kappa)$ is submodular with $c \le 0$. Then the optimal value of (equ:qpb) equals the optimal value of its SDP relaxation

Figures (1)

  • Figure 1: Worst-case $(1-\alpha)$-subquantile for ambiguity sets ${\cal Q}$ and ${\cal P}$.

Theorems & Definitions (24)

  • Theorem 1: Theorem 3.1 in Kim-Kojima_2003
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • Example 1: Model for multi-product pricing with substitutes and linear demand
  • Example 2: Robust counterpart with quadratic uncertainty
  • Example 3: Quadratic adjustable robust counterpart
  • Proposition 2
  • ...and 14 more