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Sensitivity analysis for mixed binary quadratic programming

Diego Cifuentes, Santanu S. Dey, Jingye Xu

TL;DR

The paper addresses sensitivity analysis for MBQPs under rhs changes and proves that approximating the resulting change in the optimal objective is NP-hard, even when the MBQP is binary. It leverages Burer's CPP reformulation to derive a COP dual, establishing strong duality under bounded feasibility or PSD quadratic forms, while showing duality gaps can arise otherwise and that COP-dual optimal solutions may be non-attainable. A constructive closed-form dual solution is developed from building blocks KK_i and G_j, enabling near-optimal sensitivity bounds; the authors also introduce practical refinements and demonstrate preliminary computational results across multiple instance classes. The work provides a principled framework for obtaining dual bounds to sensitivity analyses, with potential for more scalable SDP-based implementations and extensions to broader QCQP classes.

Abstract

We consider sensitivity analysis for Mixed Binary Quadratic Programs (MBQPs) with respect to changing right-hand-sides (rhs). We show that even if the optimal solution of a given MBQP is known, it is NP-hard to approximate the change in objective function value with respect to changes in rhs. Next, we study algorithmic approaches to obtaining dual bounds for MBQP with changing rhs. We leverage Burer's completely-positive (CPP) reformulation of MBQPs. Its dual is an instance of co-positive programming (COP), and can be used to obtain sensitivity bounds. We prove that strong duality between the CPP and COP problems holds if the feasible region is bounded or if the objective function is convex, while the duality gap can be strictly positive if neither condition is met. We also show that the COP dual has multiple optimal solutions, and the choice of the dual solution affects the quality of the bounds with rhs changes. We finally provide a method for finding good nearly optimal dual solutions, and we present preliminary computational results on sensitivity analysis for MBQPs.

Sensitivity analysis for mixed binary quadratic programming

TL;DR

The paper addresses sensitivity analysis for MBQPs under rhs changes and proves that approximating the resulting change in the optimal objective is NP-hard, even when the MBQP is binary. It leverages Burer's CPP reformulation to derive a COP dual, establishing strong duality under bounded feasibility or PSD quadratic forms, while showing duality gaps can arise otherwise and that COP-dual optimal solutions may be non-attainable. A constructive closed-form dual solution is developed from building blocks KK_i and G_j, enabling near-optimal sensitivity bounds; the authors also introduce practical refinements and demonstrate preliminary computational results across multiple instance classes. The work provides a principled framework for obtaining dual bounds to sensitivity analyses, with potential for more scalable SDP-based implementations and extensions to broader QCQP classes.

Abstract

We consider sensitivity analysis for Mixed Binary Quadratic Programs (MBQPs) with respect to changing right-hand-sides (rhs). We show that even if the optimal solution of a given MBQP is known, it is NP-hard to approximate the change in objective function value with respect to changes in rhs. Next, we study algorithmic approaches to obtaining dual bounds for MBQP with changing rhs. We leverage Burer's completely-positive (CPP) reformulation of MBQPs. Its dual is an instance of co-positive programming (COP), and can be used to obtain sensitivity bounds. We prove that strong duality between the CPP and COP problems holds if the feasible region is bounded or if the objective function is convex, while the duality gap can be strictly positive if neither condition is met. We also show that the COP dual has multiple optimal solutions, and the choice of the dual solution affects the quality of the bounds with rhs changes. We finally provide a method for finding good nearly optimal dual solutions, and we present preliminary computational results on sensitivity analysis for MBQPs.
Paper Structure (30 sections, 16 theorems, 112 equations, 1 figure, 18 tables)

This paper contains 30 sections, 16 theorems, 112 equations, 1 figure, 18 tables.

Table of Contents

  1. Introduction.
  2. Main results.
  3. Notation.
  4. Complexity.
  5. Strong duality.
  6. Proof sketch for part \ref{['p1']} of Theorem \ref{['thm:allstrong']}:
  7. Proof sketch for part \ref{['p2']} of Theorem \ref{['thm:allstrong']}:
  8. (i) Building block 1.
  9. (ii) Building block 2.
  10. Proof for part \ref{['p3']} of Theorem \ref{['thm:allstrong']}:
  11. Part \ref{['p4']} of Theorem \ref{['thm:allstrong']}:
  12. How good is the closed form solution of Theorem \ref{['thm:allstrong']} for sensitivity analysis?
  13. Preliminary computations.
  14. Modifications to (\ref{['eq:actual0']}).
  15. Linear penalty.
  16. ...and 15 more sections

Key Result

Theorem 1

It is NP-hard to achieve $({\Gamma_1},{\Gamma_2})$-approximation of (eq:MBQPeq) for general MBQPs for any ${\Gamma_2} \geq 1\geq {\Gamma_1} > 0$.

Figures (1)

  • Figure 1: The x-axis corresponds to $\Delta p$, and the y-axis corresponds to the optimal value of the new program or different predicted values of different methods. Figure (a) plots Shor1, Shor2, Our method, Cont for a (COMB) instance with $d=0.3$. Figure (b) plots Our method, Our method($\sigma = 2$), Our method($\sigma = 4$), Our method($\sigma = 6$), Our method($\sigma = 8$) for a (COMB) instance with $d=0.3$.

Theorems & Definitions (40)

  • Definition 1
  • Theorem 1
  • Theorem 2: Burer's reformulation burer
  • Theorem 3: Strong duality
  • Theorem 4: Local stability
  • Remark 1: Alternative proof of Burer's Theorem
  • Remark 2: Local stability not satisfied when $Q\not \succeq0$ and $\mathbb{P}$ is unbounded
  • Proposition 5
  • proof
  • Remark 3
  • ...and 30 more