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Sensitivity Analysis for Mixed Binary Quadratic Programming

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Integer Programming and Combinatorial Optimization (IPCO 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14679))

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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.

We would like to acknowledge the support from ONR grant #N000142212632.

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Notes

  1. 1.

    For convenience, we have written the dual variables with ‘negative sign’.

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Correspondence to Santanu S. Dey .

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Cifuentes, D., Dey, S.S., Xu, J. (2024). Sensitivity Analysis for Mixed Binary Quadratic Programming. In: Vygen, J., Byrka, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2024. Lecture Notes in Computer Science, vol 14679. Springer, Cham. https://doi.org/10.1007/978-3-031-59835-7_33

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  • DOI: https://doi.org/10.1007/978-3-031-59835-7_33

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