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On the Convexification of a Class of Mixed-Integer Conic Sets

Guxin Du, Rui Chen, Linchuan Wei

TL;DR

This paper addresses the convexification of mixed-integer second-order conic sets with a nonlinear right-hand side, a structure common in MIQCP and distributionally robust models. It proves that the convex hull can be exactly described by replacing the nonlinear RHS with the concave envelope of the function over the convex hull of the domain, reducing convexification to characterizing this concave envelope. The authors extend the result to general norms, discuss rank-deficient cases via a linear change of variables, and propose practical relaxations using the squared function $f^2$ to obtain tractable approximations. Computational experiments on distributionally robust chance-constrained knapsack variants demonstrate tighter relaxations and substantial speedups over naive formulations, highlighting the method's practical impact for strong MIQCP reformulations and cutting-plane algorithms.

Abstract

We investigate mixed-integer second-order conic (SOC) sets with a nonlinear right-hand side in the SOC constraint, a structure frequently arising in mixed-integer quadratically constrained programming (MIQCP). Under mild assumptions, we show that the convex hull can be exactly described by replacing the right-hand side with its concave envelope. This characterization enables strong relaxations for MIQCPs via reformulations and cutting planes. Computational experiments on distributionally robust chance-constrained knapsack variants demonstrate the efficacy of our reformulation techniques.

On the Convexification of a Class of Mixed-Integer Conic Sets

TL;DR

This paper addresses the convexification of mixed-integer second-order conic sets with a nonlinear right-hand side, a structure common in MIQCP and distributionally robust models. It proves that the convex hull can be exactly described by replacing the nonlinear RHS with the concave envelope of the function over the convex hull of the domain, reducing convexification to characterizing this concave envelope. The authors extend the result to general norms, discuss rank-deficient cases via a linear change of variables, and propose practical relaxations using the squared function to obtain tractable approximations. Computational experiments on distributionally robust chance-constrained knapsack variants demonstrate tighter relaxations and substantial speedups over naive formulations, highlighting the method's practical impact for strong MIQCP reformulations and cutting-plane algorithms.

Abstract

We investigate mixed-integer second-order conic (SOC) sets with a nonlinear right-hand side in the SOC constraint, a structure frequently arising in mixed-integer quadratically constrained programming (MIQCP). Under mild assumptions, we show that the convex hull can be exactly described by replacing the right-hand side with its concave envelope. This characterization enables strong relaxations for MIQCPs via reformulations and cutting planes. Computational experiments on distributionally robust chance-constrained knapsack variants demonstrate the efficacy of our reformulation techniques.

Paper Structure

This paper contains 2 sections, 3 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Ideal Formulation for $\mathcal{Z}$

Key Result

theorem thmcountertheorem

Suppose Assumptions assump:f and assump:col_space hold. Then the convex hull of $\mathcal{Z}$ can be characterized as with $\mathrm{conv}(\mathcal{Z})$ being a closed set.

Figures (1)

  • Figure 1: The convex hull of $\mathcal{Z}$ (a) when Assumption \ref{['assump:col_space']} does not hold, or (b) when $\boldsymbol{y}$ is further constrained.

Theorems & Definitions (4)

  • theorem thmcountertheorem
  • lemma thmcounterlemma
  • proof : of Theorem \ref{['thm:conv_characterization']}
  • corollary thmcountercorollary