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

TL;DR

This paper characterizes the convex hull of certain mixed-integer second-order conic sets with nonlinear right-hand sides, enabling stronger relaxations for related optimization problems and demonstrating practical computational benefits.

Contribution

It provides an exact convex hull description for mixed-integer SOC sets with nonlinear RHS, facilitating improved reformulations and cutting-plane methods for MIQCPs.

Findings

Convex hull can be exactly described by the concave envelope of the RHS.

Reformulations lead to stronger relaxations in MIQCPs.

Computational experiments show improved solution techniques for distributionally robust problems.

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.