Compact Disjunctive Approximations to Nonconvex Quadratically Constrained Programs

TL;DR

This paper introduces Compact Disjunctive Approximation (CDA), a novel method to approximate nonconvex MIQCPs with convex relaxations that are efficiently solvable, significantly improving solution quality over existing approaches.

Contribution

The paper presents a new compact lifted formulation for approximating the square function within MIQCPs, enabling high-precision convex relaxations with fewer variables and constraints.

Findings

CDA produces relaxations with O(n log(1/ε)) variables and constraints.

The method effectively closes gaps left by state-of-the-art solvers on difficult instances.

Numerical experiments demonstrate strong potential of the approach.

Abstract

Decades of advances in mixed-integer linear programming (MILP) and recent development in mixed-integer second-order-cone programming (MISOCP) have translated very mildly to progresses in global solving nonconvex mixed-integer quadratically constrained programs (MIQCP). In this paper we propose a new approach, namely Compact Disjunctive Approximation (CDA), to approximate nonconvex MIQCP to arbitrary precision by \textit{convex} MIQCPs, which can be solved by MISOCP solvers. For nonconvex MIQCP with $n$ variables and $m$ general quadratic constraints, our method yields relaxations with at most $O(n\log(1/\varepsilon))$ number of continuous/binary variables and linear constraints, together with $m$ \textit{convex} quadratic constraints, where $\varepsilon$ is the approximation accuracy. The main novelty of our method lies in a very compact lifted mixed-integer formulation for approximating the (scalar) square function. This is derived by first embedding the square function into the boundary of a three-dimensional second-order cone, and then exploiting rotational symmetry in a similar way as in the construction of BenTal-Nemirovski approximation. We further show that this lifted formulation characterize the union of finite number of simple convex sets, which naturally relax the square function in a piecewise manner with properly placed knots. We implement (with JuMP) a simple adaptive refinement algorithm. Numerical experiments on synthetic instances used in the literature show that our prototypical implementation (with hundreds of lines of Julia code) can already close a significant portion of gap left by various state-of-the-art global solvers on more difficult instances, indicating strong promises of our proposed approach.