Polyhedral approximation in mixed-integer convex optimization

TL;DR

This paper introduces a novel polyhedral approximation method in mixed-integer convex optimization that improves solver performance by constructing higher-dimensional approximations and leveraging disciplined convex programming.

Contribution

The authors develop a new algorithm that strengthens polyhedral approximations using higher-dimensional spaces and demonstrate its effectiveness on benchmark problems.

Findings

Solved previously unsolved instances in MINLPLIB2

Achieved superior performance over existing solvers on many benchmarks

Enabled automated model translation using disciplined convex programming

Abstract

Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers in recent years. In this paper, we intend to provide a broadly accessible introduction to our recent work in developing algorithms and software for this problem class. Our approach is based on constructing polyhedral outer approximations of the convex constraints, resulting in a global solution by solving a finite number of mixed-integer linear and continuous convex subproblems. The key advance we present is to strengthen the polyhedral approximations by constructing them in a higher-dimensional space. In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations of the convex constraints. Although our framework requires a manual translation of existing models into DCP form, after performing this transformation on the MINLPLIB2 benchmark library we were able to solve a number of unsolved instances and on many other instances achieve superior performance compared with state-of-the-art solvers like Bonmin, SCIP, and Artelys Knitro.