Extended Formulations in Mixed-integer Convex Programming

TL;DR

This paper introduces a unifying framework for extended formulations in mixed-integer convex programming, improving algorithm efficiency and solution times through conic reformulations and connections to disciplined convex programming.

Contribution

It provides the first finite-time convergence algorithm for mixed-integer conic problems and links DCP to extended formulations, enhancing practical solution capabilities.

Findings

All MICP instances are conic representable.

Conic reformulations encode problem structure effectively.

The approach solves previously open instances.

Abstract

We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representable problems, we provide the first outer approximation algorithm with finite-time convergence guarantees, opening a path for the use of conic solvers for continuous relaxations. We then connect the popular modeling framework of disciplined convex programming (DCP) to the existence of extended formulations independent of conic representability. We present evidence that our approach can yield significant gains in practice, with the solution of a number of open instances from the MINLPLIB2 benchmark library.