Title: Polyhedral approximation in mixed-integer convex optimization

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. Here, 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.

Authors:

Lubin, Miles  ^[1];

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• Search DOE PAGES for ORCID "0000-0001-6781-9633"

Yamangil, Emre ^[2]; Bent, Russell Whitford  ^[2];

• Search DOE PAGES for author "Bent, Russell Whitford"
• Search DOE PAGES for ORCID "0000-0002-7300-151X"

Vielma, Juan Pablo ^[1]

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+ Show Author Affiliations

1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:

Thu Sep 14 00:00:00 EDT 2017

Research Org.:

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA)

OSTI Identifier:

1663183

Report Number(s):

LA-UR-16-24325
Journal ID: ISSN 0025-5610

Grant/Contract Number:  

89233218CNA000001

Resource Type:

Accepted Manuscript

Journal Name:

Mathematical Programming

Additional Journal Information:

Journal Volume: 172; Journal Issue: 1-2; Journal ID: ISSN 0025-5610

Publisher:

Springer

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; Convex MINLP; Outer approximation; Disciplined convex programming