Polyhedral approximation in mixed-integer convex optimization

Journal Article · Thu Sep 14 00:00:00 EDT 2017 · Mathematical Programming

DOI:https://doi.org/10.1007/s10107-017-1191-y· OSTI ID:1663183

Lubin, Miles  ^[1]; Yamangil, Emre ^[2]; Bent, Russell Whitford  ^[2]; Vielma, Juan Pablo ^[1]

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

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.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1663183
• Report Number(s): LA-UR-16-24325
• Journal Information: Mathematical Programming, Vol. 172, Issue 1-2; ISSN 0025-5610
• Publisher: Springer
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Convex MINLP
Outer approximation
Disciplined convex programming