Piecewise polyhedral formulations for a multilinear term

Sundar, Kaarthik; Nagarajan, Harsha; Linderoth, Jeff; Wang, Site; Bent, Russell

doi:10.1016/j.orl.2020.12.002

Piecewise polyhedral formulations for a multilinear term

Journal Article · Mon Dec 07 23:00:00 EST 2020 · Operations Research Letters

DOI:https://doi.org/10.1016/j.orl.2020.12.002· OSTI ID:1760566

^[1]; ^[1]; Linderoth, Jeff ^[2]; Wang, Site ^[3]; ^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Univ. of Wisconsin, Madison, WI (United States)
Clemson Univ., SC (United States)

Herein, we present a mixed-integer linear programming (MILP) formulation of a piecewise, polyhedral relaxation (PPR) of a multilinear term using its convex-hull representation. Based on the PPR’s solution, we also present a MILP formulation whose solutions are feasible for nonconvex, multilinear equations. We then present computational results showing the effectiveness of proposed formulations on standard benchmark nonlinear programs (NLPs) with multilinear terms and compare with a traditional formulation that is built using recursive bilinear groupings of multilinear terms.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001; AC52-06NA25396

OSTI ID:: 1760566

Alternate ID(s):: OSTI ID: 1776450

Report Number(s):: LA-UR--18-22508

Journal Information:: Operations Research Letters, Journal Name: Operations Research Letters Journal Issue: 1 Vol. 49; ISSN 0167-6377

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (23)

Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems McCormick, Garth P. Mathematical Programming, Vol. 10, Issue 1 https://doi.org/10.1007/bf01580665	journal	December 1976
A polyhedral branch-and-cut approach to global optimization Tawarmalani, Mohit; Sahinidis, Nikolaos V. Mathematical Programming, Vol. 103, Issue 2 https://doi.org/10.1007/s10107-005-0581-8	journal	May 2005
Polyhedral methods for piecewise-linear functions I: the lambda method Lee, Jon; Wilson, Dan Discrete Applied Mathematics, Vol. 108, Issue 3 https://doi.org/10.1016/s0166-218x(00)00216-x	journal	March 2001
Piecewise linear relaxation of bilinear programs using bivariate partitioning Hasan, M. M. Faruque; Karimi, I. A. AIChE Journal, Vol. 56, Issue 7 https://doi.org/10.1002/aic.12109	journal	November 2009
BARON: A general purpose global optimization software package Sahinidis, Nikolaos V. Journal of Global Optimization, Vol. 8, Issue 2 https://doi.org/10.1007/BF00138693	journal	March 1996
Global optimization using special ordered sets Beale, E. M. L.; Forrest, J. J. H. Mathematical Programming, Vol. 10, Issue 1 https://doi.org/10.1007/BF01580653	journal	December 1976
Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems McCormick, Garth P. Mathematical Programming, Vol. 10, Issue 1 https://doi.org/10.1007/BF01580665	journal	December 1976
Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations Misener, Ruth; Floudas, Christodoulos A. Mathematical Programming, Vol. 136, Issue 1 https://doi.org/10.1007/s10107-012-0555-6	journal	May 2012
Some results on the strength of relaxations of multilinear functions Luedtke, James; Namazifar, Mahdi; Linderoth, Jeff Mathematical Programming, Vol. 136, Issue 2 https://doi.org/10.1007/s10107-012-0606-z	journal	October 2012
Algorithmic and modeling insights via volumetric comparison of polyhedral relaxations Lee, Jon; Skipper, Daphne; Speakman, Emily Mathematical Programming, Vol. 170, Issue 1 https://doi.org/10.1007/s10107-018-1272-6	journal	April 2018
An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs Nagarajan, Harsha; Lu, Mowen; Wang, Site Journal of Global Optimization, Vol. 74, Issue 4 https://doi.org/10.1007/s10898-018-00734-1	journal	January 2019
Global optimization of nonconvex problems with multilinear intermediates Bao, Xiaowei; Khajavirad, Aida; Sahinidis, Nikolaos V. Mathematical Programming Computation, Vol. 7, Issue 1 https://doi.org/10.1007/s12532-014-0073-z	journal	May 2014
Polyhedral methods for piecewise-linear functions I: the lambda method Lee, Jon; Wilson, Dan Discrete Applied Mathematics, Vol. 108, Issue 3 https://doi.org/10.1016/S0166-218X(00)00216-X	journal	March 2001
Approximating separable nonlinear functions via mixed zero-one programs Padberg, Manfred Operations Research Letters, Vol. 27, Issue 1 https://doi.org/10.1016/S0167-6377(00)00028-6	journal	August 2000
APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes Misener, Ruth; Thompson, Jeffrey P.; Floudas, Christodoulos A. Computers & Chemical Engineering, Vol. 35, Issue 5 https://doi.org/10.1016/j.compchemeng.2011.01.026	journal	May 2011
Tightening piecewise McCormick relaxations for bilinear problems Castro, Pedro M. Computers & Chemical Engineering, Vol. 72 https://doi.org/10.1016/j.compchemeng.2014.03.025	journal	January 2015
A piecewise McCormick relaxation-based strategy for scheduling operations in a crude oil terminal de Assis, Leonardo Salsano; Camponogara, Eduardo; Zimberg, Bernardo Computers & Chemical Engineering, Vol. 106 https://doi.org/10.1016/j.compchemeng.2017.06.012	journal	November 2017
Locally ideal formulations for piecewise linear functions with indicator variables Sridhar, Srikrishna; Linderoth, Jeff; Luedtke, James Operations Research Letters, Vol. 41, Issue 6 https://doi.org/10.1016/j.orl.2013.08.010	journal	November 2013
SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework Vigerske, Stefan; Gleixner, Ambros Optimization Methods and Software, Vol. 33, Issue 3 https://doi.org/10.1080/10556788.2017.1335312	journal	June 2017
JuMP: A Modeling Language for Mathematical Optimization Dunning, Iain; Huchette, Joey; Lubin, Miles SIAM Review, Vol. 59, Issue 2 https://doi.org/10.1137/15M1020575	journal	January 2017
A Combinatorial Approach for Small and Strong Formulations of Disjunctive Constraints Huchette, Joey; Vielma, Juan Pablo Mathematics of Operations Research, Vol. 44, Issue 3 https://doi.org/10.1287/moor.2018.0946	journal	August 2019
Jointly Constrained Biconvex Programming Al-Khayyal, Faiz A.; Falk, James E. Mathematics of Operations Research, Vol. 8, Issue 2 https://doi.org/10.1287/moor.8.2.273	journal	May 1983
Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions Vielma, Juan Pablo; Ahmed, Shabbir; Nemhauser, George Operations Research, Vol. 58, Issue 2 https://doi.org/10.1287/opre.1090.0721	journal	April 2010

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