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Recursive McCormick Linearization of Multilinear Programs.

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Τίτλος: Recursive McCormick Linearization of Multilinear Programs.
Συγγραφείς: Cardonha, Carlos1 (AUTHOR) carlos.cardonha@uconn.edu, Raghunathan, Arvind2 (AUTHOR) raghunathan@merl.com, Bergman, David1 (AUTHOR) david.bergman@uconn.edu, Nohra, Carlos3 (AUTHOR) carlos.nohra@gmail.com
Πηγή: INFORMS Journal on Computing. Nov/Dec2025, Vol. 37 Issue 6, p1650-1669. 20p.
Θεματικοί όροι: *Linear programming, *Nonlinear programming, Mixed integer linear programming, Dummy variables, Quadratic programming
Περίληψη: Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLPs). These relaxations can be obtained by using recursive McCormick linearizations (RMLs), by which an MLP is linearized by iteratively substituting bilinear products with artificial variables and constraints. This article introduces a systematic approach to identifying RMLs. We focus on identifying RMLs with a small number of artificial variables and strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation to identify minimum-size RMLs; this problem is NP-hard in general, but we show that it is fixed-parameter tractable if each monomial is composed of at most three variables. Moreover, we explore the structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best-possible LP relaxation bound. We test our algorithms by conducting numerical experiments on a large collection of MLPs. Numerical results indicate that the RMLs obtained with our algorithms can be significantly smaller than those derived from heuristic or greedy approaches, leading, in many cases, to tighter LP relaxation bounds. Moreover, our linearization strategies can be used to reformulate MLPs as quadratically constrained programs (QCPs), which can then be efficiently solved using state-of-the-art solvers for QCPs. This QCP-based solution approach is highly beneficial for hard MLP instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. [ABSTRACT FROM AUTHOR]
Copyright of INFORMS Journal on Computing is the property of INFORMS: Institute for Operations Research & the Management Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Recursive McCormick Linearization of Multilinear Programs.
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  Data: <searchLink fieldCode="JN" term="%22INFORMS+Journal+on+Computing%22">INFORMS Journal on Computing</searchLink>. Nov/Dec2025, Vol. 37 Issue 6, p1650-1669. 20p.
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  Data: *<searchLink fieldCode="DE" term="%22Linear+programming%22">Linear programming</searchLink><br />*<searchLink fieldCode="DE" term="%22Nonlinear+programming%22">Nonlinear programming</searchLink><br /><searchLink fieldCode="DE" term="%22Mixed+integer+linear+programming%22">Mixed integer linear programming</searchLink><br /><searchLink fieldCode="DE" term="%22Dummy+variables%22">Dummy variables</searchLink><br /><searchLink fieldCode="DE" term="%22Quadratic+programming%22">Quadratic programming</searchLink>
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  Data: Linear programming (LP) relaxations are widely employed in exact solution methods for multilinear programs (MLPs). These relaxations can be obtained by using recursive McCormick linearizations (RMLs), by which an MLP is linearized by iteratively substituting bilinear products with artificial variables and constraints. This article introduces a systematic approach to identifying RMLs. We focus on identifying RMLs with a small number of artificial variables and strong LP bounds. We present a novel mechanism for representing all the possible RMLs, which we use to design an exact mixed-integer programming (MIP) formulation to identify minimum-size RMLs; this problem is NP-hard in general, but we show that it is fixed-parameter tractable if each monomial is composed of at most three variables. Moreover, we explore the structural properties of our formulation to derive an exact MIP model that identifies RMLs of a given size with the best-possible LP relaxation bound. We test our algorithms by conducting numerical experiments on a large collection of MLPs. Numerical results indicate that the RMLs obtained with our algorithms can be significantly smaller than those derived from heuristic or greedy approaches, leading, in many cases, to tighter LP relaxation bounds. Moreover, our linearization strategies can be used to reformulate MLPs as quadratically constrained programs (QCPs), which can then be efficiently solved using state-of-the-art solvers for QCPs. This QCP-based solution approach is highly beneficial for hard MLP instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of INFORMS Journal on Computing is the property of INFORMS: Institute for Operations Research & the Management Sciences and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1287/ijoc.2023.0390
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      – Code: eng
        Text: English
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        PageCount: 20
        StartPage: 1650
    Subjects:
      – SubjectFull: Linear programming
        Type: general
      – SubjectFull: Nonlinear programming
        Type: general
      – SubjectFull: Mixed integer linear programming
        Type: general
      – SubjectFull: Dummy variables
        Type: general
      – SubjectFull: Quadratic programming
        Type: general
    Titles:
      – TitleFull: Recursive McCormick Linearization of Multilinear Programs.
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            – D: 01
              M: 11
              Text: Nov/Dec2025
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              Y: 2025
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