Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations
A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation...
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| Vydáno v: | Journal of global optimization Ročník 81; číslo 2; s. 293 - 321 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York
Springer US
01.10.2021
Springer Springer Nature B.V |
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| ISSN: | 0925-5001, 1573-2916 |
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| Abstract | A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude. |
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| AbstractList | A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude. |
| Audience | Academic |
| Author | Gondzio, Jacek Yıldırım, E. Alper |
| Author_xml | – sequence: 1 givenname: Jacek surname: Gondzio fullname: Gondzio, Jacek organization: School of Mathematics, The University of Edinburgh – sequence: 2 givenname: E. Alper orcidid: 0000-0003-4141-3189 surname: Yıldırım fullname: Yıldırım, E. Alper email: E.A.Yildirim@ed.ac.uk organization: School of Mathematics, The University of Edinburgh |
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| CitedBy_id | crossref_primary_10_1007_s10107_025_02210_7 crossref_primary_10_1016_j_ejco_2021_100012 crossref_primary_10_1007_s00500_023_08262_1 crossref_primary_10_1016_j_ejor_2021_10_056 crossref_primary_10_1007_s10288_021_00496_9 crossref_primary_10_1007_s10589_021_00282_7 crossref_primary_10_1007_s10796_024_10492_z crossref_primary_10_1007_s12532_024_00270_y crossref_primary_10_1016_j_cam_2023_115651 crossref_primary_10_1016_j_orp_2024_100324 crossref_primary_10_1007_s10898_024_01423_y crossref_primary_10_1109_TEVC_2024_3374519 crossref_primary_10_1080_02331934_2023_2281640 crossref_primary_10_1287_ijoc_2024_0577 crossref_primary_10_1007_s12532_025_00279_x crossref_primary_10_1016_j_ejco_2022_100037 |
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| SubjectTerms | Computer Science Integer programming Investment analysis Linear programming Mathematics Mathematics and Statistics Mixed integer Operations Research/Decision Theory Optimization Quadratic forms Real Functions Tightness |
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| Title | Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations |
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