Using general triangle inequalities within quadratic convex reformulation method

We consider the exact solution of Problem (P) which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit description of new general triangle inequalities that are derived from the ranges of the variables of (P). We show that they extend the triangle...

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Published inOptimization methods & software Vol. 38; no. 3; pp. 626 - 653
Main Author Lambert, Amélie
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 04.05.2023
Taylor & Francis Ltd
Subjects
Algorithms
Envelopes
Exact solutions
global optimization
Inequalities
lagrangian duality
Mathematics
Optimization
Optimization and Control
Quadratic convex relaxation
Quadratic equations
quadratic programming
semi-definite programming
sub-gradient algorithm
valid inequalities
Lagrangian duality
Global optimization
Semi-Definite programming
Quadratic Programming
Quadratic Convex Relaxation Valid inequalities Global optimization Semi-Definite programming Lagrangian duality sub-gradient algorithm Quadratic Programming
Quadratic Convex Relaxation
Valid inequalities
sub-gradient algorithm
Online AccessGet full text
ISSN1055-6788
1029-4937
DOI10.1080/10556788.2022.2157002

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Abstract We consider the exact solution of Problem (P) which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit description of new general triangle inequalities that are derived from the ranges of the variables of (P). We show that they extend the triangle inequalities, introduced for the binary case, to variables that belong to a generic interval. We also prove that these inequalities cut feasible solutions of McCormick envelopes, and we relate them to the literature. We then introduce (SDP), a strong semidefinite relaxation of (P), that we call 'Shor's plus RLT plus Triangle', which includes both the McCormick envelopes and the general triangle inequalities. We further show how to compute a convex relaxation whose optimal value reaches the value of (SDP). In order to handle these inequalities in the solution of (SDP), we solve it by a heuristic that also serves as a separation algorithm. We then solve (P) to global optimality with a branch-and-bound based on . Finally, we show that our method outperforms the compared solvers.
AbstractList We consider the exact solution of Problem $\QP$ which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit description of new general triangle inequalities that are derived from the ranges of the variables of $\QP$. We show that they extend the triangle inequalities, introduced for the binary case, to variables that belong to a generic interval. We also prove that these inequalities cut feasible solutions of McCormick envelopes, and we relate them to the literature. We then introduce $(SDP)$, a strong semidefinite relaxation of $\QP$, that we call ``Shor's plus RLT plus Triangle'', which includes both the McCormick envelopes and the general triangle inequalities. We further show how to compute a convex relaxation $(P^*)$ whose optimal value reaches the value of $(SDP)$. In order to handle these inequalities in the solution of $(SDP)$, we solve it by a heuristic that also serves as a separation algorithm. We then solve $\QP$ to global optimality with a \bb~based on $(P^*)$. Finally, we show that our method outperforms the compared solvers.
We consider the exact solution of Problem (P) which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit description of new general triangle inequalities that are derived from the ranges of the variables of (P). We show that they extend the triangle inequalities, introduced for the binary case, to variables that belong to a generic interval. We also prove that these inequalities cut feasible solutions of McCormick envelopes, and we relate them to the literature. We then introduce (SDP), a strong semidefinite relaxation of (P), that we call 'Shor's plus RLT plus Triangle', which includes both the McCormick envelopes and the general triangle inequalities. We further show how to compute a convex relaxation whose optimal value reaches the value of (SDP). In order to handle these inequalities in the solution of (SDP), we solve it by a heuristic that also serves as a separation algorithm. We then solve (P) to global optimality with a branch-and-bound based on . Finally, we show that our method outperforms the compared solvers.
We consider the exact solution of Problem (P) which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit description of new general triangle inequalities that are derived from the ranges of the variables of (P). We show that they extend the triangle inequalities, introduced for the binary case, to variables that belong to a generic interval. We also prove that these inequalities cut feasible solutions of McCormick envelopes, and we relate them to the literature. We then introduce (SDP), a strong semidefinite relaxation of (P), that we call ‘Shor's plus RLT plus Triangle’, which includes both the McCormick envelopes and the general triangle inequalities. We further show how to compute a convex relaxation whose optimal value reaches the value of (SDP). In order to handle these inequalities in the solution of (SDP), we solve it by a heuristic that also serves as a separation algorithm. We then solve (P) to global optimality with a branch-and-bound based on . Finally, we show that our method outperforms the compared solvers.
Author Lambert, Amélie
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  organization: Cnam-CEDRIC
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Keywords Lagrangian duality
Global optimization
Semi-Definite programming
Quadratic Programming
Quadratic Convex Relaxation Valid inequalities Global optimization Semi-Definite programming Lagrangian duality sub-gradient algorithm Quadratic Programming
Quadratic Convex Relaxation
Valid inequalities
sub-gradient algorithm
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Snippet We consider the exact solution of Problem (P) which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit...
We consider the exact solution of Problem $\QP$ which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit...
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SubjectTerms Algorithms
Envelopes
Exact solutions
global optimization
Inequalities
lagrangian duality
Mathematics
Optimization
Optimization and Control
Quadratic convex relaxation
Quadratic equations
quadratic programming
semi-definite programming
sub-gradient algorithm
valid inequalities
Title Using general triangle inequalities within quadratic convex reformulation method
URI https://www.tandfonline.com/doi/abs/10.1080/10556788.2022.2157002
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Volume 38
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