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Article Dans Une Revue Optimization Methods and Software Année : 2023

Using general triangle inequalities within Quadratic Convex Reformulation method

Résumé

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.
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Dates et versions

hal-03016403 , version 1 (20-11-2020)

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Citer

Amélie Lambert. Using general triangle inequalities within Quadratic Convex Reformulation method. Optimization Methods and Software, 2023, 38 (3), pp.626-653. ⟨10.1080/10556788.2022.2157002⟩. ⟨hal-03016403⟩
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