Semidefinite programming relaxations through quadratic reformulation for box-constrained polynomial optimization problems

Sourour Elloumi 1 Amélie Lambert 2 Arnaud Lazare 1
1 UMA - Unité de Mathématiques Appliquées
2 CEDRIC - OC - CEDRIC. Optimisation Combinatoire
CEDRIC - Centre d'études et de recherche en informatique et communications
Abstract : In this paper we introduce new semidefinite programming relaxations to box-constrained polynomial optimization programs (P). For this, we first reformu-late (P) into a quadratic program. More precisely, we recursively reduce the degree of (P) to two by substituting the product of two variables by a new one. We obtain a quadratically constrained quadratic program. We build a first immediate SDP relaxation in the dimension of the total number of variables. We then strengthen the SDP relaxation by use of valid constraints that follow from the quadratization. We finally show the tightness of our relaxations through several experiments on box polynomial instances.
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Sourour Elloumi, Amélie Lambert, Arnaud Lazare. Semidefinite programming relaxations through quadratic reformulation for box-constrained polynomial optimization problems. 2019 6th International Conference on Control, Decision and Information Technologies (CoDIT), Apr 2019, Paris, France. pp.1498-1503, ⟨10.1109/CoDIT.2019.8820690⟩. ⟨hal-02455410⟩

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