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Pré-Publication, Document De Travail Année : 2024

Upper bound hierarchies for noncommutative polynomial optimization

Résumé

This work focuses on minimizing the eigenvalue of a noncommutative polynomial subject to a finite number of noncommutative polynomial inequality constraints. Based on the Helton-McCullough Positivstellensatz, the noncommutative analog of Lasserre's moment-sum of squares hierarchy provides a sequence of lower bounds converging to the minimal eigenvalue, under mild assumptions on the constraint set. Each lower bound can be obtained by solving a semidefinite program. We derive complementary converging hierarchies of upper bounds. They are noncommutative analogues of the upper bound hierarchies due to Lasserre for minimizing polynomials over compact sets. Each upper bound can be obtained by solving a generalized eigenvalue problem.

Dates et versions

hal-04440949 , version 1 (06-02-2024)

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Igor Klep, Victor Magron, Gaël Massé, Jurij Volčič. Upper bound hierarchies for noncommutative polynomial optimization. 2024. ⟨hal-04440949⟩
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