Complete Upper Bound Hierarchies for Spectral Minimum in Noncommutative Polynomial Optimization
Résumé
This work focuses on finding the spectral minimum (ground state energy) of a noncommutative polynomial subject to a finite number of noncommutative polynomial constraints.
Based on the Helton-McCullough Positivstellensatz, the Navascués-Pironio-Acín (NPA) hierarchy is the noncommutative analog of Lasserre's moment-sum of squares hierarchy and provides a sequence of lower bounds converging to the spectral minimum, under mild assumptions on the constraint set. Each lower bound can be obtained by solving a semidefinite program.
This paper derives complementary complete hierarchies of upper bounds for the spectral minimum. They are noncommutative analogues of the upper bound hierarchies due to Lasserre for minimizing commutative polynomials over compact sets. Each upper bound is obtained by solving a generalized eigenvalue problem.
The derived hierarchies apply to optimization problems in bounded and unbounded operator algebras, as demonstrated on a variety of examples.
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