Convergent relaxations of polynomial optimization problems with non-commuting variables

Details Convergent relaxations of polynomial optimization problems with non-commuting variables Convergent relaxations of polynomial optimization problems with non-commuting variables

2009-03-25

arxiv.org/abs/0903.4368v2

SIAM J. Optim. Volume 20, Issue 5, pp. 2157-2180 (2010)

Mathematics

Optimization and Control

35 pages. v2: Improved notation and revised proof of Theorem 1

Scientific paper

10.1137/090760155

We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial inequalities as semidefinite positivity constraints. Such problems arise naturally in quantum theory and quantum information science. To solve them, we introduce a hierarchy of semidefinite programming relaxations which generates a monotone sequence of lower bounds that converges to the optimal solution. We also introduce a criterion to detect whether the global optimum is reached at a given relaxation step and show how to extract a global optimizer from the solution of the corresponding semidefinite programming problem.

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