期刊
SIAM JOURNAL ON OPTIMIZATION
卷 20, 期 5, 页码 2157-2180出版社
SIAM PUBLICATIONS
DOI: 10.1137/090760155
关键词
global optimization; semidefinite programming; positive polynomials; noncommutative variables
资金
- European QAP and PERCENT projects
- Spanish MEC [FIS2007-60182]
- Consolider-Ingenio
- Generalitat de Catalunya and Caixa Manresa
- Swiss NCCR Quantum Photonics
- EU
- Brussels-Capital region
- Institute for Mathematical Sciences
We consider optimization problems with polynomial inequality constraints in noncommuting variables. These noncommuting 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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