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Moment/Sum-of-Squares Hierarchy for Complex Polynomial Optimization
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We consider the problem of finding the global optimum of a real-valued complex polynomial on a compact set defined by real-valued complex polynomial inequalities. It reduces to solving a sequence of complex semidefinite programming relaxations that grow tighter and tighter thanks to D'Angelo's and Putinar's Positivstellenstatz discovered in 2008. In other words, the Lasserre hierarchy may be transposed to complex numbers. We propose a method for exploiting sparsity and apply the complex hierarchy to problems with several thousand complex variables. These problems consist of computing optimal power flows in the European high-voltage transmission network.
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Tight-and-cheap conic relaxation for the AC optimal power flow problem
Proposes the tight-and-cheap conic relaxation (TCR) for ACOPF, stronger than SOCR, nearly as tight as SDR on many cases, and significantly faster than the chordal SDP relaxation.
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