We study structured optimization problems with polynomial objective function and polynomial equality constraints. The structure comes from a multi-grading on the polynomial ring in several variables. For fixed multi-degrees we determine the generic number of complex critical points. This serves as a measure for the algebraic complexity of the optimization problem. We also discuss computation and certification methods coming from numerical nonlinear algebra.
Outer-Product-Free Sets for Polynomial Optimization and Oracle-Based Cuts
Rank conditions for exactness of semidefinite relaxations in polynomial optimization
Chordal and factor-width decompositions for scalable semidefinite and polynomial optimization
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