Nonconvex Robust Optimization for Problems with Constraints
INFORMS Journal on Computing
Min-max and robust polynomial optimization
Journal of Global Optimization
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We consider the optimization problems maxz∈Ω minx∈K p(z, x) and minx ∈ K maxz ∈ Ω p(z, x) where the criterion p is a polynomial, linear in the variables z, the set Ω can be described by LMIs, and K is a basic closed semi-algebraic set. The first problem is a robust analogue of the generic SDP problem maxz ∈ Ω p(z), whereas the second problem is a robust analogue of the generic problem minx ∈ K p(x) of minimizing a polynomial over a semi-algebraic set. We show that the optimal values of both robust optimization problems can be approximated as closely as desired, by solving a hierarchy of SDP relaxations. We also relate and compare the SDP relaxations associated with the max-min and the min-max robust optimization problems.