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Robust global optimization with polynomials. (English) Zbl 1134.90031

Summary: We consider the optimization problems
\[ \max_{z\in\Omega} \min_{x\in\mathbf K} p(z,x)\quad\text{and}\quad \min_{x\in\mathbf K} \max_{z\in\Omega} p(z,x), \]
where the criterion \(p\) is a polynomial, linear in the variables \(z\), the set \(\Omega\) can be described by LMIs, and \(\mathbf K\) is a basic closed semi-algebraic set. The first problem is a robust analogue of the generic SDP problem \(\max_{z\in\Omega}p(z)\), whereas the second problem is a robust analogue of the generic problem \(\min_{x\in\mathbf K}p(x)\) 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.

MSC:

90C22 Semidefinite programming
90C26 Nonconvex programming, global optimization
90C30 Nonlinear programming
90C31 Sensitivity, stability, parametric optimization
90C47 Minimax problems in mathematical programming

Software:

GloptiPoly; SeDuMi
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References:

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