Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem where f is a polynomial and as well as are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations of the optimal value function . The polynomial is obtained from an optimal (or nearly optimal) solutio...

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Bibliographic Details
Published in:Journal of global optimization 2011-09, Vol.51 (1), p.1-10
Main Author: Lasserre, J. B.
Format: Article
Language:English
Subjects:
Algebra
Computer Science
Mathematical functions
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Real Functions
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Summary:We consider the robust (or min-max) optimization problem where f is a polynomial and as well as are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations of the optimal value function . The polynomial is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the i th in the “joint + marginal” hierarchy of semidefinite relaxations associated with the parametric optimization problem , recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010 ). Then for fixed i , we consider the polynomial optimization problem and prove that converges to J * as i → ∞. Finally, for fixed ℓ  ≤ i , each (and hence ) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796–817, 2001 ; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009 ).
ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-010-9628-3