## 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
Full Text:

### References:

 [1] Basu, S., Pollack, R., Roy, M-F.: Algorithms in real algebraic geometry, Springer, Berlin, 2003 · Zbl 1031.14028 [2] Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robustness. In Wolkowicz, A., Saigal, R., Vandenberghe, L. (eds.), Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, Boston, 2000 [3] Bertsimas, Math. Prog. Series B, 98, 49 (2003) · Zbl 1082.90067 [4] Bertsimas, Oper. Res., 52, 35 (2004) · Zbl 1165.90565 [5] Goemans, J. ACM, 42, 1115 (1995) · Zbl 0885.68088 [6] Henrion, ACM Trans. Math. Soft., 29, 165 (2003) · Zbl 1070.65549 [7] Henrion, IEEE Contr. Syst. Mag., 24, 72 (2004) [8] Lasserre, SIAM J. Optim., 11, 796 (2001) · Zbl 1010.90061 [9] Lasserre, SIAM J. Optim., 12, 756 (2002) · Zbl 1007.90046 [10] Lasserre, J.B.: A unified criterion for positive definiteness and semidefiniteness. Technical report 05283, LAAS-CNRS, Toulouse, France, 2005 [11] Parrilo, Math. Progr. Ser.B, 96, 293 (2003) · Zbl 1043.14018 [12] Putinar, Indiana Univ. Math. J., 42, 969 (1993) · Zbl 0796.12002 [13] Todd, Acta Numerica, 10, 515 (2000) [14] Vandenberghe, SIAM Rev., 38, 49 (1996) · Zbl 0845.65023
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