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Robust Solutions To Uncertain Semidefinite Programs
, 1998
"... In this paper we consider semidenite programs (SDPs) whose data depends on some unknownbutbounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible values ..."
Abstract

Cited by 83 (3 self)
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In this paper we consider semidenite programs (SDPs) whose data depends on some unknownbutbounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible values of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist, as SDPs. When the perturbation is "full", our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique, and continuous (Hölderstable) with respect to the unperturbed problems' data. The approach can thus be used to regularize illconditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation and integer programming.
The ULagrangian Of The Maximum Eigenvalue Function
, 1998
"... . In this paper we apply the ULagrangian theory to the maximum eigenvalue function 1 and to its precomposition with ane matrixvalued mappings. We rst give geometrical interpretations of the Uobjects that we introduce. We also show that the ULagrangian of 1 has a Hessian which can be explicitly ..."
Abstract

Cited by 14 (3 self)
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. In this paper we apply the ULagrangian theory to the maximum eigenvalue function 1 and to its precomposition with ane matrixvalued mappings. We rst give geometrical interpretations of the Uobjects that we introduce. We also show that the ULagrangian of 1 has a Hessian which can be explicitly computed; the secondorder development of the ULagrangian provides a secondorder development of 1 along a characteristic smooth manifold: the set of symmetric matrices whose maximal eigenvalues have a xed multiplicity. The same results can be obtained when we precompose 1 with an ane matrixvalued mapping A, provided that this mapping satises a regularity condition (transversality condition). We show that the Hessian of the ULagrangian of 1 A coincides with the reduced Hessian encountered in Sequential Quadratic Programming. Finally, we use the ULagrangian to derive secondorder algorithms for minimizing 1 A. Key words. Eigenvalue optimization, convex optimization, generalize...