In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value 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 (Hölder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.

We develop an algorithm for the solution of indefinite least-squares problems. Such problems arise in robust estimation, filtering, and control, and numerically stable solutions have been lacking. The algorithm developed herein involves the QR factorization of the coefficient matrix and is provably numerically stable. keywords Indefinite least-squares problems, error analysis, backward stability. 1 Introduction Many optimization criteria have been used for parameter estimation, starting with the standard least-squares formulation of Gauss (ca. 1795) and moving to more recent works on total leastsquares (TLS) and robust (or H 1 ) estimation (see, e.g., [3, 4, 6, 7, 8, 9]). The latter formulations have been motivated by an increasing interest in estimators that are less sensitive to data uncertainties and measurement errors. They can both be shown to require the minimization of indefinite quadratic forms, where the standard inner product of two vectors, say a T b, is replaced by an...

An analysis of a class of data tting problems, where the data uncertainties are subject to known bounds, is given in a very general setting. It is shown how such problems can be posed in a computationally convenient form, and the connection with other more conventional data fitting problems is examined. The problems have attracted interest so far in the special case when the underlying norm is the least squares norm. Here the special structure can be exploited to computational advantage, and we include some observations which contribute to algorithmic development for this particular case. We also consider some variants of the main problems and show how these too can be posed in a form which facilitates their numerical solution.

The problem of tting a linear model to data, under uncertainty which can be regarded as being ellipsoidal, is considered in a very general setting. For a range of such problems, robust counterparts are established, and methods of solution are considered.

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