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• W2016985943 abstract "a b s t r a c t Given a function f ∈ C 1,1 In the literature, a parametrized optimization problem involves, usually, an objective function and constraints depending on parameters. The stability results are related to the study of various continuity properties of the solution map (with respect to the parameter), while the sensitivity analysis refers to the quantitative investigation, like, for instance, the study of Lipschitz properties of the solution map. Generally speaking, the concept of well-posedness of an optimization problem focuses on existence, uniqueness and continuity properties of the solutions (both optimal solutions and optimal value) with respect to data perturbations, and therefore deals with the stability properties (see, for instance, (1,2)). Hence well- posedness is not directly related to the different issues of sensitivity analysis, although there are obvious connections between the two research areas. On the other hand, the condition number represents a measure of the sensitivity of the solutions with respect to small changes in the problem's data, and it is usually conceived as an upper bound of the ratio of the size of the solution error to the size of the data error. At the same time, this number says how we can perturb the initial problem in order to preserve its good properties. In the literature, the ''conditioning'' is essentially related to numerical problems. A prototype result involving condition numbers is the well-known distance theorem of Eckart-Young (see (3)): via the norm of the inverse matrix it provides an upper bound on how much a given matrix may be perturbed in order to preserve its nonsingularity. This remarkable result has been extended to several problems like computation of eigenvalues and eigenvectors, and of zeros of polynomials, and to various optimization problems. It is worthwhile mentioning, in the framework of linear programming, the exhaustive paper by Renegar (4)." @default.
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• W2016985943 date "2012-01-01" @default.
• W2016985943 modified "2023-10-01" @default.
• W2016985943 title "Conditioning for optimization problems under general perturbations" @default.
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• W2016985943 doi "https://doi.org/10.1016/j.na.2011.07.061" @default.
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