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Differential stability of solutions to parametric optimization problems. (English) Zbl 0749.90077
Summary: A method for the differential stability of solutions to a class of parametric optimization problems is proposed. Any solution of the parametric optimization problem is given as a fixed point of the metric projection onto the set of admissible coefficients. A new result on the differential stability of the metric projection in Sobolev space $$H^ 2(\Omega)$$ onto a set of admissible parameters is obtained. The stability results with respect to perturbations of observations for the solutions to a coefficient estimation problem for a second-order elliptic equation are derived.

##### MSC:
 90C31 Sensitivity, stability, parametric optimization
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##### References:
 [1] Sobolev Spaces, Academic, New York, 1975. · Zbl 0314.46030 [2] Bamberger, Appl. Math. Optim. 5 pp 1– (1979) [3] ’On th uniqueness of local minima for general abstract non-linear least square problem’, Rapport de Recherche No 645, INRIA, Rocquencourt, France. [4] and , ’Stability of perturbed optimization problems with applications to parameter estimation’, Preprint. [5] Classical Potential Theory and its Probabilistic Counter Part, Springer, New York. 1985. [6] Haraux, J. Math. Soc. Japan 29 pp 615– (1977) [7] Hedberg, Acta Math. 147 pp 237– (1981) [8] Hoffmann, SIAM J. Math. Anal. 5 pp 1198– (1986) [9] Mignot, J. funct. Anal. 22 pp 25– (1976) [10] Rao, Polyhedricity of convex sets in Sobolev space H20 ({$$\omega$$}) [11] and , ’Shape sensitivity analysis of state constrained optimal control problems for distributed parameter systems’, Lecture Notes in Control and Information Sciences, Vol. 114, Springer, Berlin, 1989, pp. 236-245. [12] Sokolowski, Appl. Math. Optim. 13 pp 97– (1985) [13] Sokolowski, SIAM J. Control and Optimization 25 pp 1542– (1987) [14] Sokolowski, Control and Cybernetics. 19 pp 271– (1990) [15] ’Projections on convex sets in Hilbert space and spectral theory’, Contributions to Nonlinear Functional Analysis, Publ. No 27, Mathematical Research Center, University of Wisconsin, Madison, Academic Press, New York, 1971, 237-424. [16] Weakly Differentiable Functions, Springer, New York, 1989. · Zbl 0692.46022
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