zbMATH — the first resource for mathematics

Well-posed constrained optimization problems in metric spaces. (English) Zbl 0798.49031
Summary: Relations between different notions of well-posedness of constrained optimization problems are studied. A characterization of the class of metric spaces in which Hadamard, strong, and Levitin-Polyak well- posedness of continuous minimization problems coincide is given. It is shown that the equivalence between the original Tikhonov well-posedness and the ones above provides a new characterization of the so-called Atsuji spaces. Generalized notions of well-posedness, not requiring uniqueness of the solution, are introduced and investigated in the above spirit.

49K35 Optimality conditions for minimax problems
Full Text: DOI
[1] Tikhonov, A. N.,On the Stability of the Functional Optimization Problem, USSR Computational Mathematics and Mathematical Physics, Vol. 6, pp. 631-634, 1966.
[2] Berdishev, V. I.,Stability of the Minimum Problem under a Perturbation of the Constrained Set, USSR Mathematical Sbornik, Vol. 103, pp. 467-479, 1977.
[3] Dontchev, A., andZolezzi, T.,Well-Posed Optimization Problems, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany (to appear). · Zbl 0797.49001
[4] Furi, M., andVignoli, A.,About Well-Posed Minimization Problems for Functionals in Metric Spaces, Journal of Optimization Theory and Applications, Vol. 5, pp. 225-229, 1970. · Zbl 0188.48802 · doi:10.1007/BF00927717
[5] Lucchetti, R.,Some Aspects of the Connections between Hadamard and Tikhonov Well-Posedness of Convex Programs, Bollettino della Unione Matematica Italiana, Series C, Vol. 6, pp. 337-345, 1982. · Zbl 0507.49006
[6] Lucchetti, R., andPatrone, F.,Hadamard and Tikhonov Well-Posedness of Certain Class of Convex Functions, Journal of Mathematical Analysis and Applications, Vol. 88, pp. 204-215, 1982. · Zbl 0487.49013 · doi:10.1016/0022-247X(82)90187-1
[7] Revalski, J. P.,Well-Posedness of Optimization Problems: A Survey, Functional Analysis and Approximation, Edited by P. L. Papini, Pitagora Editrice, Bologna, Italy, pp. 238-255, 1988.
[8] Levitin, E. S., andPolyak, B. T.,Convergence of Minimizing Sequences in Conditional Extremum Problems, Soviet Mathematical Doklady, Vol. 7, pp. 764-767, 1966. · Zbl 0161.07002
[9] Revalski, J. P.,Generic Properties Concerning Well-Posed Optimization Problems, Comptes Rendus de l’Academie Bulgare des Sciences, Vol. 38, pp. 1431-1434, 1985. · Zbl 0576.49018
[10] Revalski, J. P.,Generic Well-Posedness in Some Classes of Optimization Problems, Acta Universitatis Carolinae, Mathematica et Physica, Vol. 28, pp. 117-125, 1987. · Zbl 0644.49023
[11] Beer, G., andLucchetti, R.,The Epidistance Topology: Continuity and Stability Results with Applications to Constrained Convex Optimization Problems, Mathematics of Operations Research (to appear).
[12] Bednarczuck, E., andPenot, J.-P.,On the Position of the Notions of Well-Posed Minimization Problems, Bollettino della Unione Matematica Italiana (to appear).
[13] Bednarczuck, E., andPenot, J.-P.,Metrically Well-Set Minimization Problems, Applied Mathematics and Optimization, Vol. 26, pp. 273-285, 1992. · Zbl 0762.90073 · doi:10.1007/BF01371085
[14] Atsuji, M.,Uniform Continuity of Continuous Functions on Metric Spaces, Pacific Journal of Mathematics, Vol. 8, pp. 11-16, 1958. · Zbl 0082.16207
[15] Beer, G.,On Uniform Continuity of Continuous Functions and Topological Convergence of Sets, Canadian Mathematical Bulletin, Vol. 26, pp. 52-59, 1985. · Zbl 0553.54004 · doi:10.4153/CMB-1985-004-9
[16] Beer, G., andLucchetti, R., Solvability for Constrained Problems, Preprint, Università di Milano, Quaderno n.3, 1991.
[17] ?oban, M. M., andKenderov, P. S.,Dense Gateaux Differentiability of the Sup-Norm in C(T) and the Topological Properties of T, Comptes Rendus de l’Academie Bulgare des Sciences, Vol. 38, pp. 1603-1604, 1985.
[18] ?oban, M. M., andKenderov, P. S.,Generic Gateaux Differentiability of Convex Functionals in C(T) and the Topological Properties of T, Mathematics and Education in Mathematics, Proceedings of the 15th Conference of the Union of Bulgarian Mathematicians, pp. 141-149, 1986.
[19] ?oban, M. M., Kenderov, P. S., andRevalski, J. P.,Generic Well-Posedness of Optimization Problems in Topological Spaces, Mathematika, Vol. 36, pp. 301-324, 1989. · Zbl 0679.49010 · doi:10.1112/S0025579300013152
[20] Kuratowski, K.,Topology, Vol. 1, Academic Press, New York, New York, 1966.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.