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Constrained optimization: A general tolerance approach. (English) Zbl 0714.49006

Summary: To overcome the somewhat artificial difficulties in classical optimization theory concerning the existence and stability of minimizers, a new setting of constrained optimization problems (called problems with tolerance) is proposed using given proximity structures to define the neighbourhoods of sets. The infimum and the so-called minimizing filter are then defined by means of level sets created by these neighbourhoods, which also reflects the engineering approach to constrainted optimization problems. Moreover, an appropriate concept of convergence of filters is developed, and stability of the minimizing filter as well as its approximation by the exterior penalty function technique are proved by using a compactification of the problem.

MSC:

49J27 Existence theories for problems in abstract spaces
49K40 Sensitivity, stability, well-posedness
65K10 Numerical optimization and variational techniques
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
90C48 Programming in abstract spaces
49M30 Other numerical methods in calculus of variations (MSC2010)
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References:

[1] Á. Császár: General Topology. Akademiai Kiadó, Budapest, 1978.
[2] A. V. Efremovich: The geometry of proximity. (in Russian). Mat. Sbornik 31 (73) (1952), 189-200. · Zbl 0053.11505
[3] E. K. Golshtein: Duality Theory in Mathematical Programming and Its Applications. (in Russian). Nauka, Moscow, 1971.
[4] D. A. Molodcov: Stability and regularization of principles of optimality. (in Russian). Zurnal vycisl. mat. i mat. fiziki 20 (1980), 1117-1129.
[5] D. A. Molodcov: Stability of Principles of Optimality. (in Russian). Nauka, Moscow, 1987.
[6] L. Nachbin: Topology and Order. D. van Nostrand Comp., Princeton, 1965. · Zbl 0131.37903
[7] S. A. Naimpally B. D. Warrack: Proximity Spaces. Cambridge Univ. Press, Cambridge, 1970. · Zbl 0206.24601
[8] E. Polak Y. Y. Wardi: A study of minizing sequences. SIAM J. Control Optim. 22 (1984), 599-609. · Zbl 0553.49017 · doi:10.1137/0322036
[9] T. Roubíček: A generalized solution of a nonconvex minimization problem and its stability. Kybernetika 22 (1986), 289-298. · Zbl 0626.49013
[10] T. Roubíček: Generalized solutions of constrained optimization problems. SIAM J. Control Optim. 24 (1986), 951-960. · Zbl 0603.90137 · doi:10.1137/0324056
[11] T. Roubíček: Stable extensions of constrained optimization problems. J. Math. Anal. Appl. 141 (1989), 520-135, · Zbl 0683.49007 · doi:10.1016/0022-247X(89)90210-2
[12] Yu. M. Smirnov: On proximity spaces. (in Russian). Mat. Sbornik 31 (73) (1952), 534-574. · Zbl 0152.20904
[13] J. Warga: Optimal Control of Differential and Functional Equations. Academic press, New York, 1972. · Zbl 0253.49001
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