Geodesic
Constrained optimization: A general tolerance approach
Applications of Mathematics, Tome 35 (1990) no. 2, pp. 99-128
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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 constrained 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.
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 constrained 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.
DOI : 10.21136/AM.1990.104393
Classification : 49A27, 49J27, 49J45, 49K40, 49M30, 54D35, 54E05, 65K10, 90C48, 90C99
Keywords: constrained optimization; level sets; minimizing sequences; penalty functions; compactifications; problems with tolerance 
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Roubíček, Tomáš. Constrained optimization: A general tolerance approach. Applications of Mathematics, Tome 35 (1990) no. 2, pp. 99-128. doi: 10.21136/AM.1990.104393

[1] Á. Császár: General Topology. Akademiai Kiadó, Budapest, 1978. | MR

[2] A. V. Efremovich: The geometry of proximity. (in Russian). Mat. Sbornik 31 (73) (1952), 189-200. | MR

[3] E. K. Golshtein: Duality Theory in Mathematical Programming and Its Applications. (in Russian). Nauka, Moscow, 1971. | MR

[4] D. A. Molodcov: Stability and regularization of principles of optimality. (in Russian). Zurnal vycisl. mat. i mat. fiziki 20 (1980), 1117-1129. | MR

[5] D. A. Molodcov: Stability of Principles of Optimality. (in Russian). Nauka, Moscow, 1987.

[6] L. Nachbin: Topology and Order. D. van Nostrand Соmр., Princeton, 1965. | MR | Zbl

[7] S. A. Naimpally B. D. Warrack: Proximity Spaces. Cambridge Univ. Press, Cambridge, 1970. | MR

[8] E. Polak Y. Y. Wardi: A study of minizing sequences. SIAM J. Control Optim. 22 (1984), 599-609. | DOI | MR

[9] T. Roubíček: A generalized solution of a nonconvex minimization problem and its stability. Kybernetika 22 (1986), 289-298. | MR

[10] T. Roubíček: Generalized solutions of constrained optimization problems. SIAM J. Control Optim. 24 (1986), 951-960. | DOI | MR

[11] T. Roubíček: Stable extensions of constrained optimization problems. J. Math. Anal. Appl. 141 (1989), 520-135, | DOI | MR

[12] Yu. M. Smirnov: On proximity spaces. (in Russian). Mat. Sbornik 31 (73) (1952), 534-574. | MR | Zbl

[13] J. Warga: Optimal Control of Differential and Functional Equations. Academic press, New York, 1972. | MR | Zbl

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