Constrained optimization: A general tolerance approach

Tomáš Roubíček

Aplikace matematiky (1990)

  • Volume: 35, Issue: 2, page 99-128
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Roubíček, Tomáš. "Constrained optimization: A general tolerance approach." Aplikace matematiky 35.2 (1990): 99-128. <http://eudml.org/doc/15615>.

@article{Roubíček1990,
abstract = {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.},
author = {Roubíček, Tomáš},
journal = {Aplikace matematiky},
keywords = {constrained optimization; level sets; minimizing sequences; penalty functions; compactifications; problems with tolerance; constrained optimization; problems with tolerance; minimizing filter; level sets; exterior penalty function; compactification},
language = {eng},
number = {2},
pages = {99-128},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Constrained optimization: A general tolerance approach},
url = {http://eudml.org/doc/15615},
volume = {35},
year = {1990},
}

TY - JOUR
AU - Roubíček, Tomáš
TI - Constrained optimization: A general tolerance approach
JO - Aplikace matematiky
PY - 1990
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 2
SP - 99
EP - 128
AB - 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.
LA - eng
KW - constrained optimization; level sets; minimizing sequences; penalty functions; compactifications; problems with tolerance; constrained optimization; problems with tolerance; minimizing filter; level sets; exterior penalty function; compactification
UR - http://eudml.org/doc/15615
ER -

References

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  1. Á. Császár, General Topology, Akademiai Kiadó, Budapest, 1978. (1978) MR1796928
  2. A. V. Efremovich, The geometry of proximity, (in Russian). Mat. Sbornik 31 (73) (1952), 189-200. (1952) MR0055659
  3. E. K. Golshtein, Duality Theory in Mathematical Programming and Its Applications, (in Russian). Nauka, Moscow, 1971. (1971) MR0322531
  4. D. A. Molodcov, Stability and regularization of principles of optimality, (in Russian). Zurnal vycisl. mat. i mat. fiziki 20 (1980), 1117-1129. (1980) MR0593496
  5. D. A. Molodcov, Stability of Principles of Optimality, (in Russian). Nauka, Moscow, 1987. (1987) 
  6. L. Nachbin, Topology and Order, D. van Nostrand Соmр., Princeton, 1965. (1965) Zbl0131.37903MR0219042
  7. S. A. Naimpally B. D. Warrack, Proximity Spaces, Cambridge Univ. Press, Cambridge, 1970. (1970) MR0278261
  8. E. Polak Y. Y. Wardi, 10.1137/0322036, SIAM J. Control Optim. 22 (1984), 599-609. (1984) MR0747971DOI10.1137/0322036
  9. T. Roubíček, A generalized solution of a nonconvex minimization problem and its stability, Kybernetika 22 (1986), 289-298. (1986) MR0868022
  10. T. Roubíček, 10.1137/0324056, SIAM J. Control Optim. 24 (1986), 951-960. (1986) MR0854064DOI10.1137/0324056
  11. T. Roubíček, 10.1016/0022-247X(89)90210-2, J. Math. Anal. Appl. 141 (1989), 520-135, (1989) MR1004588DOI10.1016/0022-247X(89)90210-2
  12. Yu. M. Smirnov, On proximity spaces, (in Russian). Mat. Sbornik 31 (73) (1952), 534-574. (1952) Zbl0152.20904MR0055661
  13. J. Warga, Optimal Control of Differential and Functional Equations, Academic press, New York, 1972. (1972) Zbl0253.49001MR0372708

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