| Title:
|
Constrained optimization: A general tolerance approach (English) |
| Author:
|
Roubíček, Tomáš |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
35 |
| Issue:
|
2 |
| Year:
|
1990 |
| Pages:
|
99-128 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 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. (English) |
| Keyword:
|
constrained optimization |
| Keyword:
|
level sets |
| Keyword:
|
minimizing sequences |
| Keyword:
|
penalty functions |
| Keyword:
|
compactifications |
| Keyword:
|
problems with tolerance |
| MSC:
|
49A27 |
| MSC:
|
49J27 |
| MSC:
|
49J45 |
| MSC:
|
49K40 |
| MSC:
|
49M30 |
| MSC:
|
54D35 |
| MSC:
|
54E05 |
| MSC:
|
65K10 |
| MSC:
|
90C48 |
| MSC:
|
90C99 |
| idZBL:
|
Zbl 0714.49006 |
| idMR:
|
MR1042847 |
| DOI:
|
10.21136/AM.1990.104393 |
| . |
| Date available:
|
2008-05-20T18:38:41Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104393 |
| . |
| Reference:
|
[1] Á. Császár: General Topology.Akademiai Kiadó, Budapest, 1978. MR 1796928 |
| Reference:
|
[2] A. V. Efremovich: The geometry of proximity.(in Russian). Mat. Sbornik 31 (73) (1952), 189-200. MR 0055659 |
| Reference:
|
[3] E. K. Golshtein: Duality Theory in Mathematical Programming and Its Applications.(in Russian). Nauka, Moscow, 1971. MR 0322531 |
| Reference:
|
[4] D. A. Molodcov: Stability and regularization of principles of optimality.(in Russian). Zurnal vycisl. mat. i mat. fiziki 20 (1980), 1117-1129. MR 0593496 |
| Reference:
|
[5] D. A. Molodcov: Stability of Principles of Optimality.(in Russian). Nauka, Moscow, 1987. |
| Reference:
|
[6] L. Nachbin: Topology and Order.D. van Nostrand Соmр., Princeton, 1965. Zbl 0131.37903, MR 0219042 |
| Reference:
|
[7] S. A. Naimpally B. D. Warrack: Proximity Spaces.Cambridge Univ. Press, Cambridge, 1970. MR 0278261 |
| Reference:
|
[8] E. Polak Y. Y. Wardi: A study of minizing sequences.SIAM J. Control Optim. 22 (1984), 599-609. MR 0747971, 10.1137/0322036 |
| Reference:
|
[9] T. Roubíček: A generalized solution of a nonconvex minimization problem and its stability.Kybernetika 22 (1986), 289-298. MR 0868022 |
| Reference:
|
[10] T. Roubíček: Generalized solutions of constrained optimization problems.SIAM J. Control Optim. 24 (1986), 951-960. MR 0854064, 10.1137/0324056 |
| Reference:
|
[11] T. Roubíček: Stable extensions of constrained optimization problems.J. Math. Anal. Appl. 141 (1989), 520-135, MR 1004588, 10.1016/0022-247X(89)90210-2 |
| Reference:
|
[12] Yu. M. Smirnov: On proximity spaces.(in Russian). Mat. Sbornik 31 (73) (1952), 534-574. Zbl 0152.20904, MR 0055661 |
| Reference:
|
[13] J. Warga: Optimal Control of Differential and Functional Equations.Academic press, New York, 1972. Zbl 0253.49001, MR 0372708 |
| . |
