# Constrained optimization: A general tolerance approach

Aplikace matematiky (1990)

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

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topRoubíček, 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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- T. Roubíček, A generalized solution of a nonconvex minimization problem and its stability, Kybernetika 22 (1986), 289-298. (1986) MR0868022
- T. Roubíček, 10.1137/0324056, SIAM J. Control Optim. 24 (1986), 951-960. (1986) MR0854064DOI10.1137/0324056
- 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
- Yu. M. Smirnov, On proximity spaces, (in Russian). Mat. Sbornik 31 (73) (1952), 534-574. (1952) Zbl0152.20904MR0055661
- J. Warga, Optimal Control of Differential and Functional Equations, Academic press, New York, 1972. (1972) Zbl0253.49001MR0372708

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