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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) |

### Keywords:

constrained optimization; problems with tolerance; minimizing filter; level sets; exterior penalty function; compactification### References:

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