# Penalty and Smoothing Methods for Convex Semi-Infinite Programming

TL;DR: This paper introduces a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods that subsumes well-known classical algorithms, but also provides some new methods with interesting properties.

Abstract: In this paper we consider min-max convex semi-infinite programming. To solve these problems we introduce a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods. This framework subsumes well-known classical algorithms, but also provides some new methods with interesting properties. Convergence of the primal and dual sequences are proved under minimal assumptions.

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##### Citations

49 citations

### Cites background from "Penalty and Smoothing Methods for C..."

...1 in [1] shows some particular cases where the set Ω0 is easily obtainable....

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##### References

5,933 citations

### "Penalty and Smoothing Methods for C..." refers background in this paper

...We recall here some basic notions about asymptotic cones and functions (for more details see, for instance, the books of Auslender and Teboulle [4], Rockafellar [24])....

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...We recall here some basic notions about asymptotic cones and functions (for more details see, for instance, the books of Auslender and Teboulle [4] and of Rockafellar [24])....

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2,665 citations

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### "Penalty and Smoothing Methods for C..." refers methods in this paper

...Applied to LSIP, especially Cheney and Goldstein [10] and Kelley [15] turn out to be identical or mere modifications of the dual simplex method discussed above, so that they have similar properties and drawbacks....

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...Supposing that F is 1 (as is generally the case in ordinary CSIP), we can use cutting-plane methods of Cheney and Goldstein [10], Kelley [15], Veinott [31], or Elzinga and Moore [11], and their variants (see, e.g., Reemtsen and Görner [22] for more references)....

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...To avoid slow convergence, constraint dropping rules are again given under some conditions as strict convexity on F for Cheney and Goldstein [10] and Kelley [15]....

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...Supposing that F is 1 (as is generally the case in ordinary CSIP), we can use cutting-plane methods of Cheney and Goldstein [10], Kelley [15], Veinott [31], or Elzinga and Moore [11], and their variants (see, e....

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441 citations

### "Penalty and Smoothing Methods for C..." refers background in this paper

...In [9], Chen and Mangasarian provided a systematic way to generate elements of 1....

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