Combined Entropic Regularization and Path-Following Method for Solving Finite Convex Min-max Problems Subject to Infinitely Many Linear Constraints
In this paper, we study the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints. The difficulty of such problems arises from the kinks of the max function and it is often suggested that, by imposing certain regularization function...
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| Published in | Journal of optimization theory and applications Vol. 101; no. 1; pp. 167 - 190 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York, NY
Springer
01.04.1999
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0022-3239 1573-2878 |
| DOI | 10.1023/A:1021727228957 |
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| Abstract | In this paper, we study the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints. The difficulty of such problems arises from the kinks of the max function and it is often suggested that, by imposing certain regularization functions, nondifferentiability will be overcome. We find that the entropic regularization introduced by Li and Fang is closely related to recently developed path-following interior-point methods. Based on their results, we create an interior trajectory in the feasible domain and propose a path-following algorithm with a convergence proof. Our intention here is to show a nice combination of minmax problems, semi-infinite programming, and interior-point methods. Hopefully, this will lead to new applications. |
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| AbstractList | In this paper, we study the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints. The difficulty of such problems arises from the kinks of the max function and it is often suggested that, by imposing certain regularization functions, nondifferentiability will be overcome. We find that the entropic regularization introduced by Li and Fang is closely related to recently developed path-following interior-point methods. Based on their results, we create an interior trajectory in the feasible domain and propose a path-following algorithm with a convergence proof. Our intention here is to show a nice combination of minmax problems, semi-infinite programming, and interior-point methods. Hopefully, this will lead to new applications. |
| Author | Sheu, R. L. Wu, S. Y. |
| Author_xml | – sequence: 1 givenname: R. L. surname: Sheu fullname: Sheu, R. L. – sequence: 2 givenname: S. Y. surname: Wu fullname: Wu, S. Y. |
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| Cites_doi | 10.1093/comjnl/7.2.155 10.1137/1035089 10.1007/978-94-011-1134-8 10.1007/978-3-642-82118-9 10.1137/0326071 10.1007/BF01580609 10.1016/0377-0427(94)00051-2 10.1007/BF01582293 10.1007/BF01587075 10.1002/nav.3800200310 10.1007/BF01586056 10.1137/0727095 10.1016/0898-1221(95)00025-T 10.1080/02331939408843923 10.1007/BF01581628 10.1007/BF01199466 10.1145/321371.321377 10.1080/02331939408843987 |
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| Keywords | Regularization method Interior point method Function minimization Minimax problem Optimization Semi infinite programming |
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| References_xml | – volume-title: Convex Analysis year: 1972 ident: 411912_CR21 – volume-title: Constrained Optimization and Lagrange Multiplier Methods year: 1982 ident: 411912_CR15 – volume-title: Introduction to Minimax year: 1974 ident: 411912_CR2 – volume: 7 start-page: 155 year: 1964 ident: 411912_CR13 publication-title: Computer Journal doi: 10.1093/comjnl/7.2.155 – volume: 35 start-page: 380 year: 1993 ident: 411912_CR7 publication-title: SIAM Review doi: 10.1137/1035089 – volume-title: Interior-Point Approach to Linear, Quadratic, and Convex Programming year: 1994 ident: 411912_CR23 doi: 10.1007/978-94-011-1134-8 – volume-title: Minimization Methods for Nondifferentiable Functions year: 1985 ident: 411912_CR3 doi: 10.1007/978-3-642-82118-9 – volume-title: Convex Analysis and Minimization Algorithms year: 1993 ident: 411912_CR19 – volume: 26 start-page: 1274 year: 1988 ident: 411912_CR18 publication-title: SIAM Journal on Control and Optimization doi: 10.1137/0326071 – start-page: 179 volume-title: Proceedings of the International Mathematics Conference, Kaohsiung, Taiwan, 1994 year: 1995 ident: 411912_CR10 – volume: 60 start-page: 187 year: 1993 ident: 411912_CR1 publication-title: Mathematical Programming doi: 10.1007/BF01580609 – volume: 61 start-page: 87 year: 1995 ident: 411912_CR9 publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/0377-0427(94)00051-2 – volume: 43 start-page: 257 year: 1989 ident: 411912_CR5 publication-title: Mathematical Programming doi: 10.1007/BF01582293 – volume-title: Linear Programming in Infinite-Dimensional Spaces: Theory and Applications year: 1987 ident: 411912_CR4 – volume-title: Optimization by Vector Space Methods year: 1969 ident: 411912_CR22 – volume: 44 start-page: 27 year: 1989 ident: 411912_CR26 publication-title: Mathematical Programming doi: 10.1007/BF01587075 – volume: 20 start-page: 477 year: 1973 ident: 411912_CR6 publication-title: Naval Research Logist Quarterly doi: 10.1002/nav.3800200310 – volume: 54 start-page: 295 year: 1992 ident: 411912_CR24 publication-title: Mathematical Programming doi: 10.1007/BF01586056 – volume: 27 start-page: 1621 year: 1990 ident: 411912_CR16 publication-title: SIAM Journal on Numerical Analysis doi: 10.1137/0727095 – volume: 29 start-page: 7 year: 1995 ident: 411912_CR11 publication-title: Computers and Mathematics with Applications doi: 10.1016/0898-1221(95)00025-T – volume: 28 start-page: 291 year: 1994 ident: 411912_CR8 publication-title: Optimization doi: 10.1080/02331939408843923 – volume: 19 start-page: 61 year: 1980 ident: 411912_CR12 publication-title: Mathematical Programming doi: 10.1007/BF01581628 – volume: 46 start-page: 119 year: 1997 ident: 411912_CR17 publication-title: Mathematical Methods of Operations Research (ZOR) doi: 10.1007/BF01199466 – volume-title: Convex Functions year: 1973 ident: 411912_CR20 – volume: 14 start-page: 72 year: 1967 ident: 411912_CR14 publication-title: Journal of the Association for Computing Machinery doi: 10.1145/321371.321377 – volume: 30 start-page: 235 year: 1994 ident: 411912_CR25 publication-title: Optimization doi: 10.1080/02331939408843987 |
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| Title | Combined Entropic Regularization and Path-Following Method for Solving Finite Convex Min-max Problems Subject to Infinitely Many Linear Constraints |
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