Line search methods with guaranteed asymptotical convergence to an improving local optimum of multimodal functions
•We define and analyze a pattern, called v-pattern, for general line search methods.•We derive enhanced golden section, bisection and Brent’s algorithm•The algorithms convergence using composite maps is proven under mild conditions.•We analyze the performance of the three enhanced line search method...
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| Published in: | European journal of operational research Vol. 235; no. 1; pp. 38 - 46 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
16.05.2014
Elsevier Sequoia S.A |
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| ISSN: | 0377-2217, 1872-6860 |
| Online Access: | Get full text |
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| Abstract | •We define and analyze a pattern, called v-pattern, for general line search methods.•We derive enhanced golden section, bisection and Brent’s algorithm•The algorithms convergence using composite maps is proven under mild conditions.•We analyze the performance of the three enhanced line search methods in practice.
This paper considers line search optimization methods using a mathematical framework based on the simple concept of a v-pattern and its properties. This framework provides theoretical guarantees on preserving, in the localizing interval, a local optimum no worse than the starting point. Notably, the framework can be applied to arbitrary unidimensional functions, including multimodal and infinitely valued ones. Enhanced versions of the golden section, bisection and Brent’s methods are proposed and analyzed within this framework: they inherit the improving local optimality guarantee. Under mild assumptions the enhanced algorithms are proved to converge to a point in the solution set in a finite number of steps or that all their accumulation points belong to the solution set. |
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| AbstractList | This paper considers line search optimization methods using a mathematical framework based on the simple concept of a v-pattern and its properties. This framework provides theoretical guarantees on preserving, in the localizing interval, a local optimum no worse than the starting point. Notably, the framework can be applied to arbitrary unidimensional functions, including multimodal and infinitely valued ones. Enhanced versions of the golden section, bisection and Brent's methods are proposed and analyzed within this framework: they inherit the improving local optimality guarantee. Under mild assumptions the enhanced algorithms are proved to converge to a point in the solution set in a finite number of steps or that all their accumulation points belong to the solution set. This paper considers line search optimization methods using a mathematical framework based on the simple concept of a v-pattern and its properties. This framework provides theoretical guarantees on preserving, in the localizing interval, a local optimum no worse than the starting point. Notably, the framework can be applied to arbitrary unidimensional functions, including multimodal and infinitely valued ones. Enhanced versions of the golden section, bisection and Brent's methods are proposed and analyzed within this framework: they inherit the improving local optimality guarantee. Under mild assumptions the enhanced algorithms are proved to converge to a point in the solution set in a finite number of steps or that all their accumulation points belong to the solution set. [PUBLICATION ABSTRACT] •We define and analyze a pattern, called v-pattern, for general line search methods.•We derive enhanced golden section, bisection and Brent’s algorithm•The algorithms convergence using composite maps is proven under mild conditions.•We analyze the performance of the three enhanced line search methods in practice. This paper considers line search optimization methods using a mathematical framework based on the simple concept of a v-pattern and its properties. This framework provides theoretical guarantees on preserving, in the localizing interval, a local optimum no worse than the starting point. Notably, the framework can be applied to arbitrary unidimensional functions, including multimodal and infinitely valued ones. Enhanced versions of the golden section, bisection and Brent’s methods are proposed and analyzed within this framework: they inherit the improving local optimality guarantee. Under mild assumptions the enhanced algorithms are proved to converge to a point in the solution set in a finite number of steps or that all their accumulation points belong to the solution set. |
| Author | Lisboa, Adriano Chaves Vieira, Douglas Alexandre Gomes |
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| Cites_doi | 10.1080/00150517.1966.12431364 10.1016/j.cam.2007.07.008 10.1090/S0002-9939-1953-0055639-3 10.1137/1011036 10.1007/s10107-010-0347-9 10.1007/s10957-005-6553-6 10.4236/jsea.2010.35057 10.1007/s10915-009-9314-0 10.2140/pjm.1966.16.1 |
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| Keywords | Line search Brent’s algorithm Bisection Nonlinear programming Golden section method Multimodal functions |
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| References | Hu, Huang, Lu (b0035) 2010; 42 Luenberger, Ye (b0045) 2010 Yu, Pu (b0065) 2008; 219 Avriel, Wilde (b0010) 1966; 4 Wolfe (b0060) 1969; 1 Bertsekas (b0020) 2008 Brent (b0025) 1973 Yuan (b0070) 2010; 3 Bazaraa, Shetty (b0015) 2006 Goldstein (b0030) 1965; 3 Vieira (b0055) 2012; 131 Shi, Shen (b0050) 2005; 2 Armijo (b0005) 1966; 16 Kiefer (b0040) 1953; 4 Shi (10.1016/j.ejor.2013.12.041_b0050) 2005; 2 Kiefer (10.1016/j.ejor.2013.12.041_b0040) 1953; 4 Avriel (10.1016/j.ejor.2013.12.041_b0010) 1966; 4 Wolfe (10.1016/j.ejor.2013.12.041_b0060) 1969; 1 Luenberger (10.1016/j.ejor.2013.12.041_b0045) 2010 Bertsekas (10.1016/j.ejor.2013.12.041_b0020) 2008 Hu (10.1016/j.ejor.2013.12.041_b0035) 2010; 42 Vieira (10.1016/j.ejor.2013.12.041_b0055) 2012; 131 Yuan (10.1016/j.ejor.2013.12.041_b0070) 2010; 3 Goldstein (10.1016/j.ejor.2013.12.041_b0030) 1965; 3 Bazaraa (10.1016/j.ejor.2013.12.041_b0015) 2006 Yu (10.1016/j.ejor.2013.12.041_b0065) 2008; 219 Armijo (10.1016/j.ejor.2013.12.041_b0005) 1966; 16 Brent (10.1016/j.ejor.2013.12.041_b0025) 1973 |
| References_xml | – volume: 1 start-page: 226 year: 1969 end-page: 235 ident: b0060 article-title: Convergence conditions for ascent methods publication-title: SIAM Review – volume: 3 start-page: 503 year: 2010 end-page: 509 ident: b0070 article-title: A line search algorithm for unconstrained optimization publication-title: Journal of Software Engineering and Applications – volume: 4 start-page: 265 year: 1966 end-page: 269 ident: b0010 article-title: Optimality proof for the symmetric fibonacci search technique publication-title: Fibonacci Quarterly – volume: 4 start-page: 502 year: 1953 end-page: 506 ident: b0040 article-title: Sequential minimax search for a maximum publication-title: Proceedings of the American Mathematical Society – volume: 3 start-page: 147 year: 1965 end-page: 151 ident: b0030 article-title: On steepest descent publication-title: SIAM Journal of Control – volume: 16 start-page: 1 year: 1966 end-page: 3 ident: b0005 article-title: Minimization of functions having Lipschitz conditions for partial derivatives publication-title: Pacific Journal of Mathematics – volume: 219 start-page: 134 year: 2008 end-page: 144 ident: b0065 article-title: A new nonmonotone line search technique for unconstrained optimization publication-title: Journal of Computational and Applied Mathematics – year: 2010 ident: b0045 article-title: Linear and nonlinear programming – year: 2008 ident: b0020 article-title: Nonlinear programming – volume: 42 start-page: 38 year: 2010 end-page: 53 ident: b0035 article-title: A non-monotone line search algorithm for unconstrained optimization publication-title: Journal of Scientific Computing – year: 2006 ident: b0015 article-title: Nonlinear programming – Theory and algorithms – volume: 2 start-page: 425 year: 2005 end-page: 446 ident: b0050 article-title: New inexact line search method for unconstrained optimization publication-title: Journal of Optimization Theory and Applications – volume: 131 start-page: 131 year: 2012 end-page: 161 ident: b0055 article-title: Multicriteria optimization with a multiobjective golden section line search publication-title: Mathematical Programming – year: 1973 ident: b0025 article-title: Algorithms for minimization without derivatives – volume: 4 start-page: 265 year: 1966 ident: 10.1016/j.ejor.2013.12.041_b0010 article-title: Optimality proof for the symmetric fibonacci search technique publication-title: Fibonacci Quarterly doi: 10.1080/00150517.1966.12431364 – volume: 3 start-page: 147 year: 1965 ident: 10.1016/j.ejor.2013.12.041_b0030 article-title: On steepest descent publication-title: SIAM Journal of Control – volume: 219 start-page: 134 year: 2008 ident: 10.1016/j.ejor.2013.12.041_b0065 article-title: A new nonmonotone line search technique for unconstrained optimization publication-title: Journal of Computational and Applied Mathematics doi: 10.1016/j.cam.2007.07.008 – volume: 4 start-page: 502 issue: 3 year: 1953 ident: 10.1016/j.ejor.2013.12.041_b0040 article-title: Sequential minimax search for a maximum publication-title: Proceedings of the American Mathematical Society doi: 10.1090/S0002-9939-1953-0055639-3 – volume: 1 start-page: 226 year: 1969 ident: 10.1016/j.ejor.2013.12.041_b0060 article-title: Convergence conditions for ascent methods publication-title: SIAM Review doi: 10.1137/1011036 – year: 2010 ident: 10.1016/j.ejor.2013.12.041_b0045 – volume: 131 start-page: 131 issue: 1–2 year: 2012 ident: 10.1016/j.ejor.2013.12.041_b0055 article-title: Multicriteria optimization with a multiobjective golden section line search publication-title: Mathematical Programming doi: 10.1007/s10107-010-0347-9 – volume: 2 start-page: 425 year: 2005 ident: 10.1016/j.ejor.2013.12.041_b0050 article-title: New inexact line search method for unconstrained optimization publication-title: Journal of Optimization Theory and Applications doi: 10.1007/s10957-005-6553-6 – year: 2008 ident: 10.1016/j.ejor.2013.12.041_b0020 – year: 1973 ident: 10.1016/j.ejor.2013.12.041_b0025 – volume: 3 start-page: 503 issue: 5 year: 2010 ident: 10.1016/j.ejor.2013.12.041_b0070 article-title: A line search algorithm for unconstrained optimization publication-title: Journal of Software Engineering and Applications doi: 10.4236/jsea.2010.35057 – volume: 42 start-page: 38 year: 2010 ident: 10.1016/j.ejor.2013.12.041_b0035 article-title: A non-monotone line search algorithm for unconstrained optimization publication-title: Journal of Scientific Computing doi: 10.1007/s10915-009-9314-0 – year: 2006 ident: 10.1016/j.ejor.2013.12.041_b0015 – volume: 16 start-page: 1 year: 1966 ident: 10.1016/j.ejor.2013.12.041_b0005 article-title: Minimization of functions having Lipschitz conditions for partial derivatives publication-title: Pacific Journal of Mathematics doi: 10.2140/pjm.1966.16.1 |
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| Snippet | •We define and analyze a pattern, called v-pattern, for general line search methods.•We derive enhanced golden section, bisection and Brent’s algorithm•The... This paper considers line search optimization methods using a mathematical framework based on the simple concept of a v-pattern and its properties. This... |
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| SubjectTerms | Algorithms Asymptotic methods Asymptotic properties Bisection Brent’s algorithm Convergence Decision making models Functions (mathematics) Golden section method Intervals Line search Mathematical analysis Mathematical functions Mathematical problems Multimodal functions Nonlinear programming Operational research Optimization Optimization algorithms Studies |
| Title | Line search methods with guaranteed asymptotical convergence to an improving local optimum of multimodal functions |
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