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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Vydáno v:European journal of operational research Ročník 235; číslo 1; s. 38 - 46
Hlavní autoři: Vieira, Douglas Alexandre Gomes, Lisboa, Adriano Chaves
Médium: Journal Article
Jazyk:angličtina
Vydáno: Amsterdam Elsevier B.V 16.05.2014
Elsevier Sequoia S.A
Témata:
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
Line search
Brent’s algorithm
Bisection
Nonlinear programming
Golden section method
Multimodal functions
ISSN:0377-2217, 1872-6860
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Shrnutí:•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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ISSN:0377-2217
1872-6860
DOI:10.1016/j.ejor.2013.12.041