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Nonsmooth optimization via quasi-Newton methods

Mathematical programming, 2013-10, Vol.141 (1-2), p.135-163 [Peer Reviewed Journal]

Springer and Mathematical Optimization Society 2012 ;Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013 ;ISSN: 0025-5610 ;EISSN: 1436-4646 ;DOI: 10.1007/s10107-012-0514-2 ;CODEN: MHPGA4

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Title:
Nonsmooth optimization via quasi-Newton methods
Author: Lewis, Adrian S. ; Overton, Michael L.
Subjects: Algebra ; Algorithms ; Approximation ; Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Convergence ; Dynamical systems ; Full Length Paper ; Functions (mathematics) ; Mathematical analysis ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematical models ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Methods ; Norms ; Numerical Analysis ; Optimization ; Searching ; Sequences ; Studies ; Theoretical
Is Part Of: Mathematical programming, 2013-10, Vol.141 (1-2), p.135-163
Description: We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f , not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.
Publisher: Berlin/Heidelberg: Springer Berlin Heidelberg
Language: English
Identifier: ISSN: 0025-5610
EISSN: 1436-4646
DOI: 10.1007/s10107-012-0514-2
CODEN: MHPGA4
Source: AUTh Library subscriptions: ProQuest Central
Alma/SFX Local Collection

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Nonsmooth optimization via quasi-Newton methods

Springer and Mathematical Optimization Society 2012 ;Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013 ;ISSN: 0025-5610 ;EISSN: 1436-4646 ;DOI: 10.1007/s10107-012-0514-2 ;CODEN: MHPGA4

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