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It has long been known that the gradient (steepest descent) method may fail on nonsmooth problems, but the examples that have appeared in the literature are either devised specifically to defeat a gradient or subgradient method with an exact line search or are unstable with respect to perturbation of the initial point.
Minimization of functions having Lipschitz continuous first partial derivatives
Larry Armijo · 1966
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Convergence conditions for ascent methods
Philip Wolfe · 1969
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The theory of nonlinear minimax problems
V. F. Dem’janov and V. N. Malozemov · 1971
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An extension of Davidon methods to non differentiable problems
C. Lemaréchal · 1975
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A method of conjugate subgradients for minimizing nondifferentiable functions
Philip Wolfe · 1975
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Some global convergence properties of a variable metric algorithm for minimization without exact line searches
M. J. D. Powell · 1976
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A view of unconstrained optimization
M. J. D. Powell · 1976
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Methods of descent for nondifferentiable optimization
Krzysztof C. Kiwiel · 1985
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Minimization Methods for Non-differentiable Functions
N. Z. Shor · 1985
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Practical methods of optimization
R. Fletcher · 1987
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On the limited memory BFGS method for large scale optimization
Dong C. Liu and Jorge Nocedal · 1989
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Convex analysis and minimization algorithms. I
Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal · 1993
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Nonlinear Programming
D. Bertsekas · 1999
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Incremental subgradient methods for nondifferentiable optimization
Angelia Nedić and Dimitri P. Bertsekas · 2001
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A robust gradient sampling algorithm for nonsmooth, nonconvex optimization
James V. Burke, Adrian S. Lewis, and Michael L. Overton · 2005
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Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization
Krzysztof C. Kiwiel · 2007
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Nonsmooth optimization via quasi-Newton methods
Adrian S. Lewis and Michael L. Overton · 2013
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Private communication
Yu. Nesterov · 2016
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A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles
Frank E. Curtis, Tim Mitchell, and Michael L. Overton · 2017
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Variational analysis of the Crouzeix ratio
Anne Greenbaum, Adrian S. Lewis, and Michael L. Overton · 2017
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Exact worst-case performance of first-order methods for composite convex optimization
Adrien B. Taylor, Julien M. Hendrickx, and Francois Glineur · 2017
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Yu. Nesterov · 2005
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Numerical Optimization
J. Nocedal and S. J. Wright · 2006
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Méthode générale pour la résolution des systèmes d’équations simultanées
A. Cauchy
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Gradient Sampling Methods for Nonsmooth Optimization
J. V. Burke, F. E. Curtis, A. S. Lewis, M. L. Overton, and L. E. A. Simões · 2018
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Nonsmooth variants of Powell’s BFGS convergence theorem
J. Guo and A. Lewis · 2018
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