Abstract
We study a generalization of the nonderivative discrete gradient method of Bagirov et al. for minimizing a locally Lipschitz function f on ℝn. We strengthen the existing convergence result for this method by showing that it either drives the f-values to −∞ or each of its cluster points is Clarke stationary for f, without requiring the compactness of the level sets of f. Our generalization is an approximate bundle method, which also subsumes the secant method of Bagirov et al.
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Bagirov, A.M., Karazözen, B., Sezer, M.: Discrete gradient method: Derivative-free method for nonsmooth optimization. J. Optim. Theory Appl. 137, 317–334 (2008)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)
Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, vol. 1133. Springer, Berlin (1985)
Bagirov, A.M., Nazari Ganjehlou, A.: A secant method for nonsmooth optimization. Tech. rep., Centre for Informatics and Applied Optimization, University of Ballarat, Victoria, Australia (2009)
Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15, 751–779 (2005)
Kiwiel, K.C.: Restricted step and Levenberg-Marquardt techniques in proximal bundle methods for nonconvex nondifferentiable optimization. SIAM J. Optim. 6, 227–249 (1996)
Audet, C., Béchard, V., Le Digabel, S.: Nonsmooth optimization through mesh adaptive direct search and variable neighborhood search. J. Glob. Optim. 41, 299–318 (2008)
Audet, C., Dennis, J.E. Jr.: Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17, 188–217 (2006)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Bagirov, A.M., Nazari Ganjehlou, A.: A quasisecant method for minimizing nonsmooth functions. Tech. rep., Centre for Informatics and Applied Optimization, University of Ballarat, Victoria, Australia (2009). Available at the URL http://www.optimization-online.org/DB_FILE/2009/03/2251.pdf
Kiwiel, K.C.: Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization. SIAM J. Optim. 18, 379–388 (2007)
Goldstein, A.A.: Optimization of Lipschitz continuous functions. Math. Program. 13, 14–22 (1977)
Frangioni, A.: Solving semidefinite quadratic problems within nonsmooth optimization algorithms. Comput. Oper. Res. 23, 1099–1118 (1996)
Kiwiel, K.C.: A Cholesky dual method for proximal piecewise linear programming. Numer. Math. 68, 325–340 (1994)
Bihain, A.: Optimization of upper semidifferentiable functions. J. Optim. Theory Appl. 44, 545–568 (1984)
Lemaréchal, C.: An introduction to the theory of nonsmooth optimization. Optimization 17, 827–858 (1986)
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Communicated by F. Giannessi.
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Kiwiel, K.C. Improved Convergence Result for the Discrete Gradient and Secant Methods for Nonsmooth Optimization. J Optim Theory Appl 144, 69–75 (2010). https://doi.org/10.1007/s10957-009-9584-6
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DOI: https://doi.org/10.1007/s10957-009-9584-6