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Reflections on nondifferentiable optimization, part 2, convergence

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This paper is concerned with the minimization of nondifferentiable functions. Three main results are obtained: (i) convergence of the ball-gradient algorithm, introduced by Dixon, for convex functions; (ii) convergence of the generalized gradient algorithm, as implemented by Shor and Ermol'ev, to a stationary point; and (iii) convergence of an algorithm introduced by Goldstein to a local minimum.

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    Shor, N. Z.,Generalized Gradient Methods of Minimization of Nonsmoothed Functions and Their Use in Problems of Mathematical Programming (in Russian), Mathematical Methods, Vol. 12, pp. 337–356, 1976.

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    Wolfe, P.,A Method of Conjugate Subgradients for Minimizing Nondifferentiable Functions, Nondifferentiable Optimization, Edited by M. L. Balinski and P. Wolfe, North-Holland Publishing Company, Amsterdam, Holland, 1975.

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    Ermol'ev, J. M.,Methods of Solving Nonlinear Extremal Problems, Kibernetika, Vol. 4, pp. 1–4, 1966.

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    Goldstein, A. A.,Optimization of Lipschitz Continuous Functions, Mathematical Programming, Vol. 13, pp. 14–22, 1977.

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    Dixon, L. C. W.,Reflections on Nondifferentiable Optimization, Part 1, Ball Gradient, Journal of Optimization Theory and Applications, Vol. 32, No. 2, 1980.

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    Polyak, B. T.,A General Method of Solving Extremal Problems, Doklady Akademii Nauk, SSSR, Vol. 174, pp. 593–597, 1967.

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This research was carried out while the second author was being supported by the National Research Council of Italy at the Numerical Optimization Center of the Hatfield Polytechnic.

Communicated by D. F. Lawden

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Dixon, L.C.W., Gaviano, M. Reflections on nondifferentiable optimization, part 2, convergence. J Optim Theory Appl 32, 259–275 (1980). https://doi.org/10.1007/BF00934553

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Key Words

  • Nondifferentiable optimization
  • subgradients
  • ball gradients