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Convergence of the steepest descent method for minimizing quasiconvex functions

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To minimize a continuously differentiable quasiconvex functionf: ℝn→ℝ, Armijo's steepest descent method generates a sequencex k+1 =x kt k f(x k), wheret k >0. We establish strong convergence properties of this classic method: either\(x^k \to \bar x,\), s.t.\(\nabla f(\bar x) = 0\); or arg minf = ∅, ∥x k∥ ↓ ∞ andf(x k)↓ inff. We also discuss extensions to other line searches.

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  1. Cauchy, A.,Méthode Générale pour la Résolution des Systèmes d'Équations Simultanées, Comptes Rendus de Académie des Sciences, Paris, Vol. 25, pp. 536–538, 1847.

    Google Scholar 

  2. Armijo, L.,Minimization of Functions Having Continuous Partial Derivatives, Pacific Journal of Mathematics, Vol. 16, pp. 1–3, 1966.

    Google Scholar 

  3. Burachik, R., Grana Drummond, L. M., Iusem, A. N., andSvaiter, B. F.,Full Convergence of the Steepest Descent Method with Inexact Line Searches, Optimization, Vol. 32, pp. 137–146, 1995.

    Google Scholar 

  4. Kiwiel, K. C.,An Aggregate Subgradient Method for Nonsmooth Convex Minimization, Mathematical Programming, Vol. 27, pp. 320–341, 1983.

    Google Scholar 

  5. Kiwiel, K. C.,A Direct Method of Linearizations for Continuous Minimax Problems, Journal of Optimization Theory and Applications, Vol. 55, pp. 271–287, 1987.

    Google Scholar 

  6. Mangasarian, O. L.,Nonlinear Programming, Mc-Graw-Hill, New York, New York, 1969; Reprinted by SIAM, Philadelphia, Pennsylvania, 1994.

    Google Scholar 

  7. Bereznev, V. A., Karmanov, V. G., andTretyakov, A. A.,On the Stabilizing Properties of the Gradient Method, Zhurnal Vychislitelnoi Matematiki i Matematicheskoi Fiziki, Vol. 26, pp. 134–137, 1986 (in Russian).

    Google Scholar 

  8. Polak, E.,Computational Methods in Optimization, Academic Press, New York, New York, 1971.

    Google Scholar 

  9. Bazaraa, M. S., Sherali, H. D., andShetty, C. M.,Nonlinear Programming: Theory and Algotithms, 2nd Edition, Wiley, New York, New York, 1993.

    Google Scholar 

  10. Dennis, J. E., Jr., andSchnabel, R. B.,Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1983.

    Google Scholar 

  11. Fletcher, R.,Practical Methods of Optimization, 2nd Edition, Wiley, Chichester, England, 1987.

    Google Scholar 

  12. Murty, K. G.,Linear Complementarity, Linear and Nonlinear Programming, Heldermann Verlag, Berlin, Germany, 1988.

    Google Scholar 

  13. Kiwiel, K. C.,A Linearization Method for Minimizing Certain Quasidifferentiable Functions, Mathematical Programming Study, Vol. 29, pp. 85–94, 1986.

    Google Scholar 

  14. Iusem, A. N., andSvaiter, B. F.,A Proximal Regularization of the Steepest Descent Method, RAIRO Rechérche Opérationelle (to appear).

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Communicated by O. L. Mangasarian

The research of the first author was supported by the Polish Academy of Sciences. The second author acknowledges the support of the Department of Industrial Engineering, Hong Kong University of Science and Technology.

We wish to thank two anonymous referees for their valuable comments. In particular, one referee has suggested the use of quasiconvexity instead of convexity off.

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Kiwiel, K.C., Murty, K. Convergence of the steepest descent method for minimizing quasiconvex functions. J Optim Theory Appl 89, 221–226 (1996).

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