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A minimization method for the sum of a convex function and a continuously differentiable function

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Abstract

This paper presents a method for finding the minimum for a class of nonconvex and nondifferentiable functions consisting of the sum of a convex function and a continuously differentiable function. The algorithm is a descent method which generates successive search directions by solving successive convex subproblems. The algorithm is shown to converge to a critical point.

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References

  1. Mifflin, R. Semismooth and Semiconvex Functions in Constrained Optimization, SIAM Journal on Control and Optimization, Vol. 15, pp. 959–972, 1977.

    Google Scholar 

  2. Feuer, A.,An Extension of the Method of Conjugate Subgradients to Generalized Minimax Objectives, Yale University, New Haven, Connecticut, 1976.

    Google Scholar 

  3. Vainberg, M. M.,Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Israel Program for Scientific Translations, Jerusalem, Israel, 1973.

    Google Scholar 

  4. Pshenichnyi, B. N.,Necessary Conditions for an Extremum, Marcel Dekker Company, New York, New York, 1971.

    Google Scholar 

  5. Clarke, F. H.,Generalized Gradients and Applications, Transactions of American Mathematical Society, Vol. 205, pp. 247–262, 1975.

    Google Scholar 

  6. Bertsekas, D. P., andMitter, S. K.,A Descent Numerical Method for Optimization Problems with Nondifferentiable Cost Functionals, SIAM Journal on Control, Vol. 11, pp. 637–652, 1973.

    Google Scholar 

  7. Lemarechal, C.,An Extension of Davidon Methods to Nondifferentiable Problems, Mathematical Programming Study, No. 3, pp. 95–109, 1975.

  8. Polyak, B. T.,Minimization of Unsmooth Functionals, USSR Computational Mathematics and Mathematical Physics, Vol. 9, pp. 14–29, 1969.

    Google Scholar 

  9. Wolfe, P.,A Method of Conjugate Subgradients for Minimizing Nondifferentiable Functions, Mathematical Programming Study, No. 3, pp. 145–173, 1975.

  10. Shor, N. Z.,Utilization of the Operation of Space Dilatation in the Minimization of Convex Functions, Cybernetics, Vol. 6, pp. 7–15, 1970.

    Google Scholar 

  11. Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

    Google Scholar 

  12. Feuer, A.,Minimizing Well-Behaved Functions, Proceedings of 12th Annual Allerton Conference on Circuit and System Theory, Allerton, Illinois, pp. 15–34, 1974.

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

    Google Scholar 

  14. Mifflin, R.,An Algorithm for Constrained Optimization with Semismooth Functions, Mathematics of Operations Research, Vol. 2, pp. 191–207, 1977.

    Google Scholar 

  15. Bertsekas, D. P.,Necessary and Sufficient Conditions for a Penalty Function to be Exact, Mathematical Programming, Vol. 9, pp. 87–99, 1975.

    Google Scholar 

  16. Conn, A. W.,Constrained Optimization Using a Nondifferentiable Penalty Function, SIAM Journal on Numerical Analysis, Vol. 10, pp. 760–784, 1973.

    Google Scholar 

  17. Evans, J. P., Gould, F. J., andTolle, J. W.,Exact Penalty Functions in Nonlinear Programming, Mathematical Programming, Vol. 4, pp. 72–97, 1973.

    Google Scholar 

  18. Howe, S.,New Conditions for Exactness of a Simple Penalty Function, SIAM Journal on Control, Vol. 11, pp. 378–381, 1973.

    Google Scholar 

  19. Petrzykowski, T.,An Exact Potential Method for Constrained Maxima, SIAM Journal on Numerical Analysis, Vol. 6, pp. 299–304, 1969.

    Google Scholar 

  20. Zangwill, W. I.,Nonlinear Programming via Penalty Functions, Management Science, Vol. 13, pp. 344–358, 1967.

    Google Scholar 

  21. Clarke, F. H.,A New Approach to Lagrange Multipliers, Mathematics of Operations Research, Vol. 1, pp. 165–174, 1976.

    Google Scholar 

  22. Frank, M., andWolfe, P.,An Algorithm for Quadratic Programming, Naval Research Logistics Quarterly, Vol. 3, pp. 95–110, 1956.

    Google Scholar 

  23. Levitin, E. S., andPolyak, B. T.,Constrained Minimization Methods, USSR Computational Mathematics and Mathematical Physics, Vol. 6, No. 5, pp. 1–50, 1966.

    Google Scholar 

  24. Zangwill, W. I.,Nonlinear Programming: A Unified Approach, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.

    Google Scholar 

  25. Luenberger, D. G.,Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading, Massachusetts, 1973.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Physics, Faculty of Engineering, Kyoto University, Kyoto, Japan

    H. Mine (Professor) & M. Fukushima (Graduate Student)

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  1. H. Mine
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  2. M. Fukushima
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Communicated by O. L. Mangasarian

The authors wish to express their appreciation to the referees for their careful review and helpful comments.

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Mine, H., Fukushima, M. A minimization method for the sum of a convex function and a continuously differentiable function. J Optim Theory Appl 33, 9–23 (1981). https://doi.org/10.1007/BF00935173

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  • DOI: https://doi.org/10.1007/BF00935173

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