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Optimization of Lipschitz continuous functions.(English)Zbl 0394.90088


MSC:

90C30 Nonlinear programming

Cite

References:

[1]D.P. Bertsekas and S.K. Mitter, ”A descent numerial method for optimization problems with nondifferentiable cost functionals”,SIAM Journal on Control 11 (1973). ·Zbl 0243.49012
[2]D.P. Bertsekas and S.K. Mitter, ”Steepest descent for optimization problems with nondifferentiable cost functionals”, in:Proceedings of the 5th annual Princeton conference on information and system sciences, March 1971.
[3]D.P. Bertsekas, ”Nondifferentiable optimization via approximation”,Mathematical Programming Study 3 (1975) 1–25. ·Zbl 0383.49025
[4]A. Cauchy, ”Méthode générale pour la résolution des systèmes d’équations simultanées”,Comptes Rendus Hebdomadaires des Séances de la Académie des Sciences, Paris, 25 (1847).
[5]E.W. Cheney and A.A. Goldstein, ”Newton’s method for convex programming and Tchebysheff approximation”,Numerische Mathematik I (1959) 253–268. ·Zbl 0113.10703 ·doi:10.1007/BF01386389
[6]E.W. Cheney and A.A. Goldstein, ”Tchebycheff approximation and related extremal problems”,Journal of Mathematics and Mechanics 14 (1965) 87–98. ·Zbl 0158.15102
[7]F.H. Clarke, ”Generalized gradients and applications”,Transactions of the American Mathematical Society 205 (1975) 247–262. ·Zbl 0307.26012 ·doi:10.1090/S0002-9947-1975-0367131-6
[8]R. Courant, ”Variational methods for the solution of problems of equilibrium and vibrations”,Bulletin of the American Mathematical Society 49 (1943) 1–23. ·Zbl 0063.00985 ·doi:10.1090/S0002-9904-1943-07818-4
[9]H.B. Curry, ”The method of steepest descent for non-linear minimization problems”,Quarterly of Applied Mathematics 2 (1944–45).
[10]V.G. Demyanov, ”On the solution of certain minimax problems”,Kibernetica 2 (1966).
[11]V.G. Demyanov, ”Algorithms for some minimax problems”,Journal of Computer and Systems Sciences 2 (1968) 431–433. ·Zbl 0177.23104
[12]J. Dugundji,Topology (Allyn and Bacon, Boston, 1966) p. 234, Th. 4.5.
[13]I.I. Eremin, ”The relaxation method for solving systems of inequalities with convex functions on the left sides”,Soviet Mathematics Doklady 6 (1965) 219–222. ·Zbl 0144.30601
[14]A. Feuer, ”Minimizing well-behaved functions”, in:Proceedings of the 12th annual Allerton conference on circuit and system theory, University of Illinois, October 1974.
[15]J.J-B. Fourier, ”Solution d’une question particulière du calcul des inégalites, second extrait”, Histoire de l’Académie des Sciences (1824) p. 48.
[16]J. Hadamard, ”Memoire sur le probleme d’analyse relatif a l’equilibre des plaques elastiques encastrees”, Adacemie des Sciences de l’Institut de France Ser. 2 vol. 33 (1907).
[17]A. Goldstein, ”Cauchy’s method of minimization”,Numerische Mathematik 4 (1962) 146–150, ·Zbl 0105.10201 ·doi:10.1007/BF01386306
[18]A. Goldstein, ”On steepest descent,SIAM Journal on Control Ser. A, 3 (1) (1965) 147–151. ·Zbl 0221.65094
[19]A.A. Goldstein, ”Optimization with corners”, in:Nonlinear programming 2 (Academic Press, New York, 1975) pp.215–230. ·Zbl 0395.90074
[20]L.V. Kantorovich, ”Functional analysis and applied mathematics”,Uspehi Matematičeskih Nauk 3 (1948). ·Zbl 0034.21203
[21]C. Lemarechal, ”An algorithm for minimizing convex functions”, in:Proceedings, International Federation of Information Processing Congress 74 (North-Holland, Amsterdam, 1974) pp. 552–556.
[22]C. Lemarechal, ”Note on an extension of Davidon’s method to non-differentiable functions”,Mathematical Programming Study 3 (1975) 95–109.
[23]Kaj Madsen, ”Minimax solution of non-linear equations without calculating derivatives”,Mathematical Programming Study 3 (1975) 110–126. ·Zbl 0364.90085
[24]Robert Mifflen, ”An algorithm for constrained optimization with semismooth function”, International Institute for Applied Systems Analysis, Laxenburg, Austria.
[25]J. Ortega and W. Rheinboldt,Iterative solution of nonlinear equations (Academic Press, New York, 1970). ·Zbl 0241.65046
[26]B.T. Polyak, ”Minimization of unsmooth functionals”,USSR Computation Mathematics and Mathematical Physics 9 (3) (1969). ·Zbl 0229.65056
[27]E. V. Raik, ”Fejer methods in Hilbert space”,Izvestija Akademii Nauk SSSR Serija Matematičeskaja 3 (1967) 286–293, Fig. 6. ·Zbl 0168.13202
[28]S. Saks,Theory of the integral (Dover, 1964). ·Zbl 1196.28001
[29]N.Z. Shor, ”Convergence rate of the gradient descent method with dilation of the space”,Cybernetics (1970). ·Zbl 0243.90038
[30]N.Z. Shor, ”Utilization of the method of space dilation in the minimization of convex functions”,Cybernetics (1970) 7–15.
[31]P. Wolfe, ”A method of conjugate subgradients for minimizing nondifferentiable functions”,Mathematical Programming Study 3 (1975) 145–173. ·Zbl 0369.90093
[32]P. Wolfe, ”An algorithm for the nearest point in a polytope”, IBM Research Center Rep. RC 4887 (June 1974).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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