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A descent algorithm for nonsmooth convex optimization. (English) Zbl 0545.90082
Summary: This paper presents a new descent algorithm for minimizing a convex function which is not necessarily differentiable. The algorithm can be implemented and may be considered a modification of the $\epsilon$- subgradient algorithm and Lemarechal’s descent algorithm. Also our algorithm is seen to be closely related to the proximal point algorithm applied to convex minimization problems. A convergence theorem for the algorithm is established under the assumption that the objective function is bounded from below. Limited computational experience with the algorithm is also reported.

90C25Convex programming
90C55Methods of successive quadratic programming type
65K05Mathematical programming (numerical methods)
Full Text: DOI
[1] D.P. Bertsekas and S.K. Mitter, ”A descent numerical method for optimization problems with nondifferentiable cost functionals”,SIAM Journal of Control 11 (1973) 637--652. · Zbl 0266.49023 · doi:10.1137/0311049
[2] V.F. Dem’yanov and V.N. Malozemov,Introduction to minimax (Wiley, New York, 1974).
[3] H. Everett III, ”Generalized Lagrange multiplier method for solving problems of optimal allocation of resources”,Operations Research 11 (1963) 399--417. · Zbl 0113.14202 · doi:10.1287/opre.11.3.399
[4] M. Gaudioso and M.F. Monaco, ”A bundle type approach to the unconstrained minimization of convex nonsmooth functions”,Mathematical Programming 23 (1982) 216--226. · Zbl 0479.90066 · doi:10.1007/BF01583790
[5] J.-B. Hiriart-Urruty, {$\epsilon$}-Subdifferential calculus, in: J.-P. Aubin and R.B. Vinter, eds.,Convex analysis and optimization, Research Notes in Mathematics Series, 57 (Pitman, London, 1982) pp. 43--92.
[6] C. Lemarechal, An extension of Davidon methods to nondifferentiable problems,Mathematical Programming Study 3 (1975) 95--109.
[7] C. Lemarecha, Bundle methods in nonsmooth optimization, in: C. Lemarechal and R. Mifflin, eds.,Nonsmooth optimization (Pergamon Press, Oxford, 1978).
[8] C. Lemarecha, ”Nonsmooth optimization and descent methods”, RR-78-4, International Institute for Applied Systems Analysis (Laxenburg, Austria, 1978).
[9] C. Lemarechal, ”Numerical experiments in nonsmooth optimization”, in: E.A. Nurminski, ed.,Progress in nondifferentiable optimization, IIASA, (Laxenburg, Austria, 1982) 61--84.
[10] C. Lemarechal and R. Mifflin, eds.,Nonsmooth optimization (Pergamon Press, Oxford, 1978). · Zbl 0398.90090
[11] C. Lemarechal and R. Mifflin, ”Global and superlinear convergence of an algorithm for one-dimensional minimization of convex functions”,Mathematical Programming 24 (1982) 241--256. · Zbl 0505.90064 · doi:10.1007/BF01585109
[12] C. Lemarechal, J.J. Strodiot and A. Bihain, ”On a bundle algorithm for nonsmooth optimization”, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming 4 (Academic Press, New York, 1981) pp. 245--282. · Zbl 0533.49023
[13] R. Mifflin, ”An algorithm for constrained optimization with semi-smooth functions”,Mathematics of Operations Research 2 (1977) 191--207. · Zbl 0395.90069 · doi:10.1287/moor.2.2.191
[14] R. Mifflin, ”A modification and an extension of Lemarechal’s algorithm for nonsmooth optimization”,Mathematical Programming Study 17 (1982) 77--90. · Zbl 0476.65047
[15] R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, 1970). · Zbl 0193.18401
[16] R.T. Rockafellar, ”Monotone operators and the proximal point algorithm”,SIAM Journal on Control and Optimization 14 (1976) 875--898. · Zbl 0358.90053
[17] P. Wolfe, ”A method of conjugate subgradients for minimizing nondifferentiable functions”,Mathematical Programming Study 3 (1975) 145--173. · Zbl 0369.90093