# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $ϵ$- 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.

##### MSC:
 90C25 Convex programming 90C55 Methods of successive quadratic programming type 65K05 Mathematical programming (numerical methods)
##### References:
 [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. · 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, $ϵ$-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). [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. [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. [15] R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, 1970). [16] R.T. Rockafellar, ”Monotone operators and the proximal point algorithm”,SIAM Journal on Control and Optimization 14 (1976) 875–898. [17] P. Wolfe, ”A method of conjugate subgradients for minimizing nondifferentiable functions”,Mathematical Programming Study 3 (1975) 145–173.