Abstract
In this paper we present ε-optimality conditions of the Kuhn-Tucker type for points which are within ε of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints, linear equality constraints and abstract constraints. Such ε-optimality conditions are of interest for theoretical consideration as well as from the computational point of view. Some illustrative applications are made. Thus we derive an expression for the ε-subdifferential of a general convex ‘max function’. We also show how the ε-optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.R. Colville, “A comparative study on nonlinear programming codes”, IBM Scientific Center Report 320-2949, New York (1968).
V.F. Dem'yanov and V.N. Malozemov,Introduction to minimax (Wiley, New York, 1974).
J.-B. Hiriart-Urruty, “Lipschitzr-continuity of the approximate subdifferential of a convex function”,Mathematica Scandinavica 47 (1980) 123–134.
J.-B. Hiriart-Urruty, “ε-Subdifferential calculus”, presented at the Colloquium on Convex Analysis and Optimization, Imperial College, London, 19–29 February 1980 (in support of A.D. Ioffe), preprint of the Department of Applied Mathematics, Université de Clermont II (1980).
W. Hock and K. Schittkowski,Test examples for nonlinear programming codes, Lecture Notes in Economics and Mathematical Systems 187 (Springer, Berlin, 1981).
S.S. Kutateladze, “Convex ε-programming”,Doklady Akademii Nauk SSSR 245 (1979) 1048–1050. (Translated as:Soviet Mathematics Doklady 20 (1979) 391–393).
C. Lemarechal, “An extension of Davidon methods to non differentiable problems”Mathematical Programming Study 3 (1975) 95–109.
C. Lemarechal, “Extensions diverses des méthodes de gradient et applications”, Thèse d'Etat, Université de Paris IX-Dauphine, Paris (1980).
C. Lemarechal and R. Mifflin, eds.,Nonsmooth optimization, Proceedings of a IIASA Workshop, March 28–April 8, 1977 (Pergamon Press, Oxford, 1978).
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.
V.L. Levin, “Application of E. Helly's theorem to convex programming, problems of best approximation and related questions”,Matematiceskii Sbornik 79 (1969) 250–263. (Translated as:Mathematics of the USSR-Sbornik 8 (1969) 235–247).
K. Madsen and H. Schjaer-Jacobsen, “Linearly constrained minimax optimization”,Mathematical Programming 14 (1978) 208–223.
R. Mifflin, “An algorithm for constrained optimization with semismooth functions”,Mathematics of Operations Research 2 (1977) 191–207.
R. Mifflin, “A stable method for solving certain constrained least squares problems”,Mathematical Programming 16 (1979) 141–158.
R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, NJ, 1970).
R. Saigal, “The fixed point approach to nonlinear programming”,Mathematical Programming Study 10 (1979) 142–157.
M. Valadier, “Sous-différentiels d'une borne supérieure et d'une somme continue de fonctions convexes”,Comptes Rendus de l'Académie des Sciences de Paris 268 (1969) 39–42.
P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions”,Mathematical Programming Study 3 (1975) 145–173.
Author information
Authors and Affiliations
Additional information
Research sponsored by W.T.O.C.D. CA79-356 and 79-360.
Rights and permissions
About this article
Cite this article
Strodiot, J.J., Nguyen, V.H. & Heukemes, N. ε-Optimal solutions in nondifferentiable convex programming and some related questions. Mathematical Programming 25, 307–328 (1983). https://doi.org/10.1007/BF02594782
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02594782