Abstract
We consider the problem of finding the minimum value of the upper hull ofn convex functionals on a Hilbert space, subject to convex constraints. The problem is reformulated as that of finding the minimum of the “worst” convex combination of these functionals, which eventually yields a saddle-point problem. We propose a new algorithm to solve this problem that simplifies the task of updating the dual variables. Simultaneously, the constraints can be dualized by introducing other dual multipliers. Convergence proofs are given and a concrete example shows the practical and computational advantages of the proposed algorithm and approach.
Similar content being viewed by others
References
K. J. Arrow and L. Hurwicz, Decentralization and Computation in Resource Allocation, in:Essays in Economics and Econometrics, ed. by R. W. Pfouts, 34–104, University of North Carolina Press, Rayleigh, N.C., 1960.
A. Auslender,Optimisation, Méthodes Numériques, Masson, Paris, 1976.
A. Bensoussan, J. L. Lions, and R. Temam, Sur les Méthodes de Décomposition, de Décentralisation et Applications, inMéthodes Numériques d'Analyse de Systèmes, Vol. 2, 5–190, Cahier IRIA No 11, Le Chesnay, France, 1972.
P. Bernhard and G. Bellec, On the Evaluation of Worst-Case Design with an Application to the Quadratic Synthesis Technique,Proceedings of the IFAC Symposium on Sensitivity, Adaptivity and Optimality, Ischia, Italy, 1973.
P. Bernhard and G. Cohen, Optimisation du Compromis Economique Investissement-Exploitation,Report No. E/63, Centre d'Automatique-Informatique ENSMP, Fontainebleau, France, 1979.
D. P. Bertsekas, Approximation Procedures Based on the Method of Multipliers,J. Optim. Theory and Applic. 487–510 (1977).
J. Céa,Optimisation: Théorie et Algorithmes, Dunod, Paris, 1971.
C. Charalambous and A. R. Conn, An efficient method to solve the minimax problem directly,SIAM J. Num. Anal. 162–187 (1978).
G. Cohen, Optimization by decomposition and coordination: A unified approach,IEEE Transactions on Automatic Control 23, 222–232 (1978).
G. Cohen, The auxiliary problem principle and the decomposition of optimization problems,J. Optim. Theory and Appl. 32, 277–305 (1979).
V. M. Danskin,The Theory of Max-Min, Springer-Verlag, Berlin, 1967.
S. R. K. Dutta and M. Vidyasagar, New algorithms for constrained minimax optimization,Mathematical Programming, 140–155, 1977.
V. F. Dem'yanov and V. N. Malozemov,Introduction to Minimax, Wiley, New York, 1974.
I. Ekeland and R. Temam,Convex Analysis and Variational Problems, North Holland, Amsterdam, 1976.
P. Hagedorn and J. V. Breakwell, A differential game with two pursuers and one evader,J. Optim. Theory and Appl. 18, 15–29 (1976).
L. S. Lasdon and J. D. Schoeffer, A multilevel technique for optimization,Proceedings of the Joint Automatic Control Conference, Troy, New York, 1965.
C. Lemaréchal, Nonsmooth Optimization and descent methods, RR-78-4,International Institute for Applied Systems Analysis, Austria, 1978.
C. Lemaréchal, An extension of Davidon methods to nondifferentiable problems, inMathematical Study 3, Nondifferentiable Optimization ed. by M. L. Balinski and P. Wolfe, 95–109, North Holland, Amsterdam, 1975.
D. G. Luenberger,Optimization by Vector Space Methods, Wiley, New York, 1969.
K. Madsen and H. Schjaer-Jacobsen, Linearly constrained minimax optimization,Mathematical Programming 14, 208–223 (1978).
D. D. Morrison, Optimization by least-squares,SIAM J. Num. Anal. 5:1, 83–88 (1968).
J. Medanic and M. Andjelic, Minimax solution of the multiple target problem,IEEE Transactions on Automatic Control 17 (1972).
M. Mintz and S. Hulling, Optimal evasion strategies against multiple missiles based on differential game theory,Proceedings of the IEEE Conference on Decision and Control, New Orleans, 1977.
M. Staroswiecki and Z. Bubnicki, Optimal planning of production by means of a particular cooperative equilibrium,Proceedings of the IFAC Congress, Helsinki, Finland 1978.
Author information
Authors and Affiliations
Additional information
Communicated by A. Bensoussan
This research has been supported by the Centre National de la Recherche Scientifique (CNRS-France) under Contract No. ATP-2340.
Rights and permissions
About this article
Cite this article
Cohen, G. An algorithm for convex constrained minimax optimization based on duality. Appl Math Optim 7, 347–372 (1981). https://doi.org/10.1007/BF01442126
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01442126