Abstract
Given a non-decreasing family of closed convex sets C(τ) depending on a real parameter t and an affine variety W, we consider the problem of finding the infimum τ̄ of the values τ such that C(τ) ∩ W is non-empty, and the corresponding set of solutions S = C(τ̄) ∩ W. This problem is shown to be equivalent to the minimization of a quasi-convex function over a convex set. A dual algorithm is given for solving this problem and its convergence is proved.
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References
AUSLENDER, A.: Méthodes numériques pour la décomposition et la minimisation de fonctions non-différentiables. Numerische Math. 18, (1971), 213–223.
BERTSEKAS, D.P. and MITTER, S.: A descent numerical method for optimization problems with non-differentiable cost functionals, S.I.A.M. Journal on Control, 11, (1973), 637–652.
CARASSO, C.: Un algorithme de minimisation de fonctions convexes avec ou sans contraintes : “l’algorithme d’échange”. Prépublication N° 3, Math., Univ. de Saint-Etienne, 7th IFIP Conference on Optimization Techniques, Springer-Verlag (1975).
CARASSO, C. ET LAURENT, P.J.: Un algorithme de minimisation en chaîne en optimisation convexe. Séminaire d’Analyse Numérique, Grenoble (29 janvier 1976) à paraître: S.I.A.M. Journal on Control.
CARASSO, C. ET LAURENT P.J.: Un algorithme général pour l’approximation au sens de Tchebycheff de fonctions bornées sur un ensemble quelconque. Approximations-Kolloquium, Bonn (June 8–12, 1976). Lectures Notes in Math., 556, Springer-Verlag.
CHENEY, E.W.: Introduction to Approximation Theory; McGraw-Hill (1966).
CHENEY, E.W. and GOLDSTEIN, A.A.: Newton’s method for convex programming and Tchebyscheff approximation. Num. Math. 1, (1959), 253–268.
DEMJANOV, V.F.: Algorithms for some minimax problems. Journal of Computer and System Sciences 2, (1968), 342–380.
GOLDSTEIN, A.A.: Constructive real analysis. Harper’s series in modern mathematics, Harper and Row, (1967).
LAURENT, P.J.: Théorèmes de caractérisation d’une meilleure approximation dans un espace normé et généralisation de l’algorithme de Rémès. Num. Math. 10, (1967), 190–208.
LAURENT, P.J.: Approximation et Optimisation, Hermann, Paris, (1972).
LAURENT, P.J.: Exchange algorithm in convex analysis. Conf. on Approximation Theory, the Univ. of Texas, Austin, (1973), Acad. Press.
LAURENT, P.J. and CARASSO, C.: An algorithm of successive minimization in convex programming. IX International Symposium on Mathematical Programming, Budapest, 23–27 Août 1976 (to appear). Rapport de Recherche n° 49 (Août 1976), Mathématiques Appliquées et Informatique, Université de Grenoble.
LEMARECHAL, C.: An extension of Davidon methods to nondifferentiable problems. Math. Programming Study 3, (1975), 95–109.
RIMES, E. : Sur le calcul effectif des polynomes d’approximation de Tchebycheff. C.R.A.S., Paris, 199, (1934), 337–340.
RIMES, E. : General computational methods for Chebyshev approximation. Problems with real parameters entering linearly. Izdat. Akad. Nauk Ukrainsk. SSR, Kiev (1957), Atomic Energy Commission Translations 4491.
ROCKAFELLAR, R.T.: Convex analysis, Princ. Univ. Press (1970).
STIEFEL, E.L. : Über diskrete und lineare Tschebyscheff-Approximationen. Num. Math. 1, (1959), 1–28.
STIEFEL, E.L. : Numerical Methods of Tschebycheff Approximation. In “On Numerical Approximation”, R. Langer Ed., Univ. of Wisconsin, (1959), 217–232.
STIEFEL, E.L. : Note on Jordan Elimination, Linear Programming and Tchebycheff Approximation. Num. Math. 2, (1960), 1–17.
WOLFE, P. : A method of conjugate subgradients for minimizing nondifferentiable functions. Math. Programming Study 3, (1975), 145–173.
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Carasso, C., Laurent, P.J. (1977). A Dual Algorithm in Quasi-Convex Optimization. In: Auslender, A. (eds) Convex Analysis and Its Applications. Lecture Notes in Economics and Mathematical Systems, vol 144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48298-4_7
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DOI: https://doi.org/10.1007/978-3-642-48298-4_7
Publisher Name: Springer, Berlin, Heidelberg
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