Abstract
The study of the convergence of algorithms of optimization obtained by composition or union, taken in sense of the relaxation, is done. After having recalled the Zangwill’s theorem and given two extensions we study the obtainment of generalized fixed points in the framework of the composition or the union of algorithms obtained in a free steering way for, firstly functions having a unique maximum over some particular subsets, ranges of the current point, and secondly for general functions. The validity of the different hypotheses is discussed through some examples.
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References
A. Auslender, “Méthodes numériques pour la décomposition et la minimisation de fonctions non différentiables”, Numerische Mathematik 18 (1971) 213–223.
J. Cea et R. Glowinski, “Sur les méthodes d’optimisation par relaxation”, Revue Française d’Automatique, d’Informatique et de Recherche Opérationnelle (Décembre 1973) 5–32.
R. Boyer, “Quelques algorithmes diagonaux en optimisation convexe”, Thèse 3ème cycle, Université de Provence (1974).
D. Chazan and W. Miranker, “Chaotic relaxation”, Linear Algebra and its Applications 2 (1969) 199–222.
J. Dubois, “Theorems of convergence for improved nonlinear programming algorithms”, Operations Research 21 (1973) 328–332.
J.C. Fiorot et P. Huard, “Composition et réunion d’algorithmes généraux”, Compte-Rendus Académie des Sciences Paris, tome 280 (2 juin 1975), Série A, 1455–1458—Séminaire d’Analyse Numérique No. 229, Université de Grenoble (Mai 1975).
P. Huard, “Optimization algorithms and point-to-set maps”, Mathematical Programming 8 (1975) 308–331.
P. Huard, “Extensions of Zangwill’s theorem”, this volume.
P. Huard, “A method of centers by upper-bounding functions with applications”, in: J.B. Rosen, O.L. Mangasarian, K. Ritter, eds., Nonlinear programming (Academic Press, New-York, 1970) 1–30.
B. Martinet et A. Auslender, “Méthodes de décomposition pour la minimisation d’une fonction sur un espace produit”, SIAM Journal on Control 12 (1974) 635–643.
G.G.L. Meyer, “Conditions de convergence pour les algorithmes itératifs monotones, autonomes et non déterministes”, Revue Française d’Automatique, d’Informatique et de Recherche Opérationnelle 11 (1977) 61–74.
G.G.L. Meyer, “Convergence conditions for a type of algorithm model”, SIAM Journal on Control and Optimization 15 (1977) 779–784.
G.G.L. Meyer, “A systematic approach to the synthesis of algorithms”, Numerische Mathematik 24 (1975) 277–289.
R. Meyer, “On the convergence of algorithms with restart”, SIAM Journal of Numerical Analysis 13 (1976) 696–704.
R. Meyer, “Sufficient conditions for the convergence of monotonic mathematical programming algorithms”, Journal of Computer and System Sciences 12 (1976) 108–121.
R. Meyer, “A comparison of the forcing function and point-to-set mapping approaches to convergence analysis”, SIAM Journal on Control and Optimization 15 (1977) 699–715.
J.C. Miellou, “Méthode de Jacobi, Gauss-Seidel, sur-(sous) relaxation par blocs appliquée à une classe de problèmes non linéaires”, Compte-Rendus Académie des Sciences Paris tome 273 (20 Décembre 1971) Série A, 1257–1260.
J.C. Miellou, “Algorithme de relaxation chaotique à retards”, Revue Française d’Automatique, d’Informatique et de Recherche Opérationnelle (Avril 1975) 55–82.
J.M. Ortega et W.C. Rheinboldt, Iteration solution of nonlinear equations in several variables (Academic Press, New York, 1970).
J.M. Ortega et W.C. Rheinboldt, “A general convergence result for unconstrained minimization methods”, SIAM Journal of Numerical Analysis 9 (1972) 40–43.
A.M. Ostroswki, Solution of equations and systems of equations, (Academic Press, New York, 1966).
B.T. Poljak, “Existence theorems and convergence of minimizing sequences in extremum problems with restrictions”, Soviet Mathematics Doklady 7 (1966) 72–75.
E. Polak, Computational methods in optimization, a unified approach (Academic Press, New York 1971).
F. Robert, “Contraction en norme vectorielle: convergence d’itérations chaotique pour des équations de point fixe à plusieurs variables”, Colloque d’Analyse Numérique (Gourette 1974).
F. Robert, M. Charnay et F. Musy, “Itérations chaotiques série parallèle pour des équations non linéaires de point fixe”, Aplikace Mathematiky 20 (1975) 1–37.
S. Schechter, “Relaxation methods for linear equations”, Communications on Pure and Applied Mathematics 12 (1959) 313–335.
S. Schechter, “Minimization of a convex function by relaxation”, in: Abadie, ed., Integer and nonlinear programming (North Holland, Amsterdam, 1970) 177–189.
R.S. Varga, Matrix iterative analysis (Prentice Hall, Englewood Cliffs, NJ, 1962).
W.I. Zangwill, Nonlinear programming: a unified approach (Prentice Hall, Englewood Cliffs, NJ, 1969).
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© 1979 The Mathematical Programming Society
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Fiorot, J.C., Huard, P. (1979). Composition and union of general algorithms of optimization. In: Huard, P. (eds) Point-to-Set Maps and Mathematical Programming. Mathematical Programming Studies, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120844
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DOI: https://doi.org/10.1007/BFb0120844
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