Abstract
The properties of geodesic convex functions defined on a connected RiemannianC 2 k-manifold are investigated in order to extend some results of convex optimization problems to nonlinear ones, whose feasible region is given by equalities and by inequalities and is a subset of a nonlinear space.
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Ortega, J. M., andRheinboldt, W. C.,Iterative Solution of Nonlinear Equations, Academic Press, New York, New York, 1970.
Avriel, M.,Nonlinear Programming-Analysis and Methods, Prentice-Hall, Englewood Cliffs, New Jersey, 1976.
Ben-Tal, A.,On Generalized Means and Generalized Convex Functions, Journal of Optimization Theory and Applications, Vol. 21, pp. 1–13, 1977.
Prenowitz, W., andJantosciak, J.,Join Geometries, Springer, New York, New York, 1979.
Zimmermann, K.,A Generalization of Convex Functions, Ekonomicko-Matematicky Obzor, Vol. 15, pp. 147–158, 1979.
Avriel, M., andZang, I.,Generalized Arcwise Connected Functions and Characterizations of Local-Global Properties, Journal of Optimization Theory and Applications, Vol. 32, pp. 407–425, 1980.
Martin, D. H.,Connected Level Sets, Minimizing Sets, and Uniqueness in Optimization, Journal of Optimization Theory and Applications, Vol. 36, pp. 71–93, 1982.
Horst, R.,A Note on Functions Whose Local Mimima Are Global, Journal of Optimization Theory and Applications, Vol. 36, pp. 457–463, 1982.
Hartwig, H.,On Generalized Convex Functions, Optimization, Vol. 14, pp. 49–60, 1983.
Singh, C.,Elementary Properties of Arcwise Connected Sets and Functions, Journal of Optimization Theory and Applications, Vol. 41, pp. 377–387, 1983.
Horst, R.,Global Optimization in Arcwise Connected Metric Spaces, Journal of Optimization Theory and Applications, Vol. 104, pp. 481–483, 1984.
Rapcs↼, T.,Convex Programming on Riemannian Manifold, System Modelling and Optimization, Proceedings of the 12th IFIP Conference, Edited by A. Prékopa, J. Szelezsán, and B. Strazicky, Springer-Verlag, Berlin, Germany, pp. 733–741, 1986.
Nozicka, F.,Affin-Geodätische Konvexer Hyperflächen als Lösungen eines Bestimmten Lagrange'schen Variationsproblems, Preprint No. 152, Sektion Mathematik, Humboldt University, Berlin, Germany, 1987.
Rapcsák, T.,Arcwise-Convex Functions on Surfaces, Publicationes Mathematicae, Vol. 34, pp. 35–41, 1987.
Castagnoli, E., andMazzoleni, P.,Generalized Connectedness for Families of Arcs, Optimization, Vol. 18, pp. 3–16, 1987.
Horst, R., andThach, P. T.,A Topological Property of Limes-Arcwise Strictly Quasiconvex Functions, Journal of Mathematical Analysis and Applications, Vol. 134, pp. 426–430, 1988.
Luenberger, D. G.,The Gradient Projection Methods along Geodesics, Management Science, Vol. 18, pp. 620–631, 1972.
Luenberger, D. G.,Introduction to Linear and Nonlinear Programming, Addision-Wesley Publishing Company, Reading, Massachusetts, 1973.
Hicks, N. J.,Notes on Differential Geometry, Van Nostrand Publishing Company, Princeton, New Jersey, 1965.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1969.
Szabó, Z. I.,Hilbert's Fourth Problem, Advances in Mathematics, Vol. 59, pp. 185–300, 1986.
Rapcsák, T.,Minimum Problems on Differentiable Manifolds, Optimization, Vol. 20, pp. 3–13, 1989.
Gabay, D.,Minimizing a Differentiable Function over a Differentiable Manifold, Journal of Optimization Theory and Applications, Vol. 37, pp. 177–219, 1982.
Bishop, R. L., andCrittenden, R. J.,Geometry of Manifolds, Academic Press, New York, New York, 1964.
Cottle, R. W., Giannessi, F., andLions, J. L., Editors,Variational Inequalities and Complementarity Problems, John Wiley and Sons, New York, New York, 1980.
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Communicated by F. Giannessi
This research was supported in part by the Hungarian National Research Foundation, Grant No. OTKA-1044.
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Rapcsák, T. Geodesic convexity in nonlinear optimization. J Optim Theory Appl 69, 169–183 (1991). https://doi.org/10.1007/BF00940467
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DOI: https://doi.org/10.1007/BF00940467