Abstract
In the paper, the behaviour of variable metric methods along geodesics is analyzed. First, a general framework is given in the case of Riemannian submanifolds in R n, then two general convergence theorems for a wide class of nonlinear optimization methods are proved to find a stationary point or a local optimum point of a smooth function defined on a compact set of a Riemannian manifold and the rate of a convergence is studied. These methods and theorems should be extended in such a way that penalty methods be generalized in the case of inequality constraints defined on Riemannian manifolds.
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References
Bertsekas, D. P., Constrained optimization and Lagrange multiplier methods, Academic Press, New York, London, 1982.
den Hertog, D., Roos, C. and Terlaky, T., On the classical logarithmic barrier method for a class of smooth convex programming problems, Journal of Optimization Theory and Applications 73 (1992) 1–25.
Ferreira, O. P. and Oliveira, P. R., Subgradient algorithm on Riemannian manifolds, Publicacoes Técnicas, Rio de Janeiro (1995).
Fletcher, R., A general quadratic programming algorithm, Journal of the Institute of Mathematics and Its Applications 7 (1971) 76–91.
Forgo, F., A method for solving nonlinear programming problems approximately, Szigma 1 (1969) 67–75. (in Hungarian)
Gabay, D., Minimizing a differentiable function over a differentiable manifold, Journal of Optimization Theory and Applications 37 (1982) 177–219.
Giannessi, F., Theorems of the alternative and optimality conditions, Journal of Optimization Theory and Applications 42 (1984) 331–365.
Gill, P. E., Murray, W., Saunders, M. A., Tomlin, J A and Wright, M. H., On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method, Mathematical Programming 36 (1986) 183209.
Gonzaga, C. C., Polynomial affine algorithms for linear programming, Mathematical Programming 49 (1990) 7–21.
Iri, M. and Imai, H., A multiplicative barrier function method for linear programming, Algorithmica 1 (1986) 455–482.
Jarre, F., Interior-points methods for convex programming, Applied Mathematics and Optimization 26 (1992) 287–311.
Karmarkar, N., Riemannian geometry underlying interior points methods for linear programming, Contemporary Mathematics 114 (1990) 51–76.
Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Inter-science Publishers, New-York, London, Sydney, 1969.
Lichnewsky, A, Minimization de fonctionnelle définies sur une variété par la method du gradient conjugué, Thèse de Doctorat d’État, Université de Paris-Sud, Paris, France, 1979.
Luenberger, D. G., The gradient projection methods along geodesics, Management Science 18 (1972) 620–631.
Luenberger, D. G., Introduction to linear and nonlinear programming, Addison-Wesley Publishing Company, Reading, 1973.
Mehrotra, S. and Sun, J., An interior point algorithm for solving smooth convex programs based on Newton’s method, Mathematical Developments Arising from Linear Programming, J. C. Lagarias and M. J. Todd, eds., Contemporary Mathematics 114 (1990) 265–284.
Nesterov, Y. E. and Nemirovsky, A. S., Self-concordant functions and polynomial time methods in convex programming, Report, Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, USSR, 1989.
Neto, J. X. C. and Oliveira, P. R., Geodesic methods in Riemannian manifolds, Publicacoes Técnicas, Rio de Janeiro (1995).
Ortega, J. M. and Rheinboldt, N. C., Iterative solution of nonlinear equations in several variables, Academic Press, New York, New York, 1970.
Pappalardo, M., Image space approach to penalty methods, Journal of Optimization Theory and Applications 64 (1990) 141–152.
Polak, E., Computational methods in optimization, A unified approach, Academic Press, New York, New York, 1971.
Rapcsâk, T., An exterior point algorithms for solving convex nonlinear programming problems, Alkalmazott Matematikai Lapok 1 (1975) 357–354. (in Hungarian)
Rapcsâk, T., Geodesic convexity in nonlinear optimization, Journal of Optimization Theory and Applications 69 (1991) 169–183.
Rapcsâk, T. and Thang, T. T., Nonlinear coordinate representations of smooth optimization problems, Journal of Optimization Theory and Applications 86 (1995) 459–489.
Rapcsâk, T., Tensor applications of smooth nonlinear complementarity systems, in: Variational inequalities and network equilibrium problems, eds.: F. Giannessi and A. Maugeri, Plenum Press (1995) 235–249.
Rapcsâk, T. and Thang, T. T., A class of polynomial variable metric algorithms for linear optimization, Mathematical Programming 74 (1996) 319–331.
Schwartz, J. T., Nonlinear functional analysis, Gordon and Breach Science Publishers, New York, 1969.
Tanabe, K., A geometric method in nonlinear programming, Journal of Optimization Theory and Applications 30 (1980) 181–210.
Udriste, C., Convex functions and optimization methods on Riemannian manifolds, Kluwer Academic Publishers, Boston, 1994.
Yamashita, H., A differential equation approach to nonlinear programming, Mathematical Programming 18 (1980) 155–168.
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Rapcsák, T. (1998). Variable metric methods along geodetics. In: Giannessi, F., Komlósi, S., Rapcsák, T. (eds) New Trends in Mathematical Programming. Applied Optimization, vol 13. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-2878-1_19
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DOI: https://doi.org/10.1007/978-1-4757-2878-1_19
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