Abstract
In this paper we study the order monotonicity of the metric projection P D when the set D is a closed convex set in an ordered Hilbert space or in an ordered uniformly convex Banach space. We indicate also some possible applications of this property.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alber, Ya. I. (1993) Generalized projection operators in Banach spaces: properties and applications, (Preprint) Department of Mathematics, Technion- Israel Institute of Technology
Ando, T. and Amemiya I. (1965) Almost everywhere convergence of prediction sequences in L p (1 < p <∞, Wahrscheinlichkeitstheorie verw. Geb. 4, 113–120.
Baiocchi, C. and Capelo, A. (1984) Variational and Quasivariational Inequalities. Applications to Free-Boundary Problems, John Wiley and Sons.
Bauschke, H. H. and Borwein, J. M. (1992) On the convergence of von Neumanns alternating projection algorithm for two sets, (Preprint), Faculty of Mathematics, Univ. Waterloo.
Beauzamy, B. (1985) Introduction to Banach spaces and their geometry, North-Holland Amsterdam, New York, Oxford.
Bernau, S. J. (1993) Isotone projection cones, (Preprint), Dep. Math. Sciences, The Univ. of Texas at El Paso.
Brunk, H. D. (1975) Uniform inequalities for conditional p-means given a σ-lattice, Ann. Prob. 3, 1025–1030.
Calvert, B. (1970) Nonlinear evolution equation in Banach lattices, Bull. Amer. Math. Soc. 76, 845–950.
Deutsch, Fr. (1992) The method of alternating orthogonal projections, Approximation Theory, Spline Functions and Applications, (Ed. S. P. Singh) Kluwer Academic Publishers, 105–121.
Dye, J. M. and Reich, S. (1991) On the unrestricted iteration of projections in Hilbert space, J. Math. Anal. Appl. 156, 101–119.
Dykstra, R. L. (1983) An algorithm for restricted least square regression, J. Amer. Statistical Assoc. 78 Nr.384, 837–842.
Golomb, M. and Tapia, R. A. (1972) The metric gradient in normed linear spaces, Num. Math. 20, 115–124.
Grotzinger, S. J. and Witzgall, C. (1984) Projections onto order simplexes, Appl. Math. Optim. 12, 247–270.
Haraux, A. (1977) How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan 29, 615–631.
Hiriart-Urruty, J. P. (1982) A what points is the projection mapping differentiable? Amer. Math. Monthly 89, 456–460
Holmes, R. B. (1973) Smoothness of certain metric projections on Hilbert spaces, Trans. Amer. Math. Soc. 184, 87–100.
Isac, G. and Németh, A. B. (1986) Monotonicity of metric projections onto positive cones in order Euclidean spaces, Ark. Math. 46, 568–576 and Corrigendum.
Isac, G. and Németh, A. B. (1990) Every generating isotone projection cone is latticial and correct, J. Math. Anal. Appl. 147, 53–62.
Isac, G. and Németh, A. B. (1990) Isotone projection cones in Hilbert spaces and the complementarity problem, Boll. Un. Math. Ital. 7, 773–802.
Isac, G. and Nérneth, A. B. (1990) Projection methods, isotone projection cones and the complementarity problem, J. Math. Anal. Appl. 153, 258–275.
Isac, G. and Németh, A. B. (1992) Isotone projection cones in Euclidean spaces, Ann. Sci. Math. Québec, 16, 35–52.
Isac, G. (1992) Complementarity Problems, Lecture Notes in Math., Springer-Verlag Nr. 1528.
Isac, G. (1992) Iterative methods for the General Order Complementarity Problem., Approximation Theory, Spline Functions and Applications, (Ed. S. P. Singh), Kluwer Academic Publishers, NATO ASI Series, 365–380.
McCormick, G.P. and Tapia, R. A. (1972) The gradient projection method under mild differentiability conditions, SIAM J. Control 10 Nr. 1, 93–98.
Neumann, J. Von (1950) Functional Operators, Vol. II. The Geometry of Orthogonal Spaces (this is a reprint of mimeographed lecture notes first distributed in 1933) Annals of Math. Studies Nr. 22 Princeton Univ. Press.
Opoitsev, V. I. (1979) A generalization of the theory of monotone and concave operators, Trans. Moscow, Math. Soc. Nr. 2, 243–279.
Parida, J., Sen, A. and Kumar, A. (1988) A linear complementarity problem involving a subgradient, Bull. Austral. Math. Soc. 37, 345–351.
Peressini, A. L. (1976) Ordered Topological Vector spaces, Harper & Row, New York, Evanston and London.
Phelps, R. R. (1985) Metric projection and the gradient projection method in Banach spaces, SIAM J. Control Optim. 23, 973–977.
Phelps, R. R. (1986) The gradient projection methods using Curry’s step-length” SIAM J. Control Optim. 24 692–699.
Robinson, S. M. (1992) Nonsigularity and symmetry for linear normal maps, Math. Programming 62, 415–425.
Robinson, S. M. (1992) Normal maps induced by linear transformations, Math. Oper. Research 17 Nr. 3, 691–714.
Shapiro, A. (1987) On differentiability of metric projections in R n(I): Boundary case, Proc Amer. Math. Soc. 99, 123–128.
Shi, P. (1991) Equivalence of variational inequalities with Wienner-Hoph equation, Proc. Amer. Math. Soc. 111 Nr.2, 339–346.
Zarantonello, E. H. (1971) Projections on convex sets in Hilbert spaces and spectral theory, Contribution to Nonlinear Functional Analysis, (Ed. E. H. Zarantonello) Acad Press, New York, 237–424.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Isac, G. (1995). On the Order Monotonicity of the Metric Projection Operator. In: Singh, S.P. (eds) Approximation Theory, Wavelets and Applications. NATO Science Series, vol 454. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8577-4_23
Download citation
DOI: https://doi.org/10.1007/978-94-015-8577-4_23
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4516-4
Online ISBN: 978-94-015-8577-4
eBook Packages: Springer Book Archive