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On differentiability of metric projections in $ {\bf R}\sp n$. I. Boundary case


Author: Alexander Shapiro
Journal: Proc. Amer. Math. Soc. 99 (1987), 123-128
MSC: Primary 41A50; Secondary 41A65
DOI: https://doi.org/10.1090/S0002-9939-1987-0866441-7
MathSciNet review: 866441
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Abstract: This paper is concerned with metric projections onto a closed subset $ S$ of a finite-dimensional normed space. Necessary and in a sense sufficient conditions for directional differentiability of a metric projection at a boundary point of $ S$ are given in terms of approximating cones. It is shown that if $ S$ is defined by a number of inequality constraints and a constraint qualification holds, then the approximating cone exists.


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  • [1] J. P. Aubin and A. Cellina, Differential inclusions, Grundlehren Math. Wiss., Band 264, Springer-Verlag, Berlin and New York, 1984. MR 755330 (85j:49010)
  • [2] H. Chernoff, On the distribution of the likelihood ratio, Ann. Math. Statist. 25 (1954), 573-578. MR 0065087 (16:381k)
  • [3] J. M. Danskin, The theory of max-min and its applications to weapons allocation problems, Econometrics and Operations Research, vol. 5, Springer-Verlag, Berlin and New York, 1967. MR 0228260 (37:3843)
  • [4] V. F. Demyanov and A. M. Rubinov, On quasidifferentiable mappings, Math. Operationsforsch Statist. Ser. Optimization 14 (1983), 3-21. MR 694799 (84j:49015)
  • [5] A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan 29 (1977), 615-631. MR 0481060 (58:1207)
  • [6] O. L. Mangasarian and S. Fromovitz, The Fritz John necessary optimality conditions in the presence of equality and inequality constraints, J. Math. Anal. Appl. 17 (1967), 37-47. MR 0207448 (34:7263)
  • [7] G. P. McCormick and R. Tapia, The gradient projection method under mild differentiability conditions, SIAM J. Control 10 (1972), 93-98. MR 0319578 (47:8121)
  • [8] R. R. Phelps, Metric projections and the gradient projectin method in Banach spaces, SIAM J. Control and Optim. 23 (1985), 973-977. MR 809544 (86m:90177)
  • [9] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N.J., 1970. MR 0274683 (43:445)
  • [10] A. Shaprio, Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints, Biometrika 72 (1985), 133-144. MR 790208 (87a:62037)
  • [11] -, Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs, Mathematical Programming 33 (1985), 280-299 MR 816106 (87c:90218)
  • [12] E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory, Contributions to Nonlinear Functional Analysis, Academic Press, New York, 1971, pp. 237-424. MR 0388177 (52:9014)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0866441-7
Keywords: Metric projection, normed space, distance function, directional differentiability, approximating cone
Article copyright: © Copyright 1987 American Mathematical Society

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