On differentiability of metric projections in $\textbf {R}^ n$. I. Boundary case
HTML articles powered by AMS MathViewer
- by Alexander Shapiro PDF
- Proc. Amer. Math. Soc. 99 (1987), 123-128 Request permission
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.References
- Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
- Herman Chernoff, On the distribution of the likelihood ratio, Ann. Math. Statistics 25 (1954), 573–578. MR 65087, DOI 10.1214/aoms/1177728725
- John M. Danskin, The theory of max-min and its application to weapons allocation problems, Econometrics and Operations Research, V, Springer-Verlag New York, Inc., New York, 1967. MR 0228260, DOI 10.1007/978-3-642-46092-0
- V. F. Dem′yanov and A. M. Rubinov, On quasidifferentiable mappings, Travaux Sém. Anal. Convexe 11 (1981), no. 2, 3–21 (English, with German and Russian summaries). MR 694799
- 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), no. 4, 615–631. MR 481060, DOI 10.2969/jmsj/02940615
- 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 207448, DOI 10.1016/0022-247X(67)90163-1
- G. P. McCormick and R. A. Tapia, The gradient projection method under mild differentiability conditions, SIAM J. Control 10 (1972), 93–98. MR 0319578, DOI 10.1137/0310009
- R. R. Phelps, Metric projections and the gradient projection method in Banach spaces, SIAM J. Control Optim. 23 (1985), no. 6, 973–977. MR 809544, DOI 10.1137/0323055
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683, DOI 10.1515/9781400873173
- Alexander Shapiro, Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints, Biometrika 72 (1985), no. 1, 133–144. MR 790208, DOI 10.1093/biomet/72.1.133
- Alexander Shapiro, Second order sensitivity analysis and asymptotic theory of parametrized nonlinear programs, Math. Programming 33 (1985), no. 3, 280–299. MR 816106, DOI 10.1007/BF01584378
- Eduardo H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I. Projections on convex sets, Contributions to nonlinear functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971) Academic Press, New York, 1971, pp. 237–341. MR 0388177
Additional Information
- © Copyright 1987 American Mathematical Society
- 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