On a convex set with nondifferentiable metric projection
نویسندگان
چکیده
A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with C boundary which possesses the same property.
منابع مشابه
Differentiability of the Metric Projection in Hilbert Space
A study is made of differentiability of the metric projection P onto a closed convex subset K of a Hubert space H. When K has nonempty interior, the Gateaux or Fréchet smoothness of its boundary can be related with some precision to Gateaux or Fréchet differentiability properties of P. For instance, combining results in §3 with earlier work of R. D. Holmes shows that K has a C2 boundary if and ...
متن کاملVariational inequalities on Hilbert $C^*$-modules
We introduce variational inequality problems on Hilbert $C^*$-modules and we prove several existence results for variational inequalities defined on closed convex sets. Then relation between variational inequalities, $C^*$-valued metric projection and fixed point theory on Hilbert $C^*$-modules is studied.
متن کاملCharacterization of Properly Efficient Solutions for Convex Multiobjective Programming with Nondifferentiable vanishing constraints
This paper studies the convex multiobjective optimization problem with vanishing constraints. We introduce a new constraint qualification for these problems, and then a necessary optimality condition for properly efficient solutions is presented. Finally by imposing some assumptions, we show that our necessary condition is also sufficient for proper efficiency. Our results are formula...
متن کاملError bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification
A global error bound is given on the distance between an arbitrary point in the n-dimensional real space R n and its projection on a nonempty convex set determined by m convex, possibly nondiierentiable, inequalities. The bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong Slater con...
متن کاملA Note on Suns in Convex Metric Spaces
We prove that in a convex metric space (X, d), an existence set K having a lower semi continuous metric projection is a δ-sun and in a complete M -space, a Chebyshev set K with a continuous metric projection is a γ-sun as well as almost convex.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Optimization Letters
دوره 9 شماره
صفحات -
تاریخ انتشار 2015