An Example of a Banach Space with Non-Lipschitzian Metric Projection on Any Straight Line

2021 ◽  
Vol 109 (1-2) ◽  
pp. 184-191
Author(s):  
L. Sh. Burusheva
Keyword(s):  
Banach Space ◽  
Metric Projection ◽  
Straight Line

ADAPTIVE OPERATOR EXTRAPOLATION METHOD FOR VARIATIONAL INEQUALITIES IN BANACH SPACES

2021 ◽  
Vol 5 ◽  
pp. 82-92
Author(s):  
Sergei Denisov ◽  
◽  
Vladimir Semenov ◽  
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Operations Research ◽  
Optimization Problems ◽  
Metric Projection ◽  
Feasible Set ◽  
Uniformly Smooth ◽  
Developing Area ◽  
Lipschitz Operators

Many problems of operations research and mathematical physics can be formulated in the form of variational inequalities. The development and research of algorithms for solving variational inequalities is an actively developing area of applied nonlinear analysis. Note that often nonsmooth optimization problems can be effectively solved if they are reformulated in the form of saddle point problems and algorithms for solving variational inequalities are applied. Recently, there has been progress in the study of algorithms for problems in Banach spaces. This is due to the wide involvement of the results and constructions of the geometry of Banach spaces. A new algorithm for solving variational inequalities in a Banach space is proposed and studied. In addition, the Alber generalized projection is used instead of the metric projection onto the feasible set. An attractive feature of the algorithm is only one computation at the iterative step of the projection onto the feasible set. For variational inequalities with monotone Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, a theorem on the weak convergence of the method is proved.

Approximation properties of sets with bounded complements

1981 ◽  
Vol 89 (1-2) ◽  
pp. 75-86 ◽  
Author(s):  
C. Franchetti ◽  
P. L. Papini
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Metric Projection ◽  
Special Role ◽  
Approximation Properties ◽  
Chebyshev Sets

SynopsisGiven a Banach space X, we investigate the behaviour of the metric projection PF onto a subset F with a bounded complement.We highlight the special role of points at which d(x, F) attains a maximum. In particular, we consider the case of X as a Hilbert space: this case is related to the famous problem of the convexity of Chebyshev sets.

scholarly journals Metric projections and the differentiability of distance functions

1980 ◽  
Vol 22 (2) ◽  
pp. 291-312 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Banach Space ◽  
Distance Function ◽  
Closed Subset ◽  
Metric Projection ◽  
Distance Functions ◽  
Metric Projections ◽  
Fréchet Differentiable

Let M be a closed subset of a Banach space E such that the norms of both E and E* are Fréchet differentiable. It is shown that the distance function d(·, M) is Fréchet differentiable at a point x of E ∼ M if and only if the metric projection onto M exists and is continuous at X. If the norm of E is, moreover, uniformly Gateaux differentiable, then the metric projection is continuous at x provided the distance function is Gateaux differentiable with norm-one derivative. As a corollary, the set M is convex provided the distance function is differentiable at each point of E ∼ M. Examples are presented to show that some of our hypotheses are needed.

scholarly journals Characterizations of metric projections in Banach spaces and applications

1998 ◽  
Vol 3 (1-2) ◽  
pp. 85-103 ◽  
Author(s):  
Jean-Paul Penot ◽  
Robert Ratsimahalo
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Convex Subset ◽  
Metric Projection ◽  
Nonempty Closed Convex Subset ◽  
Convex Cones ◽  
Duality Mappings ◽  
Metric Projections ◽  
General Banach Space

This paper is devoted to the study of the metric projection onto a nonempty closed convex subset of a general Banach space. Thanks to a systematic use of semi-inner products and duality mappings, characterizations of the metric projection are given. Applications to decompositions of Banach spaces along convex cones and variational inequalities are presented.

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

1995 ◽  
Vol 119 (1-2) ◽  
pp. 23-36 ◽  
Author(s):  
F. S. De Blasi ◽  
P. S. Kenderov ◽  
J. Myjak
Keyword(s):  
Banach Space ◽  
Metric Projection ◽  
Starshaped Sets

scholarly journals The Representations and Continuity of the Metric Projections on Two Classes of Half-Spaces in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Zihou Zhang ◽  
Chunyan Liu
Keyword(s):  
Banach Space ◽  
Half Space ◽  
Dual Space ◽  
Bounded Operator ◽  
Metric Projection ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Bounded Operator ◽  
Metric Projections ◽  
Necessary And Sufficient

We show a necessary and sufficient condition for the existence of metric projection on a class of half-spaceKx0*,c={x∈X:x*(x)≤c}in Banach space. Two representations of metric projectionsPKx0*,candPKx0,care given, respectively, whereKx0,cstands for dual half-space ofKx0*,cin dual spaceX*. By these representations, a series of continuity results of the metric projectionsPKx0*,candPKx0,care given. We also provide the characterization that a metric projection is a linear bounded operator.

scholarly journals Best approximation and intersections of balls in Banach spaces

1979 ◽  
Vol 20 (2) ◽  
pp. 285-300 ◽  
Author(s):  
David T. Yost
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Best Approximation ◽  
Closed Subspace ◽  
Metric Projection ◽  
Continuous Selection ◽  
Linear Map ◽  
Non Linear

Let E be a Banach space, M a closed subspace of E with the 3-ball property. It is known that M is proximinal in E, and that its metric projection admits a continuous selection. This means that there is a continuous (generally non-linear) map π: E → M satisfying ‖x−π(x)‖ = d(x, M) for all x in E. Here it is shown that the same conclusion holds under a much weaker hypothesis on M, which we call the 1½-ball property. We also establish that if M has the 1½-ball property in E, then there is a continuous Hahn-Banach extension map from M* to E*.

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