Strict lower semicontinuity of the level sets and invexity of a locally Lipschitz function

1995 ◽  
Vol 87 (3) ◽  
pp. 579-594 ◽  
Author(s):  
T. D. Phuong ◽  
P. H. Sach ◽  
N. D. Yen
Keyword(s):  
Lipschitz Function ◽  
Lower Semicontinuity ◽  
Level Sets ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz

scholarly journals Relaxed submonotone mappings

2003 ◽  
Vol 2003 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Tzanko Donchev ◽  
Pando Georgiev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Clarke Subdifferential ◽  
Monotone Mappings ◽  
Locally Lipschitz Function ◽  
Residual Subset ◽  
Locally Lipschitz ◽  
Whole Space ◽  
The Clarke Subdifferential

The notions ofrelaxed submonotoneandrelaxed monotonemappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.

scholarly journals Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

1993 ◽  
Vol 47 (2) ◽  
pp. 205-212 ◽  
Author(s):  
J.R. Giles ◽  
Scott Sciffer
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Separable Banach Space ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
Dini Derivative ◽  
Locally Lipschitz ◽  
Set Of Points ◽  
Clarke Derivative

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

BORNOLOGIES AND LOCALLY LIPSCHITZ FUNCTIONS

2014 ◽  
Vol 90 (2) ◽  
pp. 257-263 ◽  
Author(s):  
GERALD BEER ◽  
M. I. GARRIDO
Keyword(s):  
Metric Space ◽  
Lipschitz Function ◽  
Lipschitz Functions ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz Functions ◽  
The Family ◽  
Locally Lipschitz

AbstractLet$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $be a metric space. We characterise the family of subsets of$X$on which each locally Lipschitz function defined on$X$is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.

Optimality conditions for maximizing a locally Lipschitz function

Optimization ◽  
2005 ◽  
Vol 54 (4-5) ◽  
pp. 377-389 ◽  
Author(s):  
Joydeep Dutta
Keyword(s):  
Optimality Conditions ◽  
Lipschitz Function ◽  
Locally Lipschitz Function ◽  
Locally Lipschitz

Geometric conditions of differentiability for a regular locally Lipschitz function

1998 ◽  
Vol 14 (2) ◽  
pp. 209-222
Author(s):  
Shi Shuzhong ◽  
Wang Bingwu
Keyword(s):  
Lipschitz Function ◽  
Locally Lipschitz Function ◽  
Geometric Conditions ◽  
Locally Lipschitz

Parallel Iterative Methods for Nonlinear Programming Problems

2010 ◽  
Vol 159 ◽  
pp. 105-110
Author(s):  
Zhong Chen
Keyword(s):  
Nonlinear Programming ◽  
Objective Function ◽  
Convex Programming ◽  
Iterative Methods ◽  
Minimization Problem ◽  
Level Sets ◽  
Lipschitz Continuous ◽  
Positive Orthant ◽  
Locally Lipschitz ◽  
Parallel Iterative

In this paper, we present two parallel multiplicative algorithms for convex programming. If the objective function is differentiable and convex on the positive orthant of , and it has compact level sets and has a locally Lipschitz continuous gradient, we prove these algorithms converge to a solution of minimization problem. For the proofs there are essentially used the results of sequential methods shown by Eggermont[1].

Level sets of classes of mappings of two-step Carnot groups in a nonholonomic interpretation

2017 ◽  
Vol 60 (2) ◽  
pp. 391-400
Author(s):  
M. B. Karmanova
Keyword(s):  
Level Sets ◽  
Carnot Groups

Perfect level sets in many directions

1986 ◽  
Vol 12 (1) ◽  
pp. 176
Author(s):  
Malý
Keyword(s):  
Level Sets

LEVEL SETS OF A TYPICAL C n FUNCTION

1998 ◽  
Vol 24 (1) ◽  
pp. 83
Author(s):  
Darji ◽  
Morayne
Keyword(s):  
Level Sets

Hausdorff dimension of certain level sets associated with generalized Rademacher functions

Nonlinearity ◽  
2002 ◽  
Vol 15 (4) ◽  
pp. 1019-1027
Author(s):  
Min Wu ◽  
Li-Feng Xi
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets ◽  
Rademacher Functions
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