scholarly journals Reconstruction of the Clarke Subdifferential by the Lasry–Lions Regularizations

2000 ◽  
Vol 248 (2) ◽  
pp. 415-428 ◽  
Author(s):  
Pando Gr. Georgiev ◽  
Nadia P. Zlateva
Keyword(s):  
Clarke Subdifferential ◽  
The Clarke Subdifferential

Existence of solutions for a second order boundary value problem with the Clarke subdifferential

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2763-2771 ◽  
Author(s):  
Dalila Azzam-Laouir ◽  
Samira Melit
Keyword(s):  
Boundary Value Problem ◽  
Differential Inclusion ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Clarke Subdifferential ◽  
Lipschitzian Function ◽  
The Clarke Subdifferential

In this paper, we prove a theorem on the existence of solutions for a second order differential inclusion governed by the Clarke subdifferential of a Lipschitzian function and by a mixed semicontinuous perturbation.

Extensions Of Subdifferential Calculus Rules in Banach Spaces

1996 ◽  
Vol 48 (4) ◽  
pp. 834-848 ◽  
Author(s):  
A. Jourani ◽  
L. Thibault
Keyword(s):  
Banach Spaces ◽  
Clarke Subdifferential ◽  
Subdifferential Calculus ◽  
Finite Dimensional ◽  
Calculus Rules ◽  
Approximate Subdifferential ◽  
The Clarke Subdifferential

AbstractThis paper is devoted to extending formulas for the geometric approximate subdifferential and the Clarke subdifferential of extended-real-valued functions on Banach spaces. The results are strong enough to include completely the finite dimensional setting.

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.

On the Clarke Subdifferential of an Integral Functional on Lp, 1 ≤ p < ∞

1998 ◽  
Vol 41 (1) ◽  
pp. 41-48 ◽  
Author(s):  
E. Giner
Keyword(s):  
Growth Condition ◽  
Upper Bound ◽  
Directional Derivative ◽  
Integral Functional ◽  
Clarke Subdifferential ◽  
The Clarke Subdifferential

AbstractGiven an integral functional defined on Lp, 1 ≤ p < ∞, under a growth condition we give an upper bound of the Clarke directional derivative and we obtain a nice inclusion between the Clarke subdifferential of the integral functional and the set of selections of the subdifferential of the integrand.

scholarly journals Convergence of a Time Discretization for a Nonlinear Second-Order Inclusion

2017 ◽  
Vol 61 (1) ◽  
pp. 93-120
Author(s):  
Krzysztof Bartosz ◽  
Leszek Gasiński ◽  
Zhenhai Liu ◽  
Paweł Szafraniec
Keyword(s):  
Source Term ◽  
Existence Of Solutions ◽  
Time Discretization ◽  
Second Order ◽  
Clarke Subdifferential ◽  
Smooth Potential ◽  
Locally Lipschitz ◽  
Time Differential ◽  
Second Order In Time ◽  
The Clarke Subdifferential

AbstractWe study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

scholarly journals NECESSARY OPTIMALITY CONDITIONS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS

2003 ◽  
Vol 8 (2) ◽  
pp. 165-174 ◽  
Author(s):  
Davide La Torre
Keyword(s):  
Optimality Conditions ◽  
Vector Optimization ◽  
Optimization Problems ◽  
Necessary Optimality Conditions ◽  
Clarke Subdifferential ◽  
Vector Optimization Problems ◽  
Generalized Derivative ◽  
Nonsmooth Vector Optimization ◽  
Vector Case ◽  
The Clarke Subdifferential

In this paper we introduce a notion of generalized derivative for nonsmooth vector functions in order to obtain necessary optimality conditions for vector optimization problems. This definition generalizes to the vector case the notion introduced by Michel and Penot and extended by Yang and Jeyakumar. This generalized derivative is contained in the Clarke subdifferential and then the corresponding optimality conditions are sharper than the Clarke's ones.

ON FIRST-ORDER NECESSARY CONDITIONS FOR OPTIMALITY OF REGULAR GENERALIZED SEMI-INFINITE PROGRAMMING

2012 ◽  
Vol 05 (04) ◽  
pp. 1250054
Author(s):  
Nader Kanzi
Keyword(s):  
Constraint Qualification ◽  
Necessary Conditions ◽  
Natural Extension ◽  
Clarke Subdifferential ◽  
Optimal Solutions ◽  
First Order ◽  
Locally Lipschitz ◽  
Necessary Conditions For Optimality ◽  
Infinite Programming ◽  
The Clarke Subdifferential

This paper is concerned with the optimality for generalized semi-infinite programming (GSIP) with nondifferentiable and nonconvex (but being regular in Clarke sense) constraint functions. The objective function is only locally Lipschitz. We consider a lower level constraint qualification which is based on the Clarke subdifferential. This constraint qualification is a natural extension of Mangasarian–Fromovitz one to the differentiable GSIP. The main results are Fritz-John type necessary conditions for optimal solutions.

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