An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions

2012 ◽  
Vol 155 (1) ◽  
pp. 180-195 ◽  
Author(s):  
Nezam Mahdavi-Amiri ◽  
Rohollah Yousefpour
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Algorithm ◽  
Lipschitz Functions ◽  
Locally Lipschitz Functions ◽  
Locally Lipschitz

Equivalence of Some Multi-Valued Elliptic Variational Inequalities and Variational-Hemivariational Inequalities

2011 ◽  
Vol 11 (2) ◽  
Author(s):  
Siegfried Carl
Keyword(s):  
Variational Inequalities ◽  
Lipschitz Functions ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz Functions ◽  
Generalized Gradient ◽  
Elliptic Variational Inequalities ◽  
Comparison Results ◽  
Locally Lipschitz ◽  
Additional Regularity ◽  
Definition Of

AbstractFirst, we prove existence and comparison results for multi-valued elliptic variational inequalities involving Clarke’s generalized gradient of some locally Lipschitz functions as multi-valued term. Only by applying the definition of Clarke’s gradient it is well known that any solution of such a multi-valued elliptic variational inequality is also a solution of a corresponding variational-hemivariational inequality. The reverse is known to be true if the locally Lipschitz functions are regular in the sense of Clarke. Without imposing this kind of regularity the equivalence of the two problems under consideration is not clear at all. The main goal of this paper is to show that the equivalence still holds true without any additional regularity, which will fill a gap in the literature. Existence and comparison results for both multi-valued variational inequalities and variational-hemivariational inequalities are the main tools in the proof of the equivalence of these problems.

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