There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus for convex nondifferentiable functions [1]. This development started with F.H. Clarke, who introduced a generalized gradient for functions that are locally Lipschitz, but (possibly) nondifferentiable. Generalized gradients turn out to be the subdifferentials, in th...

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The main result of this note asserts that for any suba...

For nonsmooth Filippov systems, the stability of system is assumed to be proved by Lyapunov functions, such as piecewise smooth functions. This extension was based on solution and Clarke generalized gradient. However, it difficult estimate gradient a non-smooth function. In some cases, can divided into continuous discontinuous components. If Lebesgue measure components zero, function utilized p...

The classical Mountain Pass Lemma of Ambrosetti-Rabinowitz has been studied, extended and modified in several directions. Notable examples would certainly include the generalization to locally Lipschitz functionals by K. C. Chang, analyzing structure critical set mountain pass theorem works Hofer, Pucci-Serrin Tian, extension Ghoussoub-Preiss closed subsets a Banach space with recent variations...

Abstract In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential condition on portion described by Clarke generalized gradient locally Lipschitz function. First, prove new existence result for inequality employing theory pseudomonotone operators. Next, give comparis...

In recent years, many traditional optimization methods have been successfully generalized to minimize objective functions on manifolds. In this paper, we first extend the general traditional constrained optimization problem to a nonlinear programming problem built upon a general Riemannian manifold M , and discuss the first-order and the second-order optimality conditions. By exploiting the dif...

The modification of the Clarke generalized subdiNerentia1 due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Glteaux differentiability of any real function can be deduced from the GBteaux differentiability of the norm if the function has a directional derivative which attains a constant related...

In this paper, we consider a class of nonsmooth, nonconvex constrained optimization problems where the objective function may be not Lipschitz continuous and the feasible set is a general closed convex set. Using the theory of the generalized directional derivative and the Clarke tangent cone, we derive a first order necessary optimality condition for local minimizers of the problem, and define...