We show that in two dimensions or higher, the Mordukhovich-loffe approximate subdifferential and Clarke subdifferential may differ almost everywhere for real-valued Lipschitz functions. Uncountably many Frechet differentiable vector-valued Lipschitz functions differing by more than constants can share the same Mordukhovich-Ioffe coderivatives. Moreover, the approximate Jacobian associated with ...

In this paper we consider a second order evolution inclusion with a coercive viscosity operator and a multivalued term of subdifferential form. The study is motivated by the dynamic problem of frictional contact between a viscoelastic piezoelectric deformable body and a foundation. The interaction between the body and the foundation is described, due to the skin effects, by a nonmonotone possib...

In this work we continue the nonsmooth analysis of absolutely symmetric functions of the singular values of a real rectangular matrix. Absolutely symmetric functions are invariant under permutations and sign changes of its arguments. We extend previous work on subgradients to analogous formulae for the proximal subdifferential and Clarke subdifferential when the function is either locally Lipsc...

Abstract In this paper, Sobolev-type conformable fractional stochastic evolution inclusions with Clarke subdifferential and nonlocal conditions are studied. By using calculus, analysis, properties of nonsmooth sufficient for controllability the considered problem established. Finally, an example is given to illustrate obtained results.

In this paper‎, ‎we introduce and study some new single-valued gap functions for non-differentiable semi-infinite multiobjective optimization problems with locally Lipschitz data‎. ‎Since one of the fundamental properties of gap function for optimization problems is its abilities in characterizing the solutions of the problem in question‎, ‎then the essential properties of the newly introduced ...

We present here a characterization of the Clarke subdifferential optimal value function linear program as matrix coefficients. generalize result Freund (1985) to cases where derivatives may not be defined because existence multiple primal or dual solutions.

We introduce generalized definitions of Peano and Riemann directional derivatives in order to obtain second-order optimality conditions for vector optimization problems involving C 1,1 data. We show that these conditions are stronger than those in literature obtained by means of second-order Clarke subdifferential.

Abstract In this paper, for a nonsmooth semi-infinite multiobjective programming with locally Lipschitz data, some weak and strong Karush-KuhnTucker type optimality conditions are derived. The necessary conditions are proposed under a constraint qualification, and the sufficient conditions are explored under assumption of generalized invexity. All results are expressed in terms of Clarke subdif...

and Applied Analysis 3 Proof. Since f(x) defined in (2) belongs to the PDGstructured family and by Lemma 2.1 in [16] the Clarke subdifferential of F(x, ρ) at x can be formulated by ∂F (x, ρ) = ∂f (x) + ρ∂G (x) = ∂f (x) + ρ conv { ∇g j (x) | j ∈ J (x) ∪ {0}}