We present an effective algorithm for minimization of locally nonconvex Lipschitz functions based on mollifier functions approximating the Clarke generalized gradient. To this aim, first we approximate the Clarke generalized gradient by mollifier subgradients. To construct this approximation, we use a set of averaged functions gradients. Then, we show that the convex hull of this set serves as ...

A Lagrange multiplierrules that uses small generalized gradients is introduced. It includes both inequality and set constraints. The generalized gradient is the linear generalized gradient. It is smaller than the generalized gradients of Clarke and Mordukhovich but retains much of their nice calculus. Its convex hull is the generalized gradient of Michel and Penot if a function is Lipschitz. Th...

We prove the existence of solutions for the differential inclusion ẋ(t) ∈ F (t, x(t)) + f(t, x(t)) for a multifunction F upper semicontinuous with compact values contained in the generalized Clarke gradient of a regular locally Lipschitz function and f a Carathéodory function.

The paper provides topological characterization for solution sets of differential inclusions with (not necessarily smooth) functional constraints in Banach spaces. The corresponding compactness and tangency conditions for the right hand-side are expressed in terms of the measure of noncompactness and the Clarke generalized gradient, respectively. The consequences of the obtained result generali...

Nonsmooth semidefinite programming problems is concerned in this paper where the objective functions is quasidifferentiable. Necessary optimality conditions for quasidifferentiable optimization is proposed. The advantage of Kuhn-Tucker necessary optimality condition in terms of quasidifferential over that in terms of Clarke generalized gradient, is discussed.

This paper is devoted the study of a generalized hybrid dynamical system, which consists history-dependent hemivariational inequality parabolic type and nonlinear evolution equation. The unique solvability for system established via applying surjectivity multivalued pseudomonotone operators, fixed point theorem, properties Clarke gradient. As an illustrative application, dynamic frictional cont...

The classical Clarke subdifferential alone is inadequate for understanding automatic differentiation in nonsmooth contexts. Instead, we can sometimes rely on enlarged generalized gradients called “conservative fields,” defined through the natural pathwise chain rule: one application convergence analysis of gradient-based deep learning algorithms. In semialgebraic case, show that all conservativ...

Acknowledgment: We thank Philip D. Loewen for careful and critical reading of an early version of this paper. We also thank Jane J. J. Ye for her comments. Abstract. We provide a reened sensitivity analysis for nite and innnite horizon control problems where in both cases the perturbation space is L 1. Our underlying technique relies on a recent sequential description of both the generalized gr...

In this paper, we study optimal control problems on the internal energy for a system governed by class of elliptic boundary hemivariational inequalities with parameter. The has been originated steady-state heat conduction problem non-monotone multivalued subdifferential condition portion domain described Clarke generalized gradient locally Lipschitz function. We prove an existence result contro...