We consider a multivalued equation of the form Ay + ∂φ(y) 3 f in a real Hilbert space, where A is a linear operator and ∂φ represents the (Clarke) subdifferential of the function φ. We prove existence and uniqueness results of the solution by using the control variational method. The main idea in this method is to minimize the energy functional associated to the nonlinear equation by arguments ...

We consider nonlinear multistage stochastic optimization problems in the spaces of integrable functions. allow for dynamics and general objective functionals, including dynamic risk measures. study causal operators describing system derive Clarke subdifferential a penalty function involving such operators. Then we introduce concept subregular recourse establish subregularity resulting systems t...

We investigate the value function of an infinite horizon variational problem in infinite-dimensional setting. First, we provide upper estimate its Dini–Hadamard subdifferential terms Clarke Lipschitz continuous integrand and normal cone to graph set-valued mapping describing dynamics. Second, derive a necessary condition for optimality form adjoint inclusion that grasps connection between Euler...

Abstract This paper is devoted to studying a system of coupled nonlinear first order history-dependent evolution inclusions in the framework triples spaces. The multivalued terms are Clarke subgradient or convex subdifferential form. Using surjectivity result for maps and fixed point argument operator, we prove that has unique solution. We conclude with two examples an evolutionary differential...

This paper focuses on Clarke generalized Jacobian of the projection onto symmetric cones. First, we recall some basic concepts. Let ̥ : Ω ⊆ X → Y be a locally Lipschitz function on an open set Ω, where X and Y are two finite dimensional inner product spaces over the field R. Let ∇̥(x) denote the derivative of ̥ at x if ̥ is differentiable at x. The Clarke generalized Jacobian of ̥ at x is defined by...

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...

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...

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 study the mathematical program with geometric constraints such that the image of a mapping from a Banach space is included in a nonempty and closed subset of a finite dimensional space. We obtain the nonsmooth enhanced Fritz John necessary optimality conditions in terms of the approximate subdifferential. In the case where the Banach space is a weakly compactly generated Asplun...

We apply nonsmooth analysis to a well known optical inverse problem, phase retrieval. The phase retrieval problem arises in many different modalities of electromagnetic imaging and has been studied in the optics literature for over forty years. The state of the art for this problem in two dimensions involves iterated projections for solving a nonconvex feasibility problem. Despite widespread us...