Differentiation in logical form

We introduce a logical theory of differentiation for a real-valued function on a finite dimensional real Euclidean space. A real-valued continuous function is represented by a localic approximable mapping between two semi-strong proximity lattices, representing the two stably locally compact Euclidean spaces for the domain and the range of the function. Similarly, the Clarke subgradient, equivalently the L-derivative, of a locally Lipschitz map, which is non-empty, compact and convex valued, is represented by an approximable mapping. Approximable mappings of the latter type form a bounded complete domain isomorphic with the function space of Scott continuous functions of a real variable into the domain of non-empty compact and convex subsets of the finite dimensional Euclidean space partially ordered with reverse inclusion. Corresponding to the notion of a single-tie of a locally Lipschitz function, used to derive the domain-theoretic L-derivative of the function, we introduce the dual notion of a single-knot of approximable mappings which gives rise to Lipschitzian approximable mappings. We then develop the notion of a strong single-tie and that of a strong knot leading to a Stone duality result for locally Lipschitz maps and Lipschitzian approximable mappings. The strong single-knots, in which a Lipschitzian approximable mapping belongs, are employed to define the Lipschitzian derivative of the approximable mapping. The latter is dual to the Clarke subgradient of the corresponding locally Lipschitz map defined domain-theoretically using strong single-ties. A stricter notion of strong single-knots is subsequently developed which captures approximable mappings of continuously differentiable maps providing a gradient Stone duality for these maps. Finally, we derive a calculus for Lipschitzian derivative of approximable mapping for some basic constructors and show that it is dual to the calculus satisfied by the Clarke subgradient

Decidability of consistency of function and derivative information for a triangle and a convex quadrilateral

Given a triangle in the plane, a planar convex compact set and an upper and and a lower bound, we derive a linear programming algorithm which checks if there exists a real-valued Lipschitz map defined on the triangle and bounded by the lower and upper bounds, whose Clarke subgradient lies within the convex compact set. We show that the problem is in fact equivalent to finding a piecewise linear surface with the above property. We extend the result to a convex quadrilateral in the plane. In addition, we obtain some partial results for this problem in higher dimensions

Minimality properties of set-valued processes and their pullback attractors

We discuss the existence of pullback attractors for multivalued dynamical systems on metric spaces. Such attractors are shown to exist without any assumptions in terms of continuity of the solution maps, based only on minimality properties with respect to the notion of pullback attraction. When invariance is required, a very weak closed graph condition on the solving operators is assumed. The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation and an irregular form of the heat equation.Comment: 33 pages. A few typos correcte

Rothe method and numerical analysis for history-dependent hemivariational inequalities with applications to contact mechanics

In this paper an abstract evolutionary hemivariational inequality with a history-dependent operator is studied. First, a result on its unique solvability and solution regularity is proved by applying the Rothe method. Next, we introduce a numerical scheme to solve the inequality and derive error estimates. We apply the results to a quasistatic frictional contact problem in which the material is modeled with a viscoelastic constitutive law, the contact is given in the form of multivalued normal compliance, and friction is described with a subgradient of a locally Lipschitz potential. Finally, for the contact problem we provide the optimal error estimate

Clarke subgradients of stratifiable functions

We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:\mathbb{R}^{n}\to \mathbb{R}\cup\{+\infty\}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $x\in{dom}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain some Morse-Sard type theorems as well as a nonsmooth Kurdyka-\L ojasiewicz inequality for functions definable in an arbitrary o-minimal structure

Nonsmooth nonconvex stochastic heavy ball

Motivated by the conspicuous use of momentum based algorithms in deep learning, we study a nonsmooth nonconvex stochastic heavy ball method and show its convergence. Our approach relies on semialgebraic assumptions, commonly met in practical situations, which allow to combine a conservative calculus with nonsmooth ODE methods. In particular, we can justify the use of subgradient sampling in practical implementations that employ backpropagation or implicit differentiation. Additionally, we provide general conditions for the sample distribution to ensure the convergence of the objective function. As for the stochastic subgradient method, our analysis highlights that subgradient sampling can make the stochastic heavy ball method converge to artificial critical points. We address this concern showing that these artifacts are almost surely avoided when initializations are randomized

A class of differential hemivariational inequalities in Banach spaces

In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function u↦f(t,x,u) and compactness of C0-semigroup eA(t)

Subgradient sampling for nonsmooth nonconvex minimization

Risk minimization for nonsmooth nonconvex problems naturally leads to firstorder sampling or, by an abuse of terminology, to stochastic subgradient descent. We establish the convergence of this method in the path-differentiable case, and describe more precise results under additional geometric assumptions. We recover and improve results from Ermoliev-Norkin by using a different approach: conservative calculus and the ODE method. In the definable case, we show that first-order subgradient sampling avoids artificial critical point with probability one and applies moreover to a large range of risk minimization problems in deep learning, based on the backpropagation oracle. As byproducts of our approach, we obtain several results on integration of independent interest, such as an interchange result for conservative derivatives and integrals, or the definability of set-valued parameterized integrals