2.1 Differentiability and first order conditions If a function f : R → R has partial derivatives ∂if(x) = limt→0 t−1(f(x + tei) − f(x)) which exist and are continuous on R, it is a C1(R) function, and is differentiable at every x in the sense that f(x + h) − f(x) −∇f(x) · h = o(‖h‖) as h → 0. This means it can be approximated linearly, and the derivative is the linear map on R given by Df(x)(h)...

Here I discuss the use of everywhere continuous nowhere differentiable functions, as well as the proof of an example of such a function. First, I will explain why the existence of such functions is not intuitive, thus providing significance to the construction and explanation of these functions. Then, I will provide a specific detailed example along with the proof for why it meets the requireme...

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger in [8]. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Bana...

adomian decomposition method has been applied to solve many functional equations so far. in this article, we have used this method to solve the fuzzy differential equation under generalized differentiability. we interpret a fuzzy differential equation by using the strongly generalized differentiability. also one concrete application for ordinary fuzzy differential equation with fuzzy input data...

This article studies continuity and directional differentiability properties of optimal value functions, in particular at boundary points of their domain. We extend and complement standard continuity results from [12] for abstract feasible set mappings under complete convexity as well as standard differentiability results from [13] for feasible set mappings in functional form under the Slater c...

Existing methods for function smoothness in neural networks have limitations. These can make training sensitive to their hyperparameters, or constraints limit model capacity. impose too much smoothness, even areas without data, they non-meaningful constraints. The way these measure also be computationally hard. One of the main Lipschitz continuity, does not imply differentiability theoretically...

We consider optimal control problems governed by nonlinear hyperbolic conservation laws at junctions and analyze in particular the Fréchet-differentiability of the reduced objective functional. This is done by showing that the control-to-state mapping of the considered problems satisfies a generalized notion of differentiability. We consider both, the case where the controls are the initial and...

Optimal control problems with convex costs, for which Hamiltonians have Lipschitz continuous gradients, are considered. Examples of such problems, including extensions of the linear-quadratic regulator with hard and possibly state-dependent control constraints, and piecewise linear-quadratic penalties are given. Lipschitz continuous differentiability and strong convexity of the terminal cost ar...

Splines are piecewise polynomial functions that have certain “regularity” properties. These can be defined on all finite intervals, and intervals of the form (−∞, a], [b,∞) or (−∞,∞). We have already encountered linear splines, which are simply continuous, piecewise-linear functions. More general splines are defined similarly to the linear ones. They are labeled by three things: (1) a knot sequ...

In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax– Oleinik semigroups. This is equivalent to the solvability of an associated multi– time Hamilton–Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions ...