We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with denoising lower-level constraints. For problem, we able to derive M-stationarity conditions, after characterizing corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions....

In this paper, we consider Simultaneous Perturbation Stochastic Approximation (SPSA) for function minimization. The standard assumption for convergence is that the function be three times differentiable, although weaker assumptions have been used for special cases. However, all work that we are aware of at least requires differentiability. In this paper, we relax the differentiability requireme...

It is known that directional differentiability of metric projection onto a closed convex set in a finite dimensional space is not guaranteed. In this paper we discuss sufficient conditions ensuring directional differentiability of such metric projections. The approach is based on a general theory of sensitivity analysis of parameterized optimization problems.

Peano differentiability is a notion of higher order differentiability in the ordinary sense. H. W. Oliver gave sufficient conditions for the mth Peano derivative to be a derivative in the ordinary sense in the case of functions of a real variable. Here we generalize this theorem to functions of several variables.

We propose a direct method to control the first order fractional difference quotients of solutions to quasilinear subelliptic equations in the Heisenberg group. In this way we implement iteration methods on fractional difference quotients to obtain weak differentiability in the T -direction and then second order weak differentiability in the horizontal directions.

In this work, the tracking control of a class uncertain linear dynamical systems is investigated. The uncertainty considered to be represented as fuzzy numbers, and hence, these are referred systems, which presented in form differential equations (FDEs). solution an FDE found using approach called relative-distance-measure interval arithmetic under granular differentiability concept. objective ...

We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path functions constitute a proper subclass Lipschitz which admit conservative gradients, notion generalized derivative compatible with basic calculus rules. Our main result states that such inherit the differentiability property driving field. show indeed forward propagation derivatives giv...