The well known Jordan decomposition theorem gives the useful characterization that any function of bounded variation can be written as the difference of two increasing functions. Functions which can be expressed in this way can be used to formulate an exclusion test for the recent Cellular Exclusion Algorithms for numerically computing all zero points or the global minima of functions in a give...

We consider the problem of computing a critical point of a nonconvex locally Lipschitz function over a convex compact constraint set given an inexact oracle that provides an approximate function value and an approximate subgradient. We assume that the errors in function and subgradient evaluations are merely bounded, and in particular need not vanish in the limit. After some discussion on how t...

We develop and analyze a variant of Nesterov’s accelerated gradient descent (AGD) for minimization of smooth non-convex functions. We prove that one of two cases occurs: either our AGD variant converges quickly, as if the function was convex, or we produce a certificate that the function is “guilty” of being non-convex. This non-convexity certificate allows us to exploit negative curvature and ...

The present article deals with approximation results obtained by means of the Lipschitz maximal function, Ditzian–Totik modulus smoothness, and a Lipschitz-type space having two parameters for summation-integral type operators defined Mishra Yadav [22]. Further, we determine rate convergence in terms derivative bounded variation. To estimate quantitative operators, establish Voronovskaya-type G...

Techniques known as Nonlinear Set Membership prediction, Lipschitz Interpolation or KinkyInference are approaches to machine learning that utilise presupposed Lipschitz properties tocompute inferences over unobserved function values. Provided a bound on the true best Lipschitzconstant of the target function is known a priori they offer convergence guarantees as well asbounds aro...

We show that the Scott topology induces a topology for real-valued Lipschitz maps on Banach spaces which we call the L-topology. It is the weakest topology with respect to which the L-derivative operator, as a second order functional which maps the space of Lipschitz functions into the function space of non-empty weak* compact and convex valued maps equipped with the Scott topology, is continuo...

Fast-Lipschitz optimization has been recently proposed as a new framework with numerous computational advantages for both centralized and decentralized convex and nonconvex optimization problems. Such a framework generalizes the interference function optimization, which plays an essential role distributed radio power optimization over wireless networks. The characteristics of Fast-Lipschitz met...

A Lipschitz map f between the metric spaces X and Y is called a Lipschitz quotient map if there is a C > 0 (the smallest such C, the co-Lipschitz constant, is denoted coLip(f), while Lip(f) denotes the Lipschitz constant of f) so that for every x ∈ X and r > 0, fBX(x, r) ⊃ BY (f(x), r/C). Thus Lipschitz quotient maps are surjective maps which by definition have the property ensured by the open ...

In this paper, we focus on the convergence analysis for the application of the PeacemanRachford splitting method to a convex minimization model whose objective function is the sum of a smooth and nonsmooth convex functions. The sublinear convergence rate in term of the worst-case O(1/t) iteration complexity is established if the gradient of the smooth objective function is assumed to be Lipschi...

Let v be a smooth vector field on the plane, that is a map from the plane to the unit circle. We study sufficient conditions for the boundedness of the Hilbert transform Hv,ǫ f(x) := p.v. ∫ ǫ −ǫ f(x− yv(x)) dy y where ǫ is a suitably chosen parameter, determined by the smoothness properties of the vector field. It is a conjecture, due to E.M. Stein, that if v is Lipschitz, there is a positive ǫ...