Given two functions f, g : {0, 1}n → {0, 1} a mapping ψ : {0, 1}n → {0, 1}n is said to be a mapping from f to g if it is a bijection and f(z) = g(ψ(z)) for every z ∈ {0, 1}n. In this paper we study Lipschitz mappings between boolean functions. Our first result gives a construction of a C-Lipschitz mapping from the Majority function to the Dictator function for some universal constant C. On the ...

Let f be a continuous function on Rn, and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but is a function whose gradient is easily computed where it is defined. We p...

We study metric spheres ( Z , d ) $(Z, d_{Z} )$ obtained by gluing two hemispheres of S 2 $\mathbb {S}^{2}$ along an orientation-preserving homeomorphism g : 1 → $g \colon \mathbb {S}^{1} \rightarrow {S}^{1}$ where $d_{Z}$ is the canonical distance that locally isometric to off seam. show if quasiconformally equivalent in geometric sense, then $g$ a welding with conformally removable curves. al...

Abstract In present paper, an analysis of the stability behaviour ideal efficient solutions to parametric vector optimization problems is conducted. A sufficient condition for existence locally perturbed and their nearness a given reference value provided by refining recent results on theory parameterized set-valued inclusions. More precisely, Lipschitz lower semicontinuity property solution ma...

Let M be a Lagrangian manifold, let the 1-form pdx be globally exact on M and let S(x, p) be defined by dS = pdx on M. Let H(x, p) be convex in p for all x and vanish on M . Let V (x) = inf{S(x, p) : p such that (x, p) ∈ M}. Recent work in the literature has shown that (i) V is a viscosity solution of H(x, ∂V/∂x) = 0 provided V is locally Lipschitz, and (ii) V is locally Lipschitz outside the s...

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The main result of this note asserts that for any suba...

We extend the classic convergence rate theory for subgradient methods to apply to non-Lipschitz functions. For the deterministic projected subgradient method, we present a global O(1/ √ T ) convergence rate for any convex function which is locally Lipschitz around its minimizers. This approach is based on Shor’s classic subgradient analysis and implies generalizations of the standard convergenc...