A subset X of a real Hilbert space H is said to be proximally smooth provided that the function d X : H ! R (the distance to X) is continuously diierentiable on an open tube U around X. It is proven that this property is equivalent to d X having a nonempty proximal subgradient at every point of U, and that the (G^ ateaux = Fr echet) derivative is locally Lipschitz on U. The Lipschitz behavior o...

Based on a formula of Tseng, we show that the squared norm of the matrix-valued Fischer-Burmeister function has a Lipschitz continuous gradient.

Many mixed-integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly. This is, e.g., the case for problems constrained by differential equations or for models that rely on black-box simulation runs. For these problem classes, we present, analyze, and test algorithms...

In this paper, we consider a class of constrained optimization problems where the feasible set is a general closed convex set and the objective function has a nonsmooth, nonconvex regularizer. Such regularizer includes widely used SCAD, MCP, logistic, fraction, hard thresholding and non-Lipschitz Lp penalties as special cases. Using the theory of the generalized directional derivative and the t...

In this note we introduce strongly Lipschitz p-integral operators, strongly Lipschitz p-nuclear operators and Lipschitz p-nuclear operators. It is shown that for a linear operator, the Lipschitz p-nuclear norm is the same as its usual p-nuclear norm under certain conditions. We also prove that the Lipschitz 2-dominated operators and the strongly Lipschitz 2-integral operators are the same with ...

We classify ω-limit sets of autonomous ordinary differential equations x′ = f(x), x(0) = x0, where f is Lipschitz, in infinite dimensional Banach spaces as being of three types I-III. Let S ⊂ X be a Polish subset of a Banach space X. S is of type I if there exists a Lipschitz function f and a solution x such that S = Ω(x) and x ∩ S = ∅. S is of type II if it has non-empty interior and there exi...

‎In this work we prove Malliavin differentiability for the solution to an SDE with locally Lipschitz and semi-monotone drift‎. ‎To prove this formula‎, ‎we construct a sequence of SDEs with globally Lipschitz drifts and show that the $p$-moments of their Malliavin derivatives are uniformly bounded‎.

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 ...