The Lipschitz and C harmonic capacities κ and κc in R n can be considered as high-dimensional versions of the so-called analytic and continuous analytic capacities γ and α (respectively). In this paper we provide a dual characterization of κc in the spirit of the classical one for the capacity α by means of the Garabedian function. Using this new characterization, we show that κ(E) = κ(∂oE) for...

The notion of a secant for locally Lipschitz continuous functions is introduced and a new algorithm to locally minimize nonsmooth, nonconvex functions based on secants is developed. We demonstrate that the secants can be used to design an algorithm to find descent directions of locally Lipschitz continuous functions. This algorithm is applied to design a minimization method, called a secant met...

In this note we consider the computability of the solution of the initial-value problem for ordinary differential equations with continuous right-hand side. We present algorithms for the computation of the solution using the “thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of a differential equa...

Let S = {St}t≥0 be the semigroup generated on L2(R ) by a selfadjoint, second-order, divergence-form, elliptic operator H with Lipschitz continuous coefficients. Further let Ω be an open subset of R with Lipschitz continuous boundary ∂Ω. We prove that S leaves L2(Ω) invariant if, and only if, the capacity of the boundary with respect to H is zero or if, and only if, the energy flux across the b...

This paper develops a new derivative-free method for solving linearly constrained nonsmooth optimization problems. The objective functions in these problems are, in general, non-regular locally Lipschitz continuous function. The computation of generalized subgradients of such functions is difficult task. In this paper we suggest an algorithm for the computation of subgradients of a broad class ...

This note presents new results on global asymptotic stability (GAS) and global exponential stability (GES) of a general class of continuous-time recurrent neural networks with Lipschitz continuous and monotone nondecreasing activation functions. We first give three sufficient conditions for the GAS of neural networks. These testable sufficient conditions differ from and improve upon existing on...

We consider a rate independent evolution quasivariational inequality in a Hilbert space X with closed convex constraints having nonempty interior. We prove that there exists a unique solution which is Lipschitz dependent on the data, if the dependence of the Minkowski functional on the solution is Lipschitzian with a small constant and if also the gradient of the square of the Minkowski functio...

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...

This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming ...

Consider the ordinary diierential equation _ xt = F , x t ; x 0 = x 0 : 1 A solution 1 on 0; T , T T 0, to 1 is a continuously diierentiable function x : : 0 ; T !R n satisfying xt = x 0 + Z t 0 F , x s ds: We m a y ask for which functions F there exist a solution to 1 and, if so, if the solution is unique. Is it, for instance, suucient that F is a continuous function? The answer is no concerni...