The purposeof this paper is twofold. First we considera class of nondiierentiablepenaltyfunctions for constrained Lipschitz programs and then we show how these penalty functions can be employed to actually solve a constrained Lipschitz program. The penalty functions considered incorporate a barrier term which makes their value go to innnity on the boundary of a perturbation of the feasible set....

The Lipschitz constant of a finite normal–form game is the maximal change in some player's payoff when a single opponent changes his strategy. We prove that games with small Lipschitz constant admit pure ǫ-equilibria, and pinpoint the maximal Lipschitz constant that is sufficient to imply existence of pure ǫ-equilibrium as a function of the number of players in the game and the number of strate...

In the context of a strongly local Dirichlet space we show that if a function mapping the real line to itself (and fixing the origin) operates by composition on the left to map the Dirichlet space into itself, then the function is necessarily locally Lipschitz continuous. If, in addition, the Dirichlet space contains unbounded elements, then the function must be globally Lipschitz continuous. T...

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient...

A global optimization problem is studied where the objective function f(x) is a multidimensional black-box function and its gradient f ′(x) satisfies the Lipschitz condition over a hyperinterval with an unknown Lipschitz constant K. Different methods for solving this problem by using an a priori given estimate of K, its adaptive estimates, and adaptive estimates of local Lipschitz constants are...

A multiple-valued function f : X → QQ(Y ) is essentially a rule assigning Q unordered and non necessarily distinct elements of Y to each element of X. We study the Lipschitz extension problem in this context by using two general Lipschitz extension theorems recently proved by U. Lang and T. Schlichenmaier. We prove that the pair ` X,QQ(Y ) ́ has the Lipschitz extension property if Y is a Banach ...

In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient...

In this study, we propose a new solution approach for solving the Maintenance Scheduling Problem for a Family of Machines (MSPFM). After reviewing the literature, we found that Goyal and Kusy’s paper presented the only model that used a nonlinear function for the cost of operating a machine when studying the periodic maintenance scheduling problems. In our presentation of this paper, we first r...

Abstract Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{X}}$ -valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \in {\mathbb{X}}$ and for $n\geq 1$ , $X_n\;:\!=\;F(X_{n-1},\vartheta_n)$ where $\{\vartheta_n\}_{n \ge 1}$ are independent identically distributed random variables with common probability distribution $\mathfrak{p}$ $F(\cdot,\cdot)$ is continuous in the first ...

A number of global optimisation algorithms rely on the value of the Lipschitz constant of the objective function. In this paper we present a stochastic method for estimating the Lipschitz constant. We show that the largest slope in a fixed size sample of slopes has an approximate Reverse Weibull distribution. Such a distribution is fitted to the largest slopes and the location parameter used as...