Lipschitz function

Here is the space of functions having k Lipschitz continuous deriva-... 

Armijo, L. Minimization of Functions Having Lipschitz Continuous First Partial Derivatives. Pac J Math 16 1-3 (1966). 

Remark 2 The property of stability can be interpreted as a Lipschitz continuity condition on the perturbation function v(y). 

The objective function /( ) and the inequality constraint g(x) are convex since f(x) is separable quadratic (sum of quadratic terms, each of which is a linear function of xi, x2,X3, respectively) and g(x) is linear. The equality constraint h(x) is linear. The primal problem is also stable since v(0) is finite and the additional stability condition (Lipschitz continuity-like) is satisfied since f(x) is well behaved and the constraints are linear. Hence, the conditions of the strong duality theorem are satisfied. This is why... 

Lipschitz D. Medical and functional consequences of anemia i n the elderl y J Am Geriatr Soc 2003 51 S10-4S13. 

Abadie, J., and Carpentier, J. (1969), Generalization of the Wolfe Reduced Gradient Method to the Case of Nonlinear Constraints, in Optimization, R. Fletcher, Ed., Academic Press, New York. Armijo, L. (1966), Minimization of Functions having Lipschitz Continuous First-Partial Derivatives, Pacific J. Mathematics, Vol. 16, No. 1, pp. 1-3. 

Armijo, L. (1966) Minimization of functions having Lipschitz continuous first-partial derivatives. Padjic Journal of Mathematics, 16 (1), 1-3. 

Note that this assumption also implies that the potential production rate of one reservoir does not depend on e volumes produced from the other reservoirs. We will also assume for i = ,... that fi is nonnegative and nonincreasing as a function of Qi t) for all t and that the recoverable volume of each reservoir is finite. Finally, to ensure uniqueness of potential production pro files we will also assume that is Lipschitz continuous in Qi, i =... 

It follows directly that if a determines continuously differentiable times at q5 the Lipschitz exponent (LE) tjrgives an indication of how regular the function f t) is at tg. The higher the <2 the more regular the function f t) is. 

In addition, it is assumed that functions F, /, and g are Lipschitz bounded, which ensures existence and uniqueness of the solution to problem (1) [10]. Using the notation x = y, problem (1) can be written in the form... 

It is easy to show that if // > 1, the derivative of/(w) is zero, so that it is a constant. This case is not of great interest, so it is always assumed that /i < 1. For // = 1, the Hdlder condition is termed the Lipschitz condition and is obeyed by any differentiable function, and others not in this class. For // < 1, the condition implies continuity in the ordinary sense. The case // = 0, which is excluded, is consistent with discontinuity. A function obeying this condition at a point, or on a line, will be described as obeying the H(ji) condition on that set, if n is specified or otherwise just the H condition. 

A number of inequalities on the so called concentration of measure phenomenon ([Talagrand (1996)], [Ledoux (2001)], [Villani (2003)]) are available for product probability spaces. We say that the IID sequence of integrable random variables satisfies a concentration inequality if there exists a continuous function [0, oo) — [0, oo), limt oo (t) = 0, such that for every n and every Lipschitz convex function G K — R with Lipschitz constant 1, that is... 

---