Preparation des reactifs organolithiens a partir des acides ω-cyanocarboxyliques dans l'ammoniac liquide

By treating ω-cyanocarboxylic acids with lithium amide in liquid ammonia, salts, of the type [NCCH(CH₂)_nCOOLi]Li are prepared. The formation of these salts is confirmed by their reaction with N-benzolydiphenylketimine and benzophenone to give ω-diarlylate, ω-benzoylamino- and ω-hydroxy-ψ-cyanocarboxylic acids respectively.     

Primal-Dual Solution Perturbations in Convex Optimization

Solutions to optimization problems of convex type are typically characterized by saddle point conditions in which the primal vector is paired with a dual multiplier vector. This paper investigates the behavior of such a primal-dual pair with respect to perturbations in parameters on which the problem depends. A necessary and sufficient condition in terms of certain matrices is developed for the mapping from parameter vectors to saddle points to be single-valued and Lipschitz continuous locally. It is shown that the saddle point mapping is then semi-differentiable, and that its semi-derivative at any point and in any direction can be calculated by determining the unique solutions to an auxiliary problem of extended linear-quadratic programming and its dual. A matrix characterization of calmness of the solution mapping is provided as well.     

Radius Theorems for Monotone Mappings

Set-Valued and Variational Analysis - For a Hilbert space X and a mapping $F: X\rightrightarrows X$ (potentially set-valued) that is maximal monotone locally around a pair $(\bar {x},\bar {y})$ in...     

Efficient estimates of the solutions of perturbed control problems

In this paper, we obtain estimates of the solutions for a sequence of strongly convex extremal problems. As applications of our abstract results, we consider optimal control problems with various types of perturbations. We estimate the solutions of problems with perturbations in the state equation and in the control constraining set. A singularly perturbed problem and a problem with perturbed time delay parameter are studied.     

N-Closed Subsets of Nearly Compact Spaces

The aim of this paper is to present some new properties of N-closed subsets by improving some already existing results. We show that every subset of a nearly compact space (X, τ), which is generalized closed in the semi-regularization topology is always N-closed relative to X. Hausdorff spaces are characterized via N-closed sets.     

Implicit function theorems for generalized equations

We show that Lipschitz and differentiability properties of a solution to a parameterized generalized equation 0 f(x, y) + F(x), wheref is a function andF is a set-valued map acting in Banach spaces, are determined by the corresponding Lipschitz and differentiability properties of a solution toz g(x) + F(x), whereg strongly approximatesf in the sense of Robinson. In particular, the inverse map (f + F)^–1 has a local selection which is Lipschitz continuous nearx ₀ and Fréchet (Gateaux, Bouligand, directionally) differentiable atx ₀ if and only if the linearization inverse (f (x ₀) + f (x₀) (× – x₀) + F(×))^–1 has the same properties. As an application, we study directional differentiability of a solution to a variational inequality.This work was supported by National Science Foundation Grant Number DMS 9404431.     

A proof of the Lyusternik–Graves theorem

In this paper, we give a new proof of the Lyusternik–Graves theorem, based on an intermediate result regarding linear openness inspired by works of Frankowska and Ursescu.     

Error bounds for euler approximation of a state and control constrained optimal control problem 1

We examine convergence of the Euler approximation to a nonlinear optimal control problem subject to mixed state-control and pure state constraints. We prove that under smoothness, independence, controllability and coercivity conditions at a reference solution of the continuous problem, there exists a locally unique solution to the Euler approximation, for sufficiently fine discretization, which converges to the reference solution with rate proportional to the mesh size.