In this paper, we consider the width invariant trapezoidal and triangularapproximations of fuzzy numbers. The presented methods avoid the effortful computation of Karush-Kuhn-Tucker Theorem. Some properties of the new approximation methods are presented and the applicability of the methods is illustrated by examples. In addition, we show that the proposed approximations of fuzzy numbers preserv...

These notes characterize maxima and minima in terms of first derivatives. I focus primarily on maximization. The problem of minimizing a function f has the same solution (or solutions) as the problem of maximizing −f , so all of the results for maximization have easy corollaries for minimization. The main result of these notes is the Karush-Kuhn-Tucker (KKT) Theorem, recorded as Theorem 3 in Se...

We consider Karush-Kuhn-Tucker (KKT) systems, which depend on a parameter. Our contribution concerns with the existence of solution of the directionally perturbed KKT system, approximating the given primaldual base solution. To our knowledge, we give the first explicit result of this kind in the situation where the multiplier associated with the base primal solution may not be unique. The condi...

In this note we give an elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets. 2006 P...

In this paper we characterize the local maxima of a continuous global optimization formulation for finding the independence number of a graph. Classical Karush-Kuhn-Tucker conditions and simple combinatorial arguments are found sufficient to deduce several interesting properties of the local and global maxima. These properties can be utilized in developing new approaches to the maximum independ...

Abstract-A second-order dual to a nonlinear programming problem is formulated. This dual uses the Ritz John necessary optimality conditions instead of the Karush-Kuhn-Tucker necessary optimal&y conditions, and thus, does not require a constraint qualification. Weak, strong, strictconverse, and converse duality theorems between primal and dual problems are established. @ 2001 Elsevier Science Lt...

In this paper, we consider a nonsmooth multiobjective semi-infinite programming problem with vanishing constraints (MOSIPVC). We introduce stationary conditions for the MOSIPVCs and establish the strong Karush-Kuhn-Tucker type sufficient optimality conditions for the MOSIPVC under generalized convexity assumptions.

The author regrets that some terms are missing or misspecified in the Karush–Kuhn–Tucker conditions in Sections 3.7–3.10 on page 346. Corrections are given below. add multiplier p m c m (discount rate in tree node m multiplied by the probability of its occurrence) to d d (number of days in demand season d).