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...

This chapter is mainly about William Karush and his role in the Karush-KuhnTucker theorem of nonlinear programming. It tells the story of fundamental optimization results that he obtained in his master’s thesis: results that he neither published nor advertised and that were later independently rediscovered and published by Harold W. Kuhn and Albert W. Tucker. The principal result – which concer...

Abstract: In this work, a nonsmooth multiobjective optimization problem involving generalized invexity with cone constraints and Applications (for short, (MOP)) is considered. The Kuhn-Tucker necessary and sufficient conditions for (MOP) are established by using a generalized alternative theorem of Craven and Yang. The relationship between weakly efficient solutions of (MOP) and vector valued s...

In this paper, we consider using the neural networks to efficiently solve the second-order cone constrained variational inequality (SOCCVI) problem. More specifically, two kinds of neural networks are proposed to deal with the Karush-Kuhn-Tucker (KKT) conditions of the SOCCVI problem. The first neural network uses the FischerBurmeister (FB) function to achieve an unconstrained minimization whic...

In this paper we analyze the application of Newton's method to the solution of systems of nonlinear equations arising from equivalent forms of the rst{order Karush{Kuhn{Tucker necessary conditions for constrained optimization. The analysis is carried out by using an abstract model for the original system of nonlinear equations and for an equivalent form of this system obtained by a reformulatio...

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...