Optimality Conditions for Non-Smooth Optimization Problem
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Date
2017-06
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Addis Ababa University
Abstract
Optimization is a mathematical problem with many real world applications. The goal is to determine minimizers or maximizers of a multivariable real function, under a restriction domain. The differentiability assumptions play a vital role in nonlinear programming, because most of methods of finding the optimum point in nonlinear programming starts by finding the gradient of the function and then the stationary points For unconstrained optimization problems, checking the positive definiteness of the Hessian matrix at stationary points, one can conclude whether those stationary points are optimum points or not if the objective functions is differentiable. Similarly ,if the objective function and functions in the constraint set are differentiable, the well -known optimality condition called Karush-Kuhn-Tucker (KKT) condition leads to find the optimum point(s) of the given optimization problem. But, since finding the gradient of the function for non-differentiable functions is not possible, we treat the problem by finding the sub gradient of the directional derivative. Consequently, the optimization procedures for the optimization problems on which functions in the problem are not differentiable is different from the optimization procedures for the optimization problems in which the objective functions as well as functions in constrains are differentiable. The main purpose of this project focuses on finding the optimality conditions for optimizations problems without any differentiability assumptions. The sub-gradient of a directional derivative approach are used to solve nonsmooth optimization problem of convex type. we establish the existence of optimization problem specially for non-linear programing (NLP) ,We introduce “A cone Approach on the Karush-Kuhn-Tucker optimality conditions, constraint qualification” and we discussed in this paper some of the constraint qualifications as well as some relation between them and also to show and also to observe the weakest of these constraint qualifications with respect to the concept of cones and their polar.
Key words and phrases: Optimality conditions, constraint qualifications, nonlinear programming, unconstrained optimization, positive definiteness, Hessian matrix, smooth optimization, nonsmooth optimization, directional derivative, sub- gradient, sub-differentiable, polar cones, the tangent cone, Slater constraint qualification, and the Quasi-Regularity constraints
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Keywords
Optimality conditions, constraint qualifications, nonlinear programming, unconstrained optimization, positive definiteness, Hessian matrix, smooth optimization, nonsmooth optimization, directional derivative, sub- gradient, sub-differentiable, polar cones, the tangent cone, Slater constraint qualification, and the Quasi-Regularity constraints