Weak Subdifferential in Nonsmooth Analysis and Optimization

Şahlar F Meherrem,Refet Polat

doi:10.1155/2011/204613

Abstract

Some properties of the weak subdifferential are considered in this paper. By using the definition and properties of the weak subdifferential which are described in the papers (Azimov and Gasimov, 1999; Kasimbeyli and Mammadov, 2009; Kasimbeyli and Inceoglu, 2010), the author proves some theorems connecting weak subdifferential in nonsmooth and nonconvex analysis. It is also obtained necessary optimality condition by using the weak subdifferential in this paper.

Highlights

Nonsmooth analysis had its origins in the early 1970s when control theorists and nonlinear programmers attempted to deal with necessary optimality conditions for problems with nonsmooth data or with nonsmooth functions such as the pointwise maximum of several smooth functions that arise even in many problems with smooth data, convex functions, and max-type functions.For this reason, it is necessary to extend the classical gradient for the smooth function to nonsmooth functions.The first such canonical generalized gradient was the generalized gradient introduced by Clarke in his work 1
It is necessary to extend the classical gradient for the smooth function to nonsmooth functions
We investigate the relationships between the Frechet lower subdifferential and weak subdifferentia and we prove some theorems related to the weak subdifferential

Summary

Introduction

Nonsmooth analysis had its origins in the early 1970s when control theorists and nonlinear programmers attempted to deal with necessary optimality conditions for problems with nonsmooth data or with nonsmooth functions such as the pointwise maximum of several smooth functions that arise even in many problems with smooth data, convex functions, and max-type functions. For this reason, it is necessary to extend the classical gradient for the smooth function to nonsmooth functions.

Main Results

Necessary Optimality Conditions via the Weak Subdifferential

Conclusion