Adil Bagirov Napsu Karmitsa - Introduction to Nonsmooth Optimization

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Part I
Nonsmooth Analysis and Optimization

Introduction

Convexity plays a crucial role in mathematical optimization. Especially, convexity is the most important concept in constructing optimality conditions. In smooth (continuously differentiable) optimization theory, differentiation entails locally linearizing the functions by the gradients, leading to a lower approximation of a convex function. These ideas can be generalized for nonsmooth convex functions resulting in the concepts of subgradients and subdifferentials . A subgradient preserves the property of the gradient, providing a lower approximation of the function, but in the nonsmooth case it is not unique anymore. Thus, instead of one gradient vector we end up with a set of subgradients called subdifferentials.

Unfortunately, convexity is often too demanding an assumption in practical applications, and we have to be able to deal with nonconvex functions as well. From a practical point of view, locally Lipschitz continuous functions are proved to be a suitable and sufficiently general class of nonconvex functions. In a convex case, differentiation is based on the linearization of a function. For nonconvex functions, the differentiation can be seen as convexification. All of the concepts of convex analysis can be generalized for a nonconvex case in a very natural way. As in the convex analysis, the links between nonconvex analysis and geometry are strong: subgradients and generalized directional derivatives have a one-to-one interpretation to the tangent and normal cones of epigraphs and level sets.

Our aim here is to present the theory of nonsmooth analysis for optimization in a compact and easy-to-understand form. The reader is assumed to have some basic knowledge of linear algebra, elementary real analysis, and smooth nonlinear optimization. To give as self-contained a description as possible, we define every concept used and prove almost all theorems and lemmas.

This part is organized as follows: after recalling some basic results and notions from smooth analysis, we concentrate on Convex Analysis in Chap.. We start our consideration with geometry by defining convex sets and cones. The main resultand foundation for the forthcoming theoryis the separation theorem stating that two convex sets can always be separated by a hyperplane. Next we turn to analytical concepts, by defining subgradients and subdifferentials. Finally, we show that those geometrical and analytical notions can be connected utilizing epigraphs, level sets, and the distance function.

Chapter is devoted to Nonconvex Analysis. We generalize all of the convex concepts to locally Lipschitz continuous functions. We also show that nonsmooth analysis is a natural enlargement of classical differential theory by generalizing all of the familiar derivation rules such as the mean-value theorem and the chain rule. The price we have to pay when relaxing the assumptions of differentiability and convexity is that instead of equalities, we only obtain inclusions in most of the results. However, by posing some mild regularity assumptions, we can turn the inclusions back into equalities. After the analytical consideration, we generalize, in a similar way to the convex case, all of the geometrical concepts, and give links between analysis and geometry. At the end of the chapter, we recall the definitions of quasidifferentials, codifferentials, and limiting subdifferentials all of which can be used to generalize the subdifferential of convex functions to the nonconvex case.

Chapter concentrates on the theory of nonsmooth optimization. Optimality conditions are an essential part of mathematical optimization theory, heavily affecting, for example, the development of optimization methods. We formulate the necessary conditions for locally Lipschitz continuous functions to attain local minima both in unconstrained and constrained cases. These conditions are proved to be sufficient for convex problems, and the found minima are global. We formulate both geometrical and analytical conditions based on cones and subdifferentials, respectively. We consider both general geometrical constraint sets and analytical constraints determined by functions and inequalities. In the latter case, we end up to generalizing the classical Fritz John (FJ) and Karush-Kuhn-Tucker (KKT) conditions.

The question, Can we still attain sufficient optimality conditions when relaxing the convexity assumptions? has been the initiating force behind the study of generalized convexities. In Chap. , we define and analyze the properties of the generalized pseudo- and quasiconvexities for locally Lipschitz continuous functions. Finally, we formulate relaxed versions of the sufficient optimality conditions, where the convexity is replaced by generalized pseudo- and quasiconvexities.

The last chapter of this part, Chap. , deals with Approximations of Subdifferentials. We introduce the concept of continuous approximations to subdifferential, and a discrete gradient that can be used as an approximation to the subgradient at a given point.

Springer International Publishing Switzerland 2014

Adil Bagirov , Napsu Karmitsa and Marko M. Mkel Introduction to Nonsmooth Optimization 10.1007/978-3-319-08114-4_1

1. Theoretical Background

Adil Bagirov 1

(1)

Centre for Informatics and Applied Optimization, School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat, VIC, Australia

(2)

Department of Mathematics and Statistics, University of Turku, Turku, Finland

Adil Bagirov (Corresponding author)

Email:

Napsu Karmitsa

Email:

Marko M. Mkel

Email:

In this chapter, we first collect some notations and basic results of smooth analysis. We also recall fundamentals from matrix calculus.

1.1 Notations and Definitions

All the vectors are considered as column vectors and, correspondingly, all the transposed vectors are considered as row vectors (bolded symbols are used for vectors). We denote by the usual inner product and by the norm in the -dimensional real Euclidean space . In other words,

where and are in and are the th components of the vectors and

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