Asymptotically Attainable Elements: Model Examples

Abstract

The present chapter deals with the simplest examples of problems reacting in different ways to the perturbation of a system of conditions. In some cases, the influence of “small” perturbations is “weak”, in other cases, on the contrary, there is an abrupt reaction of a “system” with respect to infinitesimal perturbations of the system of conditions. It is interesting, however, that both of these possibilities may, in a certain sense, exist simultaneously. Thus, roughness or precision takes place “in certain directions”. Classification of these “directions” represents important theoretical and applied interest. This book is devoted to the investigation of this question. In this chapter, the above-mentioned phenomenon is discussed on the basis of examples. However, reading carefully, one can see a certain system. First of all we shall try to clarify the difference between the properties of roughness (or asymptotic nonsensitivity) and stability. Besides, we shall try to illustrate some kinds of generalized elements which are the closure of asymptotics of realized solutions. Here we do not attempt to construct examples of problems representing practical interest. Rather, we focus on “caricature” effects of the incorrectness of natural statements. To make examples visual and intelligible, we introduce restrictions on the considered problems in the form of inequalities. Restrictions of such a kind are typical for mathematical programming (MP) problems.