3 The Study of Smooth Optimization Problems
Marius Durea, Radu Strugariu · 2014
This chapter plays a central role in this monograph.In its first section, we present smooth optimization problems and deduce existence conditions for minimality.We take this opportunity to prove and discuss the Ekeland Variational Principle and its consequences.We then obtain necessary conditions for optimality as well as sufficient optimality conditions of the first and second-order for smooth objective functions under geometric restrictions (i.e., restrictions of the type x ∈ M, where M is an arbitrary set).The second section is dedicated to the investigation of optimality conditions under functional restrictions (with equalities and inequalities).The main aim is to deduce Karush-Kuhn-Tucker conditions and to introduce and compare several qualification conditions.Special attention is paid to the case of convex and affine data.Subsequently, we derive second-order optimality conditions for the case of functional restrictions.The last section of this chapter includes two examples which show that, for practical problems, the computational challenges posed by the optimality conditions are sometimes not easy to solve.