Connectedness and Necessary Conditions for an Extremum

This monograph is a study of necessary conditions of an extremum in which topological connectedness plays a major role. The synthesis of the well-known Dybrovitskii-Milyutin approach, based on functional analysis, and topological methods permits the derivation of the so-called alternative conditions of an extremum: if the Euler equation has the trivial solution only at an extreme point, then some inclusion is valid for the functionals belonging to the dual space. Also, the present approach gives a transparent answer to the question why the Kuhn-Tucker theorem establishes the restrictions on the signs of the Lagrange multipliers for the inequality constraints but why this theorem does not establish any analogous restrictions on the multipliers for the equality constraints. Examples from mathematical economics illustrate the alternative conditions of any extremum. Parallels are drawn between these examples and the problems of static equilibrium in classical mechanics. This volume should be of use to mathematicians and graduate students interested in the areas of optimization, optimal control and mathematical economics.