Optimality conditions in nonsmooth optimization
Bevil M. Glover · Bulletin of the Australian Mathematical Society · 1994
The purpose of this thesis is to establish, by a variety of techniques from convex and nonsmooth analysis, first order necessary and sufficient optimality conditions for general mathematical programming problems defined in abstract spaces and involving nonsmooth mappings.The conditions obtained are, in essence, generalizations of the classical Fritz John and Karush-Kuhn-Tucker theorems from finite dimensional differentiable programming.In order to obtain verifiable optimality conditions involving subdifferentials for abstract cone-constrained programming problems it is necessary to develop new tools in nonsmooth analysis.In the first chapter this route is taken by establishing infinite dimensional chain rules for Lipschitz mappings (between Banach spaces) which possess a sub differential multifunction satisfying a weak* closure condition.This condition is shown to be satisfied by the class of compactly Lipschitzian mappings which are then investigated and characterized by the upper semicontinuity and compactness properties of their generalized Jacobians.This generalized Jacobian is an extension, for vector-valued Lipschitz mappings, of the Clark sub differential for real-valued locally Lipschitz functions.Both metric regularity and local solvability conditions are developed as applications of these results.In the second chapter a Fritz John-type theorem is obtained using Ekeland's variational principle.This result is then used to obtain Karush-Kuhn-Tucker conditions for Lipschitz programming problems without recourse to a constraint qualification.Applications of these results to concave minimization and certain stochastic programming problems are outlined and discussed.It is well known that one approach to the derivation of necessary optimality conditions for differentiable programming problems is via a solvability theorem applied to a certain 'linearization' of the problem.This approach is extended using suitable generalizations of a fundamental solvability theorem known as Farkas Lemma.In the third chapter, various unifying generalizations of this result are presented with applications to global convex maximization, least squares problems and Lagrangian duality.In Chapter