Necessary optimality conditions for nonsmooth problems with operator constraints (Q1114602)

The authors extend Neustadt's theory of upper convex approximations (UCA's) to operators between normed linear spaces; ``upper convexity'' is defined in terms of the ordering induced by a closed convex cone in the range space. A scalarized version of this concept, called the weak UCA, is also introduced, and used to define a set of generalized subdifferentials. Necessary conditions for optimality in an abstract mathematical program are then given, both in terms of UCA's and in terms of the corresponding generalized subdifferentials. These problems are distinguished by their infinite-dimensional inequality constraints - G(x)\(\in L\), where G:E\(\to F\) is a nonlinear operator between normed linear spaces E and F, and L is the closed convex cone of ``nonnegative elements'' of F. The authors give necessary conditions in Fritz John form, and then describe constraint qualifications under which they hold in ``normal'' (KKT) form.