Prof. Refail Kasimbeyli
Eskişehir Technical University, Türkiye
On the separation theorem in nonconvex analysis, generalized derivatives and optimality conditions
This talk presents a conical supporting surfaces approach in nonconvex analysis. By using the conical supporting surfaces, we extend the concept of dual cones, introduce the so-called augmented dual cones, and show that the supporting and separating philosophy (of the convex analysis) based on hyperplanes, can be extended to a nonconvex analysis by using elements of these cones. The special class of monotone sublinear functions is introduced with the help of elements of augmented dual cones. By using the conical supporting surfaces, we present a nonlinear separation theorem in nonconvex analysis. Conical supporting and separating surfaces are defined as level or sublevel sets of a special class of monotone sublinear and / or superlinear functions. It is shown that the usual definitions of directional derivatives and subdifferentials in convex analysis can be generalized by using the concept of conical surfaces. We generalize the classical directional derivative concept and introduce the notion of the radial epiderivative. The concepts of weak subdifferentials and radial epiderivatives are used to obtain new necessary and sufficient optimality conditions for global optimums in nonconvex optimization, and to develop new global solution methods in nonsmooth and nonconvex single-objective optimization and in vector optimization.