Various types of nonsmooth invex functions and applications
Giuseppe Giorgi, Angelo Guerraggio · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1994
Invex functions were introduced by Ilanson in 1981 as a generalization of differentiable convex functions: let x {contained_in} R{sup n} be open and let f : X {yields} R; if there exists a vector valued function {eta}(x, x{sup 0}) : X {times} X {yields} R{sup n} such that f(x) - f(x{sup 0}) {>=} < {del} f(x{sup 0}), {eta}x, x{sup 0}, lforallx, x{sup 0} {element_of} X, f is called invex. By means of this notion Hanson established the Kuhn-Tucker sufficient optimality criteria, the weak duality and the strong duality for a nonlinear optimization problem involving differentiable invex functions. The aim of this paper is to consider various proposals for the extension to a nonsmooth setting of the above definition. In particular we consider functions endowed with generalized directional derivatives in the sense of Clarke, Demyanov and Rubinov, Pshenichnyi, Jeyakumar and Ye. From these extensions we obtain some {open_quotes}parallel{close_quotes} results with respect to the differentiable case. Moreover we will show the various relationships among the nonsmooth invex functions examined. Finally a sufficient optimality condition is obtained for a nonlinear programming problem involving nonsmooth invex functions, in the sense of Demanov and Rubinov.