Application Of Uv-decomposition Theory To Nonlinear Optimization

In nonsmooth optimization, second-order expansion theory is significant both for deriving optimalty optimality conditions and developing algorithms . Therefore, the study concerning the theory of the second-order properties of nonsmooth functions have been paid much attention.Lemarechal, Miffilin, Sagastizabal and Oustry(2000) introduced the UV—theory, which opens a way to defining a suitable restricted second-order derivative of a convex function f at a nondifferentible point x. The basic idea is to decompose R~n into two orthogonal subspaces U and V depending on x so that f's nonsmoothness near the point is concentrated essentially in V. A certain Lagangian associated with the convex functin was introduced, called U—Lagrangian. When f satisfies certain structural properties, it is possible to find smooth trajectories, via the intermediate function , yielding α second-order expansion for f.Under a set of conditions such as V— optimality , feasibility and transervality, we obtain a set of relative weak sufficient conditions for the existence of U—Hessiian. the existence of optimal solution set W(u) of the U—Lagrangian and the characterization of the associated smooth trajectary x + u + W(u) tangential to U, so that the second order expansion of f can be develped.In the paper, the UV—decomposition theory is applied to NLP. The results on the penalty function of constrained minimization with a finit number of constraints are gen-eralied. The UV—decomposition theory to the exact penalty functions in NLP for a nonlinear complementary problem with a basic index set and a feasible basic index set are established.