Study On Second-order Optimality Conditions In Hilbert Spaces

Optimality conditions are usually used to study the properties of optimal solutions to optimization problems.It is of great significance to the study of optimization theory,and it also plays an important role in solving optimization problems,designing optimization algorithms,and determining the termination conditions of algorithms.At present,the first-order optimality conditions have been extensively studied by many authors.Many authors have obtained some second-order optimality conditions using various second-order directional derivatives.In finite-dimensional spaces,Eberhard and Wenczel and Mordukhovich discussed three different types of second-order optimality conditions which are based on generalized second-order directional derivatives,graphical derivatives of proximal subdifferential and second-order proximal subdifferential defined via coderivative of proximal subdifferential,and the equivalence among these optimality conditions for paraconcave functions is also proved.However,the equivalence result on second-order optimality conditions may not hold in the Hilbert space.So it is natural to study the second-order optimality conditions in Hilbert space.In this paper,based on this,mainly considers the second-order lower Dini-directional derivative,the secondorder mixed graphical derivative and the second-order mixed proximal subdifferential of the nonsmooth function defined in Hilbert space,as well as the three defined by these generalized second-order derivatives class second-order optimality conditions,and prove the equivalence among these second-order optimality conditions for paraconcave functions.The second-order optimality conditions and equivalence theorems studied in this paper are the extends and generalizes of the research results of the second-order optimality conditions in finite-dimensional spaces.