Tuyển tập những báo cáo nghiên cứu khoa học hay nhất của trường đại học vinh tác giả: 3. Thái Doãn Chương, Dưới vi phân giới hạn của hàm giá trị tối ưu trong một số bài toán "bệnh tật" quy hoạch trơn. | LIMITING SUBGRADIENTS OF THE MARGINAL FUNCTION IN SOME PATHOLOGICAL SMOOTH PROGRAMMING PROBLEMS THAI DOAN CHUONG a Abstract. In this paper we show that the results of Mordukhovich Nam and Yen 6 on differential stability in parametric programming can be used to derive upper estimates for the limiting subgradients of the marginal function in some pathological smooth programming problems proposed by Gauvin and Dubeau 2 . 1. INTRODUCTION Let p X X Y R be a function taking values in the extended real lineR -rc rc G X Ya set-valued mapping between Banach spaces. Consider the parametric programming problem minimize p x y subject toy G x . The extended-real-valued function 1-2 n x inf p x y y G x is said to be the marginal function or the value function of . The solution map M of the problem is defined by M x y G x n x x y . For we say that p is t he objective function and G is the constraint mapping. Continuity and differentiability properties of n in the case where X Rn Y Rm p is a smooth function . a C1-function and G x is the set of all x satisfying the parametric inequality equality system gi x y 0 i 1 . p hj x y 0 j 1 . q gj X X Y R i 1 . p and hj X X Y R j 1 . q are smooth functions were studied firstly by Gauvin and Tolle 3 Gauvin and Dubeau 2 Their results and ideas have been extended and applied by many authors see Mordukhovich Nam and Yen 6 where the case p is a nonsmooth function and G is an arbitrary set-valued map between Banach spaces is investigated and the references therein. We will show that the results of 6 on differential stability in parametric programming can be used to estimate the set of limiting subgradients . the limiting subdifferential of the marginal function in six pathological smooth programming problems proposed by Gauvin and Dubeau 2 Thus general results on differentiability Nhận bài ngày 30 3 2007. sửa chữa xong ngày 23 5 2007. properties of the marginal function of are very useful even for the classical .
