The General Concept of Cone Approximations in Nondifferentiable Optimization
General optimization problems connected with necessary conditions for optimality have been studied by many authors in recent years. Since Clarke (1975) introduced the notion of a generalized gradient and the corresponding tangent cone, numerous papers hav
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1.
INTRODUCTION
General optimization problems connected with necessary conditions for optimality have been studied by many authors in Since Clarke (1975) introduced the notion of a generalized gradient and the corresponding tangent cone, numerous papers havebeen published which extend standard smooth and recent years.
convex optimization results to the general case. In this paper we show how necessary optimality conditions may be constructed for local solutions of nonsmooth nonconvex optimization problems involving inequality constraints. We shall use the approach developed by Dubovitskij and Miljutin (1965), which is closely connected with appropriate cone approximations of sets and differentiability concepts (to obtain multiplier conditions) .
Having studied the properties
of numerous published cone approximations (see Thierfelder 1984), we propose a general definition of a local cone approximation K and introduce the corresponding K-directional derivative and Using these notions K-subdifferential of a functional f :X~ R. it is possible to derive general multiplier conditions which turn out to be true generalizations of the Kuhn-Tucker theory for smooth and convex optimization problems. 2.
LOCAL CONE APPROXIMATIONS
Let [X,T] be a locally convex Hausdorff space and [X*,cr*] be the topological dual space of X endowed with the weak * (star) topology. We consider the problem
V. F. Demyanov et al. (eds.), Nondifferentiable Optimization: Motivations and Applications © Springer-Verlag Berlin Heidelberg 1985
171
(P):f 0 (x) -+min, xes :=Gexjfi(x) ~o, iEI :={1, .•• ,m}} where the f i: X-+ R, i E { 0} U I, are extended real-valued functionals. The definition of an abstract local cone approximation is fundamental to the following considerations since it can be used to replace an arbitrary set by a simple structured set. Moreover, the K-directional derivative leads to generalized differentiability for an extended real-valued functional.
Definition 2. 1 .
The mapping K: 2x x X-+ 2X is called a local cone
approximation if a cone K(M,x) is associated with each set M C X and each point x EX such that
(i)
K(M-x,O) = K(M,x)
(ii)
K(MfiU,x) = K(M,x) V UEU(x)
(iii)
K(M,x) =X if xE int M
= ro
M.
(iv)
K (M,x)
(v)
K(. > 0
1 .) 1
fK(x 1 y)
VtE (0 1 A) :x+tyEM}
then we obtain lim f(x+ty)-f(x)
t+O
f' (x 1 y) Vy EX
t
oKf(x) = {x*EX*Ix*(y) ~f' (x 1 y) VyEX} This example shows that the notions introduced above are proper generalizations of the corresponding notions from convex analysis. Now we shall derive some basic propositions. Theorem 3.1. (1)
Let f:X~R 1 xEX 1
epi fK (x 1
• )
oo.
Then
= { (y 1 I;) IVE > 0 :3: ~ E R: Is - ~I < E and (y 1 ~)
(2)
lf(x) I
- 0 there exists a ~ < ~ +
(y,~)EK(epi
f,
E
such that
(x,f(x)))
(3 • 1 )
Since (0,
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