The stability of a method of searching for saddle points

U.S.S.R. Comput.Maths.Math.Phys.,Vol.25,No.6,p.lO0,1985

O041-5553/85 $iO.OO+O.OO Pergamon Journals Ltd.

Printed in Great Britain

THE STABILITY OF A METHODOF SEARCHINGFOR SADDLEPOINTS P.A.

DOROFEYEV

There have been many papers (see, e.g., /1-3/) on the subject of algorithms for determining saddle points. The problem of the stability of one of these methods /3/ is discussed below. Consider a finite function /(z)=/(y.z). defined and convexly-concave in an open convex neighbourhood of the set X=Y×Z, where YcE~, and ZcE,, are convex compacta having internal points. We shall denote the subdifferential of the convex function ](Y,z') at the point y'eZ by 0vI(~'.,') . and the superdifferential of the condave function /(~'.z) at the point z'EZ by 8./(~',z'). We shall introduce ~(~)= max/(y,z),

Y" = { ~ ' ~ Yl~(~')ffi mio ~(v)= I'), ~aY

zeZ

where X

Y" is a set of corresponding components of the saddle points of the function

lqe shall determine the the equation

~-neighbourhood,

~0.

of the arbitrary set

AcE,

/(x) at using

[A]~ = { b • E . I inf lia-bll,~ =}. aiA

Consider the following modification of the Arrow-Gurwitz method /I/. and the point z0eX. For kmO we will assume ~A+I:~Y (YA--~A, ga),

Here

at, nz

When (see /3/).

We shall fix

Zt+t : ~ Z (Zh+~*Oh).

are the operators of projection on the sets Y and Z respectively, e--0

Theorem.

any limiting point of the sequence {y~} belongs to If

emO

~ is a limiting point of the sequence

Y"

{z,) , such that

and

ghe[0#f(~A,*A)].. /(zt)--/(y~,zh)~/"

Uminf/(x~)--/($), then

-edz
dy, dz

are the diameters of the sets Y and Z. REFERENCES

i. 2. 3.

ARROW K°, GURWITZ L. and UDZAVA H., Investigations using linear and non-linear programming. Moscow,Izd-vo inostr, lit., 1962. DEM'YANOV V.F., The subgradient method and saddle points. Vestn. LGU, 7,17-23,1981. NURMINSKZI E.A., Numerical methods of solving deterministic and stochastic minimax problems. Kiev, Naukova dumka, 1979.

Translated by H.Z.

*Zh.vNchisl.Mat.mat. Fiz.,25,11,1739,1985. ** The full text of this paper is deposited in VINITI, iOO

1985, 2262-85 DEP. p.lO.