On a method of solving systems of non-linear inequalities

ON A METMlDOF SOLVINGSYSTEllS OF NON-LINEAR INEQUALITIES M. A. GRRMANOVand Sofia,

SPIRIDONOV

Bulgaria

(Received

1.

V.S.

3 July

1965)

STATEMENT OF THE PROBLEM

GIVEN the system of m inequalities:

To find the point are satisfied.

X”(xlo,

x2’,

. . . . x,,O), for which all

inequalities

(1)

2. Let U denote the set of all points of the n-dimensional Euclidean space E, for which inequalities (1) are satisfied. We assume that I! is not empty. If all the functions vk are linear, there are familiar relaxation methods (see [l], [2]) for finding one point X0 of the set M. These methods are extended in 1131to the case when the 9k are convex. The present note describes another extension of the relaxation method and a modification, more suitable for computations. 3. Let the functions gJk (k = 1, 2, . . . . m) of system (1) be convex and have continuous first derivatives with respect to all xl (1 = 1, 2, * . . ) n), and Xi is an arbitrary point of En. Let K denote the set of subscripts k (k = 1, 2, . . . , m) for which Pk(Xi) > 0. We now consider all the orthogonal trajectories of the surfaces Tk(X) = Ck (k cz K) which PUSS through Xi. If Pik are the points of intersection of these trajectories with the surfaces qk(X) = 0, a point can be found on them for which the length of arc XiPik is maximal. We denote this point by xitl We can thus *

Zh.

vlchisl.

obtain Mot.

the sequence mot. Fiz.

4,

of points 2, 335 - 336,

194

1966.

Solving

non-linear

systems

of

Xl,

.**a

X2,

inequalities

Xi.4 Xi+l,

195

...

(2)

We have: Theorem

I

There

are

two possibilities

for

(a)

the

sequence

is

finite

(b)

the

sequence

is

infinite,

This we fix

method the

4.

is

a direct

parameter

The method

and its

last

extension

para.

of

2.

continuous

to

all

. . . . n),

and Xi

then

evaluate

2,

qk(Xi)

(k

the

= 1,

to X0 E

relaxation

3 can be modified

and have

(1 = 1.

term Xp E M; V.

method

[II

of

(if

it).

. . . . m) be convex rl

(2):

but convergent

h = 1 in

of

sequence

2,

as follows.

first

is

Let qk (k

derivatives

an arbitrary

with

point

= 1,

respect

in E,.

We

. . . . m) and find

‘pk”(&) = max qk (Xi).

I

If

pk’(Xi)

< 0,

trajectory sects of

of

the

points

that

gonal

E

h!. If . gk”(X)

to

of

sequence

computational para.

3.

point

point

of

from Xi to Xi+1

Xi+l.

Theorem view,

In a computer

Passes 1 also

this

solution

We thus holds

method the

can be realized

the

through

orthogonal

Xi.

It

inter-

obtain a sequence for this.

is more convenient

motion

along

by steps

the

with

ortho-

the

Tk’(X).

the

qk be linear,

an arbitrary along

(2).

we consider

> 0,

= C, which

similar

If all the ok (k = 1, 2, fication of that of [ll (if all

qk’(Xi)

cpkO(X) = 0 at the

trajectory

gradient

Xi

surface

surface

From the than

then

the

the

point hyperplane

. . . . m) are linear, this method is we fix the parameter A = 1 in it).

but not of

normalized.

En,

may not

pk(X)

= 0.

be the

Then the

(Pk(Xi),

same as the

Let ~kOl(Xi)

where Xi

distance

= 0 be the

a modiFor, let is

from Xi

hyperplane

for

which cPn”(Xi) = max (pk(Xi) > 0. k

Then the

is

the

qk’(X)

orthogonal

same as the = 0.

trajector perpendicular

The corresponding

through

Xi to the hyperplane

dropped point

Xi+1

from Xi onto of

the

the

sequence

gk’(X) plane is

the

= C

M.A.

196

orthogonal

projection

Gernanov

of

Xi

on

and

gk’(X)

V.S.

Spiridonov

= 0. Translated

by

D.E.

grown

REFERENCES 1.

MOTZKIN. T.S. and SCHOENBERG, 1.1. The relaxation method for liuear inequalities, Can. J. Math. 6, 3, 393 - 404, 1954.

2.

AGMON, S. The relaxation method for linear inequalities, Can. J. - 392, 1954. lath. 6, 3, 382

3.

EREMIN, 1.1. Extension sf the Motzkin - Agmon relaxation method, 1965. Usp. mat. Nauk, 20, 2, 183 - 187,