The search for the minima of several unimodal functions

U.S.S.R. Comput. Maths. Math. Phys. Vol. 19, pp. 258- 263 0 Pergamon Press Ltd. 1980. Printed in Great Britain.

0041-5553/79/1001-0258$0750/O

THE SEARCH FOR THE MINIMA OF SEVERAL UNIMODAL FUNCTIONS* A. G. KOROTCHENKO Gor’kii (Received 23 October 1978)

THE PROBLEM of the optimal distribution of a specified resource over stages of calculations both in the search for the minima of several unimodal functions, and also in finding the least of them, is solved. 1. In this paper we consider the problem of the optimum distribution of a specified resource over stages of calculations in the search for the minima of m unimodal functions p&c), respectively. defined in the intervals [cc bj], i=i, 2, . . . , m, rn>Z In what follows this problem will be called problem 1. As well as problem 1 we also consider problem 2 of the optimal distribution of a resource over the stages of calculations for finding the least of the minima of the functions cpr(x).When solving problem 2 we will assume that the intervals [ai, bj] do not intersect. By a resource in this paper we mean the total time consumption or the cost of calculations of the functions cpi(x). We note that the problem of finding the least of the minima of several unimodal functions, defined on non-intersecting intervals, arises, for example, in finding the global minimum of the function cp(x),x E [a, b] , in the case where all the intervals of its unimodality are separate, Moreover, in finding all the local minima of the function &) in the same conditions we have the problem of finding the minima of several unimodal functions. It is assumed that any function cpi(x)can be evaluated at an arbitrary point of the corresponding interval [ai, bi] . The search for the minimum of the unimodal function cpi(x),defined in the interval [a, bi] will be carried out by the golden section algorithm, which is fairly simple to realize and is close to optimal, see [ 1,2] . This algorithm consists of a calculation (by a definite rule) and a comparison of the values of the function cpi(x).Thereby, the length of the localization interval of the minimum of the function ++(x) is successively reduced, that is, the interval which, in considering the results of the calculations, contains the argument reducing the function cpl(x)to a minimum. Suppose the function cp&x)is calculated at Vipoints. We denote by Sl(vi) the value of the criterion estimating the error in determining the coordinate xi* of the minimum of the function ~pi(x)and equal to the length of the interval of localization of the minimum of this function, obtained after calculating it at vi points by the golden section algorithm. It follows from [ 1,2] that the quantities S,(vi) have the form lZh. vjkhisl. Mat. mat. Fir., 19,5, 1337-1340,

1979.

258

Short communications

a,=&--a,.

Si (VI) =Oiiti-v’l

2,. . . , m,

vm) the value of the criterion

WedenotebyIV(ul,..., the coordinates

i=l,

xi* of the minima of all the unimodal

greatest of the intervals of localization

259

estimating

functions

of the minimum

z=(1+1’5)/2.

(1.1)

the error in determining

pr-(x) in an equal length of the

of the function

cpi
max Si (vi).

W(Vl,. . . v,)= )

(1.2)

1CiCm

After calculating

each function

minima of the functions of these functions,

cp,(x) at “i points the coordinate

x* of the least of the

p,(x) can be found in any of the intervals of localization

i = 1,2,

of the minima

. . . , m. It was stated above that when solving problem 2 the intervals

[UT bj] of definition of the functions vi(x) are assumed to be non-intersecting. Therefore, as the criterion estimating the error in determining the coordinate x* of the least minimum of the functions cpr(x), we can take the sum of the lengths of the intervals of localization of the minimum of the functions cpr-(x).We denote by Q(vI , . . . , vm) the value of the criterion introduced in this way. We obtain ml

Issi(Vi).

Q(v,,...,vm)=

(1.3)

isi

Therefore, we have m stages of calculations value of the minimum

of the function

golden section algorithm,

such that at the i-th stage the approximate

~pi(x) is found by calculating its values at ui points by the the calculations by formulas

i = 1, 2, . . . , m. Before beginning

(1.2), (1.3) we can estimate the error in determining error in determining

the least minimum

i= 1,2,.

at each point of the function

Cpi(x) amounts

. . ,m.

It is also assumed that before beginning of the function

qr(x) and the

of these functions.

Suppose the time (or cost) of the calculations tOPi*

the minima of all the functions

the search the number Vi of permissible

vi(x) is not specified, and the total time of the calculations

calculations

is restricted by the

inequality

pivi+. . . +p,,,v,,,
the optimal integral values vi = u/ in problem W(Yi’, . . . , vm’)=

min W(v~,...,vm), V,,...,Ym

(1.4) 1 such that (1.5)

and vi = vi” in problem 2, where “)= min Q(v~i,...,v~). Q(v,",...,vm V‘,...,%

Here condition

(1.4) must be satisfied.

(1.6)

260

A. G. Koror~h~nko

In addition, we will consider that the quantities vi must satisfy the constraints i=l, 2. . . . ,

via2,

m.

(1.7)

The constraints (1.7) are explained by the fact that to shorten the interval of localization of the minimum of a unimodal function it must be calculated at not less than two points. 2. We first consider the case where the time consumption of one calculation of each function ~~x~isthe~e,thatis,~i=l,~=I,2 ,_._, M. It follows from (1. I), (I.:!j, (1.4) and (1 S) that to determine the quantities I’~‘,. . . , Y, ‘ in this case it is necessary to solve the problem in integers

min W(vi, . . . , V,) -%...,Vm

(2.1)

with the satisfaction of the constraints (1.7) and

vi+. . . +v,dv.

(2.2)

We fast solve problem (2.1) with the condition that the quantities rl, . , . , vm can take any values (not necessarily integral), ~tisfying only the constraint (2.2). Considering (I .l), (1.2) it is easy to see that with this condition the function W(pt ) . . . , vm) attains its minimum value for vi = wi, where the quantities wi have the form

(2.3) oi=wlf(In

z)-’

i=2,3

In(a,/aj),

~ . . . . m..

We denote by pi the integral part of the number wp i= 1,2, . . . , nr - 1, and let Without loss of generality we will assume that the following relations pm=N-&-. *. -pm-i. hold:

i=l. 2, *.*, nz-l.

S:(pi)PSi+lfpz+l),

(2.4)

If relations (2.4) are not satisfied, then for them to be satisfied it is sufficient to renumber the functions +$x). We note by I the integral part of the number (p, - w,).ItiseasytoseethatO
if the following equations are satisfied: @,=!-I,.

i=l,

2, .

..)

m.

(2.6)

But if even one index i is found such that wi # pi, then &~=~<+I,

i=l

,2,...,

r+l,

(2.7)

Short communications

/.L,‘=pi,

i=r+2,.

p*‘=:Y-pi’-.

. . , m-i,

261

, -p;_i.

(2.8)

We will consider that the total number of N of calculations of the functions L&C),i = 1, 2 , . , m, specified before the beginning of the search is sufficiently great, namely that the f&owing condition is satisfied:

We prove that the vector (pl ‘, . . . , pm’) is the solution of the integer-valued problem (2.1) with the constraints (1.7) and (2.2). Using relations (2.3), (2.9), it is easy to show that the vector (pl’, , . . , p(,‘j satisfies the constraints (1.Q (2.2). vm) be an arbitrary integer-v~ued vector, not identical with the vector Let(q,..., (i-$, * * . , pm’) and satisfying the constraints (1.7), (2.2). Then (1.2) and (2.1) imply that to prove the assertion formulated it is sufficient to prove the validity of the relation max Si (Vi) i6i6m

3

max SC(CL{‘).

(2.10)

j
We note that the functions Si(Yj) used in (2.10) are defined in (1. I ). In the case where Eqs. (2.6) are satisfied, relation (2.10) is obvious. We consider the case where for at least one index i the Eqs. (2.6) are not satisfied. By the definition of the vector (vvl, . . . , vm) its components can be represented in the form where the rj are integers such that at least one of them is non-zero. V,=!.k*‘+Ti,i=l, 2,. .., m, Since the vectors (vl, . . , vm) and (yl’, . . . , pm’) satisfy the constraint (2.2), an index k can be found for which ~2-1, 1Gksm. Taking into account (1.1), (3.4) and (2.7>, (2.8), we obtain that

which implies the validity of (2.10). It can be similarly proved that the vector (pl’, . . , pm’) yields a minimum value of the function Q(vl, , vm) defined in (1.3), with the constraints (1.7), (2.2), that is, this vector is the solution of problem 2. Therefore, as the quantities p1 ‘, . . . , v, ‘, and also as the quantities Ye”, . . , , urn”, we can choose the corresponding components of the vector (,ul’, . . . , pm’), which are defined by (2.Q (2.7), (2.8).

A. G. Korotchenko

262

We note that the solution of problems 1 and 2 found is not in general unique. In the general case, when the time taken to calculate the functions &), i = 1, 2, . . . , m, general different, we find an approximate solution for problem 1 and also for problem 2 for fairly large N

is in

We will consider that when solving problem 1 the relation m

1n

1

Iis

N >> -

In t

Pir

);r:

i-i

f-2

holds, where pi is the time taken for one calculation of the function pi(X). Then the approximate solution of problem 1 has the form

(3.1) u,,=w,+

(In T)-; In

i=2$3 ,...rm,

(ail%),

where the square breackets denote the integral part of the number. In solving problem 2 we will assume that the following relation is satisfied:

rir>- 1

In ‘c

nz

I2

2- -

piIn?+ Pi

*=1

(

1

In 7

m

min lnl~ien

Ui Pi. pi

)IZ

i=*

The approximate solution of problem 2 has the form Vi =[qi],

Qi=

(N-$$pjlnz)

(2 j=l

i +-hail In-c

i=I,2,.

pj)-’ j=1

(3.2)

. . , m.

pi

Using (3.1), (3.2), it can be shown that if pi = 1, i = 1,2, . . . , m, the approximate solutions of problems 1 and 2 are identical. 4. As an example illustrating this approach to the optimal distribution of a specified resource over the stages of calculations, we consider the problem of a search for the minimum of and two unimodal functions: (p,(z), ZE [O. 51 ~(4, s=P3, 361. We will assume that the time taken to calculate these functions is the same. Let the total number of permissible calculations of the functions Q(X), Q(X) be 20. Then v1 ’ = 8, v2’ = 12. The value of the criterion W(v, ’, v2’), estimating the error in determining the minima of the functions cpl(x), rp2(x), is 0.1724. But if, for example, we put v1 = 6, v2 = 14, then W(vl, v2) = 0.4509. For “I = 11, v2 = 9 we obtain W(vl, v2) = 0.6383.

Short communications

263

In conclusion we note that the optimal distribution of a specified resource over the stages of calculations may be useful in those cases where the calculation of the functions cp,(x), i = 1, 2 9 * ** , m, requires time-consuming calculations or experiments. The author is indebted to F. L. Chernous’ko and to the reviewer for useful comments. Translated by J. Berry. REFERENCES 1.

KIEFER, J. Sequential minimax search for maximum. Proc. Amer. Math. Sot., 4,3,502-506,

2.

WILDE, D. J. Optimum seeking methods (Metody poiska ekstremuma), “Nauka”, Moscow, 1967.

U.S.S.R. Comput. Maths. Math. Phys. Vol. 19, pp. 263-266 0 Pergamon Press Ltd. 1980. Printed in Great Britain.

1953.

0041-5553/79/1001-0263$07.50/O

COMPUTATION OF THE TWO-DIMENSIONAL STATIONARY EQUATION OF NEUTRON TRANSFER BY THE QUASI-DIFFUSION METHOD* N. N. AKSENOV and V. Ya. GOL’DIN Moscow (Received 6 September

1978)

IT IS shown by the example of computations in two-dimensional plane geometry, that the quasi-diffusion method retains its efficiency for two-dimensional problems. 1. The solution of two-dimensional problems for the neutron transfer equation involves great difficulties. These are due to the fact that the complexity of the solution of the twodimensional problems hinders the use of rapidly convergent iterative methods and leads to a large volume of calculations. The quasi-diffusion method was used successfully in one-dimensional problems [ 11. The aim of this paper is to apply this method to the two-dimensional transfer equation. It is shown below that the quasi-diffusion method retains its efficiency in two-dimensional problems. This result is very important, since it has not been possible to obtain a theoretical estimate of the rate of convergence. 2. We consider the problem of the solution of the stationary neutron transfer equation by the example of two-dimensional plane geometry. Confining ourselves to the case of an isotropic scattering indicatrix and isotropic sources, the equation can be written in the form

~tae-Q

*Zh. vj&hisl. Mat. mat. Fiz., 19,5, 1341-1343,

Up

1979.

fg(r,P’)dR’+F

>