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Optimisation in stochastic models
OPTIMISATION IN STOCHASTIC MODELS* L.
S.
GURIN
(MOSCOW) (Received
20
February
This paper is a direct continuation of in referring numbering of the formulae; the in [ll.
1964)
[I].
This also involves the to formulae (1) - (15). we mean
If we take condition (4) as the end of the computation. then. as already indicated in (1). the accuracy will depend not only on E but also on ?I and other parameters. In order to guarantee a definite degree of accuracy in this case it is necessary to make E a function of these parameters. Moreover a reasonable value should be chosen for a in formula (4). In order shall start With the
to obtain definite again from heuristic symbols
recommendations considerations.
on these
cluestions
we
adopted
f (20. YO)- f @l* YI) = fn @o, Yo)- fN (Q, t/l) + 5 ($)
= Af + 4 (y)
,
whence M if 6% yo) -
f (21, yl)la = (Af)2 + g,
Comparing with formula (4). we obtain the resuit mean value of the square of the difference will On the other
hand,
for
the
function
that for be 39.
(9)
[f(so, YO)--f(x~. Y,)]* = 16iiz(M2z,* + m*y$). Taking
l
the means,
we obtain
Zh. Vych. Mat. 4. No. 6,
1134 - 1137, 228
1964.
(16) a = 2/3
the
Optinisation
in
stochastic
229
nodels
Hence it is possibleto obtain the dependenceof E on A, and that of M on In,and with these we may expect the computationto end with a mean error M(x2+ Y")< e'.
($8)
In fact, taking the mean of xo2 + yo2 in accordancewith condition (171, i.e. assuming
and taking
CJJ to
be uniformlydistributedin (0, WV
we obtain
Comparing(19) and (18). +e obtain for the given E' E=
64Pe" 3L-i12‘ .W / nz~I i:
(3-O)
In Table I the results are given of a calculationfor the function fk
Y) = 5xz+4xy+&2,
consideredin [l], for E' = 0.01, x0 = 100, y. = 0,
k
=
0,
n
=
4;
results
TABLE I
64(l) 213 5.4 8.2 4.9 7.6 4.2 5.0 3.9 4.0 4 5 4 4 _I__.._I____J__I__/---L-J0.320.400.220.300.220.300.320.380.440.410.75~.64
L.S.
230
are given corresponding position of the results
Gurin
to various different values of A and - h. The in each cell of the table is as follows: 10-M (ZN)
I
10-b (ZN)
1~~ (12+ ya) loos The number of cases
(out of
(cl9 + y”) .
100) of divergence
is given
in brackets.
Analysis of Table 1 shows that it is indeed true that RIfz2 + y*) scarcely depends on h at all and. as a rule, &RX* + y*f < E*. Further, it is evident that hopt and A may be chosen in accordance with the formulae (11) and (15) obtained earlier, using E‘ in place of E, i.e. to take
These formulae were obtained in the following way: with A given in accordance with (21). formula (21) for A 1s obtained from formula (15). Substituting A and h into formula (111, we find that the appropriate inequality is amply satisfied. In fact,
both the latter
expressions
exceed
Mu and (11)
gives
The increase of A by a factor J-6 is brought about in order to to possible guarautee greater stability for the process in relation vergence. Thus we may regard on its parameters for clarified.
di-
the dependence of the convergence of the process given coefficients of quadratic form as fully
However in t&e solution of practical problems these coefficients are unknown. This creates a new problem which may be solved by two methods: 1) in searching for an extremum for the values ating the second derivatives of the function f(x.
of
f~(x,
yf,
by evalu-
y) and in accordance
Optimisslfion
them calculating
with 2)
selecting
of the point We shall
A
(x,,
in
stochastic
M, UI, and then also
and A directly yn).
consider
231
models
h, A;
by analysing
the nature
of the motion
method (1).
We shall introduce heuristic considerations at the start in the choice of the tiral step for evaluating the second derivatives. We have
Hence h* must be of the same order
as the mean square deviation
{,
i.e.
(22)
where the constant
A
has the
vnlue aefined
above.
In order not to return to this question, we note that the calculations described below were carried out for various values of P and it turned out that it is best to take u = 1. The algorithm for method (1) takes the following form: we are given arbitrary A and A; we then carry out a calculation according to the algorithm described above for the given A and A; on each occasion when it is necessary to increase N a times we carry out a re-computation of A and A. evaluating first of all the second derivatives through the second differences (in this we make the trial steps in conformity with (22)). and then determining M, II and, finally, h and A in accordance with (21). TABLE 2
lU-BM(Zh’) 27.2 io-%((ZN) 82 HJOM(zw+y*) 0.65
2.54 2.0 0.31
uJOs(z~+y~)
0.28,
0.92
3.31 4.0 0.23 0.26
6.02 9.8 0.29 0.41
Moreover, two further refinements have been introduced into the algorithm in order to avoid overloading the discharge network during
L.S.
232
operations
on the electronic
Curin
computer:
a) if it turns out that the step for the antigradient obtained in accordance with the formulae is greater than a certain constant L, it should be taken as equal to L (we took L = 100); b) if it turns out that III< 8. where 6 > 0 is a sufficiently small number (we took 6 = 0.011, then we should proceed to the usual multiplication of N by TVand recompute h and A. It is easy to see that with a positive-definite quadratic successful choice of 6 no cyclic process will occur.
form and a
A calculation carried out under the same conditions as in Table 1 gave results which are set out in Table 2. We shall compare the last column of Table 2 with Table 1, where n was equal to 4. In Table 1 for a successful choice of h and A lesser values of M( EN) were obtained. but for many A and A larger values were obtained. This means that in the finishing process the approximation to optimal h and A proceeds slowly; this is due to the difficulty of evaluating the second deriva(a difficulty demanding, furthertives of f(~, y) by random realisations more, supplementary realisations). However even the better results from Table 1 are only 1% times better than the corresponding results of Table 2. Therefore we may regard method (1) of evaluating the quadratic form as satisfactory. It may be expected that method (2) will turn out to be better only for the less accurate optimisation of f(x, y). But in this case we should bring in for comparison other methods of optimisation also, for example, the method of random search. We note further that Table 2 confirms the conclusion drawn in (1) that an optimal n exists near to n = 1.4. Without embarking on a more accurate determination of the value of n, we shall merely emphasise the following essential fact: uopt > 1.
Translated
by
R.F.B.
Qreig
REFERENCE 1.
in stochastic models (Optimizatsiya v GURIN, L. S., Optimisation stokhasticheskikh modelyakh), Zh. Vych. Mat. 4, No. 2. 367-370.1964.
S.
GURIN
(MOSCOW) (Received
20
February
This paper is a direct continuation of in referring numbering of the formulae; the in [ll.
1964)
[I].
This also involves the to formulae (1) - (15). we mean
If we take condition (4) as the end of the computation. then. as already indicated in (1). the accuracy will depend not only on E but also on ?I and other parameters. In order to guarantee a definite degree of accuracy in this case it is necessary to make E a function of these parameters. Moreover a reasonable value should be chosen for a in formula (4). In order shall start With the
to obtain definite again from heuristic symbols
recommendations considerations.
on these
cluestions
we
adopted
f (20. YO)- f @l* YI) = fn @o, Yo)- fN (Q, t/l) + 5 ($)
= Af + 4 (y)
,
whence M if 6% yo) -
f (21, yl)la = (Af)2 + g,
Comparing with formula (4). we obtain the resuit mean value of the square of the difference will On the other
hand,
for
the
function
that for be 39.
(9)
[f(so, YO)--f(x~. Y,)]* = 16iiz(M2z,* + m*y$). Taking
l
the means,
we obtain
Zh. Vych. Mat. 4. No. 6,
1134 - 1137, 228
1964.
(16) a = 2/3
the
Optinisation
in
stochastic
229
nodels
Hence it is possibleto obtain the dependenceof E on A, and that of M on In,and with these we may expect the computationto end with a mean error M(x2+ Y")< e'.
($8)
In fact, taking the mean of xo2 + yo2 in accordancewith condition (171, i.e. assuming
and taking
CJJ to
be uniformlydistributedin (0, WV
we obtain
Comparing(19) and (18). +e obtain for the given E' E=
64Pe" 3L-i12‘ .W / nz~I i:
(3-O)
In Table I the results are given of a calculationfor the function fk
Y) = 5xz+4xy+&2,
consideredin [l], for E' = 0.01, x0 = 100, y. = 0,
k
=
0,
n
=
4;
results
TABLE I
64(l) 213 5.4 8.2 4.9 7.6 4.2 5.0 3.9 4.0 4 5 4 4 _I__.._I____J__I__/---L-J0.320.400.220.300.220.300.320.380.440.410.75~.64
L.S.
230
are given corresponding position of the results
Gurin
to various different values of A and - h. The in each cell of the table is as follows: 10-M (ZN)
I
10-b (ZN)
1~~ (12+ ya) loos The number of cases
(out of
(cl9 + y”) .
100) of divergence
is given
in brackets.
Analysis of Table 1 shows that it is indeed true that RIfz2 + y*) scarcely depends on h at all and. as a rule, &RX* + y*f < E*. Further, it is evident that hopt and A may be chosen in accordance with the formulae (11) and (15) obtained earlier, using E‘ in place of E, i.e. to take
These formulae were obtained in the following way: with A given in accordance with (21). formula (21) for A 1s obtained from formula (15). Substituting A and h into formula (111, we find that the appropriate inequality is amply satisfied. In fact,
both the latter
expressions
exceed
Mu and (11)
gives
The increase of A by a factor J-6 is brought about in order to to possible guarautee greater stability for the process in relation vergence. Thus we may regard on its parameters for clarified.
di-
the dependence of the convergence of the process given coefficients of quadratic form as fully
However in t&e solution of practical problems these coefficients are unknown. This creates a new problem which may be solved by two methods: 1) in searching for an extremum for the values ating the second derivatives of the function f(x.
of
f~(x,
yf,
by evalu-
y) and in accordance
Optimisslfion
them calculating
with 2)
selecting
of the point We shall
A
(x,,
in
stochastic
M, UI, and then also
and A directly yn).
consider
231
models
h, A;
by analysing
the nature
of the motion
method (1).
We shall introduce heuristic considerations at the start in the choice of the tiral step for evaluating the second derivatives. We have
Hence h* must be of the same order
as the mean square deviation
{,
i.e.
(22)
where the constant
A
has the
vnlue aefined
above.
In order not to return to this question, we note that the calculations described below were carried out for various values of P and it turned out that it is best to take u = 1. The algorithm for method (1) takes the following form: we are given arbitrary A and A; we then carry out a calculation according to the algorithm described above for the given A and A; on each occasion when it is necessary to increase N a times we carry out a re-computation of A and A. evaluating first of all the second derivatives through the second differences (in this we make the trial steps in conformity with (22)). and then determining M, II and, finally, h and A in accordance with (21). TABLE 2
lU-BM(Zh’) 27.2 io-%((ZN) 82 HJOM(zw+y*) 0.65
2.54 2.0 0.31
uJOs(z~+y~)
0.28,
0.92
3.31 4.0 0.23 0.26
6.02 9.8 0.29 0.41
Moreover, two further refinements have been introduced into the algorithm in order to avoid overloading the discharge network during
L.S.
232
operations
on the electronic
Curin
computer:
a) if it turns out that the step for the antigradient obtained in accordance with the formulae is greater than a certain constant L, it should be taken as equal to L (we took L = 100); b) if it turns out that III< 8. where 6 > 0 is a sufficiently small number (we took 6 = 0.011, then we should proceed to the usual multiplication of N by TVand recompute h and A. It is easy to see that with a positive-definite quadratic successful choice of 6 no cyclic process will occur.
form and a
A calculation carried out under the same conditions as in Table 1 gave results which are set out in Table 2. We shall compare the last column of Table 2 with Table 1, where n was equal to 4. In Table 1 for a successful choice of h and A lesser values of M( EN) were obtained. but for many A and A larger values were obtained. This means that in the finishing process the approximation to optimal h and A proceeds slowly; this is due to the difficulty of evaluating the second deriva(a difficulty demanding, furthertives of f(~, y) by random realisations more, supplementary realisations). However even the better results from Table 1 are only 1% times better than the corresponding results of Table 2. Therefore we may regard method (1) of evaluating the quadratic form as satisfactory. It may be expected that method (2) will turn out to be better only for the less accurate optimisation of f(x, y). But in this case we should bring in for comparison other methods of optimisation also, for example, the method of random search. We note further that Table 2 confirms the conclusion drawn in (1) that an optimal n exists near to n = 1.4. Without embarking on a more accurate determination of the value of n, we shall merely emphasise the following essential fact: uopt > 1.
Translated
by
R.F.B.
Qreig
REFERENCE 1.
in stochastic models (Optimizatsiya v GURIN, L. S., Optimisation stokhasticheskikh modelyakh), Zh. Vych. Mat. 4, No. 2. 367-370.1964.