If the set S = {x: f (CC)
GRADIENT METHODS FOR TEE MINIMISATION OF FUNCTIONALS* B. T.
POLYAK
(MOSCOW) (Received
2
July
1962)
Let f(t) be a functional defined in the (real) Hilbert problem consists in finding its minimum value f* = inf minimum point z* (if such exists). We shall assume everywhere below that differentiable, i.e. for any x, y f lx +
Y> =
f (4
+
the function
@ (47
ld +
where h(x) E H is the gradient of the functional and is assumed to depend continuously on x. Gradient minimisation sequence ~9, x1, . . . , n”,
is continuously
0 (II Y II),
f(x)
methods consist in constructing . . . according to the formula zn+i = Cl? -
space H. The f(x) and some
(1)
at the point
x
the minimising
a& (sn) .
The size of the step a,>0 can be chosen in various ways. Thus, in the descent a, simple iteration method a, = const. In the method of fastest is chosen to minimise the functional at each step, i.e.. a,, gives the minimum 9 (a) = f (zn - ah (rn)). the minimisation problem is When f(n) is a quadratic functional, equivalent to the solution of the linear equation h(x) = 0. This case has received detailed study, and a survey of this work can be found in [I]. Although gradient methods are widely used in the practical solution
l
Zh.
uych.
rat.
3, No. 4, 643-653, 864
1963.
for
sininisation
865
of extremal problems for non-quadratic functionals also (mainly in finite-dimensional spaces, aa in 1~1, for instance) little study has been made of the questions of convergence and rate of convergence in this case. There are several results for a finite-dimensional space in [31, 141, for a Hilbert space in [51 and for a Banach space in Id. In most of the research (in td, I61 and [71 in particular) the problem of the extreme is considered as an auxiliary problem associated with the solution of the equation h(x) = 0. The conditions for the convergence are therefore expressed in the form of restrictions on h(x). We shall use another approach: we take the extremumproblem to be the initial problem, the equation h(x) = 0 being simply a necessary condition for an extremum. For this reason we lay certain restrictions on the functional itself (as a rule these are simple restrictions, such as boundedness below) and not only on its gradient. Since the majority of applied problems are formulated with the use of variational principles, testing these conditions does not usually cause any difficulty. Let us first give a theorem which is useful for the proof of the convergence of many methods of descent (not necessarily gradient methods). Theorea 1. Let Z be a topological space, f(z) a continuous functional on it, P an operator mapping 2 into 2, where f (Pz) < f (2) and f (J%) is a semi-continuous function of z from above. Then if the sequence z”, z1 = Pz”, . . ., P+l = i%P, . . . has a limit point z*, f (Pz’) = f (2.). Proof.
Obviously f (z*) \ 0. It follows from the fact thlt-f(Pz) is semi-continuous above that there is a neighbourhood U of the point I* such that f (Pz) - f (Pz*) i 0
>
11 h
1).
(0) /i2[(‘2
[,I h
(0) II - R II5 (0 -
x (0)/II2dt >
0
-
eRf)2 dt = -!!!L$K(4~n2_+)>0.27~.
0
Thus Ijh(t)\lz
