Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Some Minimax Problems in Optimization Theory*
V. F. Demyanov
1. Many problems arising in engineering, economics and mathematics are of
the form : Minimize a function <p{x) subject to x e Q where <p{x) is one of the following functions :
(1)
<p{x) = max/(x, y)9
y^G
(2)
<p{x) = max/(;e, y)9
(3)
<p{x) = max min f{x9 y9 z)9
(4)
<p{x) = max
y^G(x)
y^Gi(x) z^Giix)
min ••• max min f{x,yÌ9 '"9yk9zÌ9
yi^Gnix) ZIGGHU)
~'9zk)
y»^Gu(x) zt^Guix)
and sets G{x)9 G(j{x) depend on x9 G is a given set.
Such problems often appear in the engineering design theory. In recent years
much attention was paid to the problems described. We mention only some books
dealing with minimax theory [1], [5], [7], [9], [13]. It seems possible to claim that at
present the minimax theory is formed. The minimax theory deals with the following problems :
1. Investigation of the properties of the functions (1)—(4) including their directional differentiability. For various types of functions conditions for the function
to be directionally differentiable are obtained and formulae for the first and higher
order directional derivatives are found (see for example [6], [8], [9], [10], [18], [20],
[21], [22]).
2. Necessary and sufficient conditions and their geometric interpretation [2], [3],
[9].
3. Steepest-descent directions and their applications to constructing numerical
* Not presented in person.
© 1975, Canadian Mathematical Congress
335
336
V. F. DEMYANOV
methods. Numerical methods of thefirstorder (of the gradient type). These problems have been widely disucssed and studied for the function (1). For this case the
first order methods [13] as well as various second order methods (see for example
[25]) have been worked out. Some useful estimations have been obtained [12], [16].
Active research is under way to obtain numerical methods for minimizing the
function (2) (see [24]) and the function (3) [17]. But there is too much to be done in
this field. The main problem for the immediate future is to develop software for
minimax problems and its practical applications.
Sometimes it is possible making use of special properties of the problem to
develop an effective method for its solution.
4. Saddle points. The problem offindingsaddle points is a special case of minimax
problems. For this case it is possible to construct methods where it is not necessary to calculate the value of the function (1) at each step (see surveys [11], [14]).
5. Optimal control problems with a minimax criterion function.
6. Nonlinear approximation problems [15], [19].
Now we discuss Problems 1 and 5 in detail.
2. Let <p{x) = maxye(Kx)f{x9 y) where xeEmyeEm.
Fix xQ and g e E„. Let
T{y) = {VeEm | g aQ > 0:y + aveG{x0 + ag)Vae[09 a0]}.
The closure of T{y) we denote by r{y) = r{xQ, y, g). It is known [9], [22] that under
some additional conditions the function cp{x) is differentiable at the point xQ w.r.
to the direction of g and
^Xo)
0g
= lim—-[<p{xQ + ag) - <p{xQ)] = sup
«-+0 «
ye£(*0) v^r{y)l
sup
'&'<&*)}
Higher order derivatives we define as follows. Suppose that / ^ 2 and that it is
already known that
<p(x0 + ag) = <p(x0) + S ^ ^ - ^
+ o(«'-i)
where dk<p/dgk are derivatives of the function <p w.r. to the direction of g at the
point XQ. Then the limit
dl<p{x0) r
/! T , , .
/ x tì 9*p(*o) ock
if it exists is called the /th derivative of the function <p{x) at the point xQ w.r. to
the direction of g.
Now let us introduce the set ri{y, vÌ9 •••, vt_{) of feasible directions of the /th
order. Suppose that sets rl{y)9 r2{y, vx)9 •••, ri~\y, vh •••, v,_2) have already been
defined. Fix y e G{x0)9 vx e rl{y)9 •••, V/_x e ri~\y9 vl5 •••, V/_2) and define the set
V{y9 vl9 - , V/.0 = \vteEm | 3 a0 > 0: y 4- JS a* • v* e tf(*0 + ag)Vae[09 a0]}.
The closure of P{y9 vh •••, v,_x) let us denote by P{y9 vx, ••-, v/_i) and call it the set
of feasible directions of the /th order (or course P and f' depend on xQ and g).
SOME MINIMAX PROBLEMS IN OPTIMIZATION THEORY
337
Suppose that the function / is / times continuously differentiate. Then for any
a e [I : /] the following expansion is valid :
(5)
/ ( *o + ag, y + S akVk + <>{a°) j
V
k=l
\
k
= /(*b, y)+& Afa,y9
k=i
g, vu ", VÙJTKl + o{aa)
where A k does n o t depend on a and is a function of the derivatives of the function
/ o f order ^ k. Suppose that for any sequence {ys}9 ys e R{xQ + asg)9 as -> H- 0,
there exists a subsequence {yS{} which can be written as
/-i
J>„ = J + I ! aj v* + «(• vw + o(a{)
where j ; e G{x0), h e rl{y\ •••, V i e / ^ ( A v b •••, Vz), % e T 7 '^, vi, - , Vi)>
at- vti -> 0 as / -> oo.
THEOREM 1. Le* / ^ 2. If there exists the first derivative of the function <p{x) at the
point XQ w.r. to the direction of g and (5) is true, then under the assumptions above
there exists the derivative of any order a e [2: n] and
(6)
%ig
=
SU
P
SU
P
^ ^ 0 ' y* & vi> ' " ' v * )
wÄere >4ff(x0, y, g, Vi, •••, vff) is taken from (5); T^" 1 IJ the set of elements of [y, vi,
•••, vff_i] iSt/c/i / t o supremum in the formula for the {a—l)th derivative is achieved
at points [y9 Vi, •••, v„_i]. Note that Ta~l is not empty for G G [2 : n ].
3. Minimax problems in optimal control. Let x{t) =f{x9 u, t)9 x{0) = JC0, where
x = {xÌ9 •••, xn)9 f = {fÌ9 •••, / „ ) , u = {ul9 •••, ur) a n d the functions f a n d 3 / , / 3 *
are continuous in all variables. By XJ let us denote the class of piecewise continuous controls u{t) such that u{t) e W c= Er for any t e [0, T]. Let /(w, z) =
$g{x, u9 t9 z) dt where ze Z a Ep, the functions g and dg/dx are continuous in
all variables. N o w let us consider the problem
(7)
max I{u, z)
> min.
Under some additional conditions the following result is valid.
THEOREM
2 (SEE [4]). For a control w* e U to be an optimal one it is necessary that
T
min max \[Hz{u9 z) — Hz{u*9 z)] dz = 0
« e t / ZœR(U*)
0
where R{u) = {z e Z\l{u9 z) = max üGZ I{u9 v)}9
d(pJs)
dz
=
_ (df{x{z9u*)9u*9z)\*
r(T)
M _ dg{x{z9 a*), u*9 z9 z)
\
dx
) w }
dx
'W
References
}
= Q
1. A. Auslander, Problèmes de minimax via l'analyse convexe et les inégalités variationnelles.
Théorie et algorithmes, Lecture Notes in Economies and Math. Systems, Springer-Verlag, Berlin
and New York, 1972.
2. V. V. Beresnev, Necessary conditions for extremum in a convex maximin problem on connected
338
V. F. DEMYANOV
sets, Kibernetika (Kiev) No. 2 (1972), 87-91. (Russian) MR 46 # 3079.
3. V. G. Boltjanskiï and I. S. Cebotaru, Necessary conditions in a minimax problem, Dokl. Akad.
Nauk SSSR 213 (1973), 257-260 = Soviet Math. Dokl. 14 (1973), 1665-1668.
4. T. K. Vinogradova and V. F. Demyanov, On the minimax principle in optimal control probìemsy Dokl. Akad. Nauk SSSR 213 (1973), 512-514 = Soviet Math. Dokl. 14 (1973), 1720-1722.
5. Ju. B. Germeïer, Introduction to the operations research theory, "Nauka", Moscow, 1971.
(Russian)
6* E. G. Gol'steïn, Duality theory in mathematical programming and its applications "Nauka",
MOSCOW, 1971. (Russian) MR 48 #893.
7. J. M. Danskin, Jr., The theory of max-min and its application to weapons allocation problems,
Econometrics and Operations Research, vol. 5, Springer-Verlag, New York, 1967. MR 17 # 3843.
8. B. S. Darhovskiï and E. S. Levitin, On differential properties of minimax in one distribution
resources problem, Z. Vycisl. Mat. i Mat. Fiz. 13 (1973), no. 5,1219-1227. (Russian)
9« V. F. Demyanov, Minimax: Directional differentiability, Leningrad Univ. Press, Leningrad,
1974. (Russian)
10. V. F. Demyanov and A. B. Pevnyï, First and second marginal values of mathematical programming problems, Dokl. Akad. Nauk SSSR 207 (1972), 277-280 = Soviet Math. Dokl. 13 (1972),
1502-1506.
11.
, Numerical methods of searching saddle points, TL. Vycisl. Mat. i Mat. Fiz. 12 (1972),
no. 5,1099-1127. (Russian)
12.
, Certain estimates in minimax problems, Kibernetika (Kiev) No. 1 (1972), 107-112.
(Russian) MR 46 #1354.
13. V. F. Demyanov and V. N. Malozemov, Introduction to minimax, "Nauka", Moscow, 1972.
(Russian)
14. S. I. Zuhovickiï, R. A. Poljak and M. E. Primak, Concave n-person games (numerical
methods), Ékonom. i Mat. Metody 7 (1971), 880-900. (Russian) MR45 #6460.
15. V. N. Malozemov and A. B. Pevnyï, Alternation properties of solutions of nonlinear minimax
problems, Dokl. Akad. Nauk SSSR 212 (1973), 37-39 = Soviet Math. Dokl. 14 (1973), 1303-1306.
16. A. B. Pevnyï, Minimization of a maxmin function, Z. Vycisl. Mat. i Mat. Fiz. 12 (1972), no.
1,227-230. (Russian) MR 46 # 1089.
17.
, On convergence rate of some methods for finding minimax and saddle points, Kibernetika (Kiev) No. 4 (1972), 95-98. (Russian)
18. B. N. Psenionyï, Convex multivalued mappings and their conjugates, Kibernetika (Kiev) No.
1972), 94-102. (Russian) MR 46 #7926.
19. J. R. Rice, The approximation of functions. Vol. 1 : Linear theory, Addison-Wesley, Reading,
Mass., 1964. MR 29 #3795.
20. A. I. Sotskov, Necessary conditions for a minimum for a type of nonsmooth problem, Dokl.
Akad. Nauk SSSR 189 (1969), 261-264 = Soviet Math. Dokl. 10 (1969), 1410-1413. MR 41
#4353.
21. Ju. N. Tjurin, A mathematical statement of a simplified model of industrial planning, Economics and Math. Methods 1 (1965), 391-410. (Russian)
22. W. W. Hogan, Directional derivatives for extremal value functions, Working Paper no. 177,
Western Management Science Inst., Los Angeles, Calif., 1971.
23. A. B. Pevnyï, On thefindingof minimax under connected constraints, Optimization (Novosibirsk) 10 (27) (1973), 22-29. (Russian)
24. E. F. Voïton and L. N. Poljakova, On one class of extremal problems with connected constraints, Optimization (Novosibirsk) 10 (27) (1973), 41-46. (Russian)
25. L. V. Vasiljev, Onfindingthe steepest descent direction in minimax problems, Sixth All-Union
Conf. on Extremal Problems, Tallinn, April 16-19, 1973, Abstracts of Reports, vol. I, pp. 61-62.
(Russian)
LENINGRAD STATE UNIVERSITY
LENINGRAD, U.S.S.R.