On Some Theoretical and Computational Aspects of the Minimax Problem

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REPORT R-423 JULY 1969

COORDINATED SCIEN CE LABORATORY

ON SOME THEORETICAL

AND COMPUTATIONAL ASPECTS

OF THE MINIMAX PROBLEM

JURAJ MEDANIC

UNIVERSITY OF ILLIN O IS - URBANA, ILLIN O IS

Juraj Medanic

Coordinated Science Laboratory University of Illinois

Urbana, Illinois

This wor k was supported in part by the Joint Services Electronics Program (U.S„ Army, U.S. Navy, and U.S. Air Force) under contract DAAB 07-67-C-0199; and in part by Air Force AF-AFOSR 68-1679; and in part by NSF Grant GK-3893.

Reproduction in whole or in part is permitted for any purpose of the United States Government.

This document has been approved for public release and sale; its distribution is unlimited.

OF THE MINIMAX PROBLEM*

by

Juraj Medanic**

Coordinated Science Laboratory University of Illinois

Urbana, Illinois

This work was supported by the Joint Services Electronics Program (U.S. Army, U.S. Navy, and U.S. Air Force) under Contract No. DAAB-07-67-C-0199; in part by Air Force AF-AFOSR 68-1579; and in part by NSF Grant GK-3893.

•irk

This report contains both a brief summary of the pertinent theory of the minimax problem in finite dimensional spaces and the development of an elimination type algorithm for the practical computation of the minimax solution. Because algorithms for finding saddle point type solutions are easier to implement it is of considerable interest to know when they can be employed in minimax problems. A lemma giving sufficient conditions for the minimax to be a local saddle point is presented in the paper. However, the most interesting minimax problems do not admit saddle point solutions and other algorithms must be sought. Two lemmas are given in the paper which guarantee the convergence of the elimination algorithm proposed in the paper for the computational solution of the minimax problem. A shortcoming of the algorithm is that its application is restricted to minimax problems in which the max function is pseudo-convex.

OF THE MINIMAX PROBLEM

Introduction

The minimax problem consists in determining x* and possibly

y*

J(x*,y*) = min max J(x,y) . x e X ye Y

(1)

Solutions of many problems in decision theory, in the theory of games and differential games, in control under uncertainty and operations research rests on the possibility of computing the above defined minimax solution.

The degree of difficulty is associated with the nature of the admissible subsets X and Y of the respective spaces X and Y and on the dependence of J on x and y. In decision theory and the theory of games x and y are

frequently probability distributions over appropriate sample spaces and X and Y are collections of admissible probability distributions; in differen­

tial games X and Y are usually subsets of spaces of piecewise continuous, or measurable functions; in control problems involving uncertainty and operations research problems X and Y may be subsets of finite dimensional spaces. Theoretical aspects of the problems are fairly well understood but practical computational procedures are lacking even for the simplest case where X and Y are finite dimensional subsets of E^ and E g , respectively, and J(x,y) is a real valued function defined on XxY. In this paper we review briefly the basic properties of the minimax solution and summarize the computational difficulties involved in solving minimax problems in finite dimensional spaces.

It is well known that if there exists a global saddle point (x*,y*), i.e. if J(x*,y) < J(x*,y*) < J(x,y*) for all (x,y)eXxY then the minimax solution coincides with the saddle point solution. When this is the case, the search for the minimax solution is relatively simple, since (i) most methods in nonlinear programming may be employed to obtain the saddle point solution and (ii) by appropriate sequencing of minimizing and maximiz­ ing iteration steps both upper and lower bounds on the saddle point may be

obtained and used as a stopping criterion. The existence of a global saddle point

can, however, be ascertained only for the class of convex-concave* functions. To justify the use of saddle point algorithms in the solution of minimax problems, it is sufficient to have the minimax solution be a local saddle point of J(x,y) in XxY. It is therefore natural and important to ask the question: When is a local saddle point of J(x,y) the minimax solution in XxY? The first contribution of this paper is a lemma stating sufficient

conditions for the minimax solutions to be a local saddle point. If a function satisfies the conditions in the lemma, saddle point algorithms can be used to find the minimax solution. Some of the most interesting minimax problems,

however, do not admit a saddle point solution. The second contribution of this paper is an algorithm which obtains the minimax solution in a class of minimax problems where the minimax solution is not a saddle point and local methods cannot be used to obtain it.

Applications

Many fields of system theory in the design and planning stage utilize in some manner solutions based on the minimax philosophy. For the

purposes of this paper it is sufficient to point out some practical fields of application. The first two problems are discussed in the book by

Danskin [ l] and have been selected because they represent a class of realistic problems and also because the desired solution is truly a

minimax (more precisely maxmin) solution. The third example is not a true minimax problem but represents a common approach in the design of some control systems in the presence of uncertainty.

Consider first the "weapons allocation" problems. Suppose a set of n weapon systems is given and is to be used in the possible counter­ attack of n different targets. The amount of the i-th type of system

(measured in proportion of the budged) is x^ and its effectiveness against the specified i-th target is b^. The value of the i-th target on a relative scale is v_^. It is assumed the defender of the targets will strike first, with the full knowledge of the system x = ( X p . . . , x n) and with ample time to

develop an optimal counter against the selected system x. If he applies y^ units of attack to the i-th weapons system the original number of weapons

"a iy i

is reduced to x^ = x^e . The damage the residual quantity x! can do to

b.xl 1

the i-th target in the counterattack is v^(l - e 1 1) and suppose the damage adds from target to target. Then the total damage resulting to the enemy defending the targets, after the enemy has attacked and the residual weapons have been used to counterattack is,

J ( x 9y) n . S v . (1 i= 1 r - a . x . l l e - b . y .

it

for x is naturally given by the maxmin solution J(x,y) - b . y . l l n “a ix i e max min 2 v. (1 - e ) x G X yeY i

= 1

under the constraints

(

2

)

2 x . i=l 1

x. > 0, y. > 0

l — y l — (3)

where c and c are constants.

x

y

Consider next the "residual value of a target" problem. Suppose now that one target is defended by x^ definse units of the i-th type of weapons and attacked by y. attack units of the enemy's i-th type of weapons,

-k.x./y. 1 I

1

l

The quantity e is the probability that an individual attack unit of the i-th type gets through, the ratio x^/y^ being the amount of defense with the i-th type weapon against an individual (enemy's) i-th unit and k^ the effectiveness of such defense; a. is the probability that a surviving attack

-k.x./y.

unit destroys the target. The quantity 1 - a.e 1 1 1 is then the proba--k.x./y. y. bility that the target survives the attack and finally (1 - a^e 1 1 1) 1 is the probability the target survives the attack by weapons of the i-th type. The total probability of the target to survive the attack by all available types of weapons, called the residual value of the target, is

J(x ,y)

2

V

1

selecting x constructs his defense first and that the player selecting y moves in full knowledge of what the x-player has selected. Hence, once again, a true maxmin problem.

Many related and interesting problems can be found in Danskin [l]. All are distinguished by the specific form of J(x,y) which is separable with respect to the pairs ( ^ » y ^ • That is, in general, the criterion has the form

n

J(x,y) = J i V v y p (

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which has allowed Danskin to analyze the features of the solution by

extensive application of a lemma due to Gibbs. Although Danskin emphasizes the theoretical aspects of the problem some examples have been solved by hand but no indication is given of an existing computer algorithm. Indeed, it seems that outside of an outright search procedure, or analytic calcula­ tion of the max function no organized way of computing the maxmin solution is made available. However, quantitative solutions of realistic problems can only be obtained by computerized methods. Moreover, even qualitative solutions are hard to analyze in the absence of specific properties of J(x,y) .

In control theory the design of compensating or control circuits in linear control systems is a classical and fairly developed field. Many different criteria can be devised and are employed in accepted synthesis

procedures. The minimax criterion (and its modifications) also possess its advantages if there is some uncertainty concerning the values of plant

as in Fig. 1, where the time constant T of the compensating circuit and the over all gain constant K are adjustable parameters. On the other hand, the motor time constant T^, the damping factor § and the natural frequency (0^ of the driving servo may be uncertain parameters varying within specified

bounds (for example if the same compensating system is to be used on a batch of servo units). Given fixed values for T^, 5, and U)^ a frequently used design criterion is to minimize the integral squared error,

joo

J(T,K,Tm ,|,a)n) = ^ J' E(s)E(-s)ds (6)

J -joo

where in this problem

E (s)

T s3 + <2?0) T + 1) s2 + (id2T + 2?u> )s +

w2

_____ m_____ n m ______ n m n ______ n_______ T s4 + (2§tt) T + l)s3 + (U)2T + 2§U) ) s 2 + (U)2 + TK) s + K

m n m n m n n

(7)

Given only bounds for T^, §, and 0)^, a feasible modification of the design criterion is to select values T* and K* such that

J(T*,K*,T

*,£>*,&

m ? ’ n

In case the minimax solution to problems with uncertainty is judged to be too pessimistic, modified versions such as the performance sensitivity [

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In this paper, a method for solving the minimax problem based on the contraction of the admissible domain of minimizing parameters to an

arbitrary small neighborhood about the minimax value will be presented. The following, Section 3, of this paper is devoted to the presentation of the theoretical aspects of the minimax problem and justification of the algorithm. The algorithm itself is presented in Section 4 and problems discussed in this section are solved in Section 5.

2. Theory

Consider a real valued function J(x,y) defined on the subsets X and Y of the respective finite dimensional spaces and E g . It is assumed that

(a) X and Y are closed, bounded (hence compact) and convex and that

3 J

(b) J(x,y) together with the partial derivatives ^ is jointly continuous in x and y.

The solution of the minimax problem consists in obtaining x* and y* such that

J(x*,y*) = min max J(x,y) . (2.1)

xeX yeY

In this paper an algorithm will be presented which solves the problem under the additional assumption

(c) J(x,y) is convex in X for all yeY. Let the max function be defined by

F(x) = max J(x,y) (2.2)

yeY

denoted by Y(x) and called the answering set. The minimax problem, therefore, consists in finding the minimum in X of F ( x ) . All difficulties encountered in the solution of the minimax problem are related to the nature of F(x) . Let us therefore summarize the basic properties of F(x) in the form of a

lemma:

Lemma 1. The max function F(x) is continuous in X and possesses at any xeX a directional derivative D^F(x) in the arbitrary direction characterized by the unit direction vector Y. The directional derivative is upper semi-continuous in the pair (Y,x), continuous in Y alone for all xeX and defined by

D F ( x ) = max |^(x,y) *Y. (2.3)

Y y0Y(x) °X

The answering set Y(x) is upper semi-continuous with respect to inclusion.

Continuity of F(x) and upper-semiconductivity of Y(x) follows from the continuity of J(x,y) and the definition of F(x) [

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Lemma 2. There exists a global minimum of F(x) in X.

Existence of the global minimum follows by the theorem of Weierstrass from the continuity of F(x) and compactness of X.

To characterize the local minima of F ( x ) , let the directional derivative be used to make a first order expansion of F(x) about some

nominal point x in the direction Yer(x) , where T(x) is the cone of admis­ sible unit direction vectors at xeX:

F(x + Yh) = F(x) + Dy F(x) -h + 0(h) (2.4)

and h is the step size, h = ||x-x||, x = x + Yh. For h sufficiently small the

first order approximation is valid for all Yer(x) . in order that the point x be a local minimum of F(x) it is necessary that an arbitrary small step

in any direction does not further decresas the value of the max function, i.e.,

F ( x + yh) “ F(x) > 0, for all Y,

||v||

y ep i a ) [ F ( x + Y h ) - F(x)] > 0. (

2

6

In view of the definition of the directional derivative (2.3), the expansion (2.4) and the definition of h this leads to the following necessary condition for local minimax solutions:

Lemma 3 . In order that x* be a local minimax solution it is necessary and sufficient that

min max § “ (x*,y)-Y > 0. (2.7)

Y®r (x) yeY(x*)

[1,4,5,6], In case the admissible domain X is given by a set of inequality constraints which satisfy the Kuhn-Tucker constraint qualification [

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Lemma 4 . In order that x* be the local minimax solution of J(x,y) under the set of inequality constraints cp(x) < 0 and satisfying the Kuhn-Tucker constraint qualification it is necessary that there exist a vector V > 0 such that

min ( max (x*,y) + V*|“£:)*Y > 0. (2.8)

Y yeY(x*) dX

Moreover, if cp^(x*) < 0, = 0.

The necessary conditions in lemmas 3 and 4 tend to draw a parallel between the minimax problem and minimization problems of nonlinear program­ ming. Formally, the basic difference between the two is that in minimization problems F(x) is usually continuous and continuously differentiable whereas in the minimax problem the most interesting case occurs when it is only continuous. The necessary conditions in minimax problems are thereby more difficult to employ constructively and, in particular, the multiplier

theorem is not as revealing as in minimization problems. Additional computa­ tional burden and principal difficulties are involved in the solution of minimax problems because F(x) is not explicitely known and instead of an r-dimensional search in X one must carry out an r+s-dimensional search in

t XxY.

t

It is for this reason why both necessary conditions and the expression for D^F(x) are stated in terms of the function J(x,y).

It is not difficult to see that local methods based on recursive relations X n+1 V n " X n + X hn n yn + Tlk n n n (2.9)

such as gradient and Newton-Raphson type algorithms (e.g., the algorithm in [

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(x*,y*) the saddle point condition is satisfied:

J(x*,y) < J(x*,y*) < J(x,y*), for all (x,y) eN(x*,y*) . (2.10)

A new result in terms of a sufficient condition for the minimax solution to be a local saddle point is given by the following lemma:

Lemma 5 . If for all xeX the answering set Y(x) is convex then there exists a local saddle point (x*,y*) which is the minimax solution of J(x,y) in XxY.

The proof is given in the Appendix. It is interesting to note the relation

of the above lemma to the saddle point theorem of Ky Fan (reduced to finite dimensional spaces) [ll]. Let, analogous to Y(x) , X(y) be the set of

points xeX which minimizes J(x,y) for a given yeY. Then,

Lemma 6 . If for all xeX and yeY the answering sets Y(x) and X(y) are convex then there exists a pair (x*,y*)®XxY which is a global saddle point of J(x,y) in XxY, i.e. satisfies

J(x*,y) < J(x*,y*) < J(x,y*), for all (x,y)eXxY. (2.11)

It is well known that when (2.11) holds minmax J(x,y) = maxmin J(x,y) = J ( x *,y*)). Consequently, in view of lemma 5, omitting the convexity requirement on X(y) has the effect of reducing the minimax solution from a global saddle point to a local saddle point.

In particular, convexity of the set Y(x) is assured if J(x,y) is concave in y, and the above lemmas then indicate that the minimax is at

least a local saddle point. If concavity can not be assumed the solution is generally not a saddle point and algorithms based on local properties of J(x,y) only, in general, will not converge. To the authors best knowledge,

two algorithms have been proposed for the solution of the general minimax problem in finite dimensional spaces. The first [2] is based on the proper sequencing of minimizing and maximizing iterations, the second [6] on the utilization of local maxima in Y and the generalized definition of the gradient of F ( x ) . Both algorithms are, however, best suited for the type of problems analyzed by Damjanov [l3] where the set Y is a finite set. Both algorithms have extensive computational requirements, Heller's having

the advantage in that most of the tasks required can be performed by

standard routines for global maximization, linear and quadratic programming. Among the weaker points of both algorithms is the stopping criteria for estimating the distance from the current position to the minimax solution.

In this paper we present an algorithm for an important subclass of

minimax problems which do not satisfy the condition in lemma 5. The proposed algorithm may be shown to converge to the minimax solution whenever the

special case when J(x,y) is convex in X (assumption (c)). The use of c o n ­ vexity property allows great simplifications in computational requirements. Moreover, the nature of the algorithm, which consists of reducing the

domain X to an arbitrary neighborhood of the minimax point x*, inherently eliminates the difficult question of estimating the distance to the minimax solution. In this section two lemmas are given which guarantee the conver­ gence by the proposed algorithm to the minimax solutions.

Denote X by X and consider an arbitrary x eX , where X is a

J

o

J

subset of X . Maximize J(x ,y) to obtain y such that

o n

J

F(x ) = J(x ,y ) = max J(x ,y) . (2.11)

v n n ,yn „ n ,y

yeY

Expand J(x,yn) about x^ and use the convexity assumption to obtain

J(x,y ) > J(x ,y ) + ^ (x ,y ) (x-x ) , for all xeX. (2.12)

\ \ n ’-'n' ox v n ’-'n v n7 * v

'

Define the set

H (y ) = {x : (x ,y ) (x-x )) < 0} (2.13) n w n ox v n ,yn v n —

and define the recursive relation

X . = H (y ) fix , x eX , n = 0,1,... . (2.14)

n+1 n w n n* n n* ’

The following lemma justified the use of the elimination algorithm:

The following modification of the above contraction process is of practical interest. Suppose is a subset of Xq obtained by the modified

process and let x^eX^. Find any y^eY such that

J(x ,y. ) > J(x i jy 0 , x .eX

v

n iyk

N n-l’^n-l *

n-1 n-1

If such a y^ does not exist take y^ = y^, where y^ is given by (2.11). Define

H (y ) = (x : (x ,y ) (x-x ) < O]

n k ox n

n ' —

and redefine the recursive relation as

X n = H (y.)flX ,

n+1 n w k n (2.17)

Lemma 8. The series [x } contracts to the minimax solution x*eX.

--- n

The proofs are given in the Appendix.

3. Algorithm

Construction of an algorithm based on lemmas 7 and 8 consists in resolving the following tasks (a) maximization over Y to obtain Y(x^) ,

(b) reduction of X^, (c) selection of the next nominal point x^+ ^ and (d) determination of a viable stopping criterion. The following algorithm was developed and applied to solve the examples in the following section. The basic steps of the algorithm are presented first and the modified algorithm acutally constructed is then described. In order to give the

only one element or that only one element from Y ( x p was utilized in the algorithm. Hence, at the beginning of the n-th elimination the reduced domain X is defined by the set of linear constraints,

n

J

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g*x £ 8^x i » i = 1,..., n + m - 1 (3.1)

where for i = l,...,m, the inequalities correspond to the constraints originally defining X while the rest have been added in the previous n-1 eliminations. The original constraints are specified by input data which give g^ and x^, i = l,...,m while for i = m f l , ...,n-l, x^ is the nominal point in the i-.th elimination, y^ is an element of Y ( x p and

d J , v

-k

8 i = si (V y i> • (3.2)

Let x^ be the nominal point obtained in the n-1 elimination. Maximize J(x ,y) to obtain y eY(x ) and find g which defines the set

n J,/

J n

Hn (y ) by the linear inequality

x < g *

n x .n (3.3)

Contraction of X^ to X ^ ^ consists in adjoining (3.3) to the set of constraints (3.1).

To select ’x ^ ^ e X ^ observe that all points on the ray

x = x - \ g , \ > 0 (3.4)

n n

belong to for sufficiently small X, such that none of the constraints in (3.1) is violated. It is proposed that the nominal point be selected as

the midpoint between x^ and the first intersection of the ray (3.4) with the constraint set (3.1). The value of X corresponding to the point of intersection is X n min X . i 1 min i n

8

and X > 0. Hence, the new nominal point is

x t = x - %X g .

n+1 n n n (3.6)

If, as assumed, the admissible domain X = Xq , defined by (3.1) for i = m, was bounded and the point x is chosen in the interior of X , the same

o o ’

properties will hold for x^ and X^, and a meaningful elimination is

performed in each iteration whenever gn + -^

4

n+1 n n n°n

The step size between x and x .. is selected as the basic

n n+1

stopping criterion. The procedure is interrupted when the setp size is below a prespecified bound q, i.e. when

* J | g n l l < q • (3.7)

Before the search is completely terminated an additional stopping criterion is applied which contains an estimate of the hypervolume of X . The

n

following criterion was used in solving the examples in the following t

section. If after N eliminations (3.7) becomes satisfied, solve the

+

A different criterion is used in Ref. C15] and has been applied with success in [l4].

following nonlinear progamming problem: Maximize

s =

(3.8)

(3.1)

(3.3)

Gx < z . (3.9) Let Stop if S o max xeXn x-x (3.10) S < d. o — (3.11)

If not, take a step, basically in the direction connecting x^ to x, but modified so as to insure that the new nominal point x^+ ^ is not on the boundary of X^. In the algorithm used here this was achieved by taking

xn+1 to be a convex combination of three points ( x ^ ^ x ^ x ) . Continue the basic elimination procedure.

Modifications made in the actual implementation of the algorithm relate to the maximization over Y and were induced by the result of lemma 8. The procedure used to maximize was to replace the set Y by a grid of points covering Y. The step of the grid is adjustable and a finer grid is used when maximizing for the first time. The coarser grid is used in all

subsequent maximizations whenever the grid search produces a point J(x ,y ) n k

In general, the grid search, that is maximization over Y, will

produce either the set Y(x ) or, with the coarse step, the set Y (x ) of

n n

points satisfying (2.15). In any case, the obtained set need not be a

singleton and the following modification of the algorithm was made in order to utilize the possibility of greater elimination. For every y^eY(x^) find

^ T

the corresponding X ^ given by (3.5) where now g^ = gn (yk) = ^ ^Xn ,yk ^ * ’

find the step sizes

Cqk 3 , qk = ^ n J l8 n (y k) II * L e t § n an d

n n+1

adjoining to (3.1) the set of constants

Sn (yk}* X - 8n (yk> * V V ?( x n} (3'12)

except those hyperplanes in (3.12) which would make x lie outside of X ,.

n+1 n+1

The general flow diagram of the elimination algorithm is presented in Figure 2.

4. Solution of Examples

The solution of minimax problems by the algorithm presented in the previous section demands only that a subroutine supplying J(x,y) and ^ (x,y) be added to the basic program. The application is now illustrated

in the examples cited in Section 3.

The constraint set in the first two examples contains an equality constraint n £ x. i=l L c

X

and the best way to handle the constraint in these examples is to use it to reduce the dimensionality of the space of x, i.e. to place

x = c - S x . . (4.2)

n x i=1 i

The problem of finding the maxmin solution, rewritten in the form of a minimax problem then takes the form

J(x,y) = n -2 v . (1 i=l 1 -<2.y, Q 1 1 - p . x. e e ) x^ > 0, i = 1,...,n-l c x n S x. > 0 i=l 1 “ (4.3) (4.4) (4.5)

and the gradient has the form

-sT -a.y. -ß.x.e Oj Q 1 1 1 1 t— = -v.p.e e OX. 1 1 1 -Q'.y. l l + v ß e n n

- a y

where x^ in (4.3) and (4.6) is a dependent variable defined by (4.2). The maximization over Y was handled by prescribing a "grid" of points and selecting the maximum of J(x,y) for fixed x over the grid, refining the step of the grid as necessary in accordance with the modification of the algorithm as discussed in the previous section. For a specific example with the numerical values n = 4,

< H-* II 1 _i = 4 II

CM

CO

CM

and for c x = Cy = t^ie final result in terms of the coordinates of the last nominal point was

x^ = .0000 x 0 = .1891 x 0 = .2942 x. = .5167

2 3 4 (4.7)

and was attained in 22 iterations. The final function value was J = -2.23525 and the radius of the constraints sphere, i.e., of the bound Sq was .00717.

Therefore, the minimax solution is within the sphere of radius Sq = .00717

with the center at the last nominal point.

In the second example, the equality constraint was handled in the same way; the relevant expressions defining the minimax problem have the form n -k x /y y i J(x,y) = - £ v (1 -

a e

L

y,

i

— = -v.k.Q'.e

(l-Ck'.e

)

y , - i

y - i

+ Vnknane

(X “ ane

* )

> i = l,...,n-l

where again x^ in (4.7) and (4.10) is a dependent variable given by (4.2). For the example with numerical values n = 4. c = 1, c = 2 .

y x

V1 " 1

II

1

O'

1

CM

II

CM>

CM

5

*2

v 3 = 3

CO

II

7

^3

4 = 4

k4 =

11

^4

the result in terms of the coordinates of the final nominal point was

X]_ = .71352 x2 = .38893 x3 = .48833 x4 = .40922

and was attained in 102 iterations. The final function value was J = -9.960 and the radius of the constant sphere was S0 = .8117*10 < d = 10 In both examples as well as others with different data, it was found that the time consuming stage of the algorithm is the maximization which occurs

(a) when the set Y(x) is sought and (b) when the second stopping criterion has to be applied. As indicated (a) was handled by a grid with varying

grid step, while (b) was handled by Rosenbrock's rotating coordinate algorithm modified for use on CDC 1604 by Heller [6]. In the second example the need

to perform (b) arose in approximately 107o of iterations while the need to perform (a) occurs in every iteration. One can therefore see the advantages of the time-saving modification implemented into the algorithm and based on lemma 8.

In the third example, standard tables were used to compute the explicit expression for J(T,K,Tm ,§ ,

00

observed, however, that the obtained functional was not convex in the pair T,K for all points (T^,? ,a)n) . Therefore a mapping

3 = TK (4.12)

was employed and the space of minimizing parameters transformed into

x = |K|

lK I

with the result that J(x,y) was convex in x for all yeY. The expression for J(x,y) was of the form

6 ! 62 J = - + - (4.14) where 6 1 ■ c32( -do2d3 + d od ld 2> + <c22 ' 2c lc3>dod ld4

6 = ( c .2 - 2c c„)d d,d. + c 2 (-d1d 2 + d,d,d )

d = K

d 1 =

d_ = UD T + 2§0J

d . = 2§(J0T + 1

"N

^

from which it is simple to find ^ and . The following bounds were used for the variables:

400 < K < 5 0 0 160 < Y < 300 g < oo < 11 — n — .7 <

Î

was obtained in 54 iterations, with the final value for J = .3283 and the _3

radius of the constraint sphere SQ < 5.10 . The envelope of the step responses corresponding to various ((1^,5,^) for K*, T* are not presented since nothing essentially new can be visualized from them aside from the usual property of the minimax solution, i.e. that the dispersion of the

output signal at any instant of time is less than with other pairs (K,T).

5. Conclusions

In this paper, the intention was to present some definitive results on the minimax problem in finite dimensional spaces. The section on theory contains a brief summary of results which are important in characterizing the minimax solution. Lemma 5 which gives sufficient conditions for a local saddle point to be the minimax solution is a new result and extends the class of minimax problems in which local saddle

point methods may be used for the minimax search. Lemmas 7 and 8 are also new results in so far as the minimax problem is concerned, but the result

in lemma 7 is an extension of the result used by Wilde [lO] in contour gradient schemes for finding the minimum of a convex function in a convex d o m a i n .

The algorithm based on lemmas 7 and 8 was found to be relatively efficient although its merits can be better judged after comparisons with other algorithms. Presently, the author is aware of only two algorithms [2,6] and a comparative study was not undertaken. The algorithm, as constructed, is only one possible way of performing the basic tasks of selecting nominal points, maximizing over Y and judging when to stop. All these tasks were resolved in the simplest manner. Moreover, the method of maximizing over Y is tailored to the result in lemma 8 and complete maximi­ zation is carried out only when necessary. This advantage is not possessed by other algorithms, in which complete maximization in each iteration is mandatory. If, in the elimination algorithm, maximization over Y is

necessary, the fact that the answering set Y(x) is not a singleton does not

hinder, but only helps since it allows greater elimination per iteration. Furthermore, the algorithm possesses the desirable property of faster

solution time with good initial guess xqSX since the closer J ( x Q ,yo) = F(xq)

is to the minimax value the smaller number of complete maximizations will be necessary. Finally, the fact that the algorithm reduces the admissible domain to an arbitrary, prespecified, neighborhood of the minimax solution

inherently eliminates the difficult question of estimating the distance from the minimax solution. This is one of the more difficult questions in

minimax search, since one may find ways to construct a monotonically decreasing sequence {f(x.)} of values of the max function but nothing can

be said about the speed of convergence nor can the necessary condition (2.7)

be used as a simple stopping criterion.

It should be mentioned that the algorithm is not restricted to minimax problems where X is defined by linear constraints as in (3.1).

Nonlinear boundaries of X may, for the purpose of implementing the algorithm, be represented by a suitable number of supporting hyperplanes. Since this representation is carried out only for the purpose of selecting nominal points it does not affect the final minimax value. Moreover, due to the nature of the algorithm, the original nonlinear constraints are constantly replaced by linear constraints, i .e., hyperplanes through nominal points.

A serious limitation of the elimination method is that J(x,y) must be convex in X. It is unfortunate since it restricts the generality of the method and excludes direct application to the general minimax problem.

References

1. Danskin, J. M . , The Theory of M a x - M i n . Series Economitrics and Operations Research, Vol. V, Springer-Verlag, New York, 1967. 2. Salmon, D., "Minimax Controller Design," IEEE Trans, on Automatic

C o n t r o l . Vol. AC-13, No. 3, July, 1968.

3. Medanic, J., "Segment Method for the Control of Systems in the Presence of Uncertainty," Proc. J A C C , Boulder, Colorado, 1969.

4. Psheniehnii, "A Dual Method in Extremum Problems I and II," K y b e r n e t i c a , Vol. 4, Nos. 4 and 33.

5. Demjanov, V. F., K i b e r n e t i k a . Vol. 2, No. 6, 1966.

6. Heller, J. E., "A Gradient Algorithm for Minimax Design," Report R - 4 0 6 ,

7. Kuhn, H. W. and Tucker, A. W. , "Nonlinear Programming," Proc. Second Berkeley S y m p . , Math. Stat. and Prob., J. Neyman, ed., Univ. of Calif. Press, 1951.

8. Bram, J., "The Lagrange Multiplier Theorem for Max-Min with Several Constraints," SIAM Journal. Vol. 14, pp. 665-667, 1966.

9. Koivuniemi, A. J., "Parameter Optimization in Systems Subject to Worst (Bounded) Disturbances," IEEE Trans, on Automatic C o n t r o l . Vol. AC-11, July, 1966.

10. Wilde, D. J., Foundations of Optimization, Prentice-Hall, 1967. 11. Ky, Fan, "Minimax Theorems," Proc. Natl. Acad. S c i . . Vol. 39, 1953. 12. Kakutani, S., "A Generalization of Brouwer's Fixed Point Theorem,"

Duke Math. J . , Vol. 8, No. 3, 1941.

13. Demjanov, V. F„, "Algorithms for Some Minimax Problems," J, Computer

and System Sciences, Vol. 2, pp. 342-380, (1968).

14. Ciric, V., Ph.D. thesis, Rice University, Houston, Texas, May, 1969. 15. Medanic, J., "Elimination Algorithm for Computation of the Minimax,"

Fourth Allerton Conference on Circuit and System Theory, University of Illinois, October, 1968.

Appendix

This appendix contains the proofs of lemmas 5, 7, and 8. The proofs of the other lemmas may be found in the literature as noted with each lemma. The result of lemma 5 may be anticipated; however, this author has never encountered it in the form presented in this paper, nor is he aware that the problem solved by the lemma has ever been paused.

To prove lemma 5 introduce the following definitions. Definition 1 . For an arbitrary xeX let

“ (y,V) = (x,y) -Y (A.2)

where the domain of definition of y is the answering set Y(x) and the domain of definition of y is T (x) , the cone of admissible unit direction vectors IMI = 1 with the vertex at xeX.

Definition 2 . Let the domain of definition of

y

Definition 3 . Let the answering set Y(Y), Y eF(x) be the subset consisting of yeY(x) which maximize against the given Y. Let the answering set F ( y ) , yeY(x) be the subset consisting of Y®r(x) which minimize against the given y.

If for all xeX, the answering set Y(x) is convex, as in lemma 5, then Proposition 1 . For all yeY(x), Yef(x) the answering sets F(y) and Y(Y) are convex.

and convexity of f ( x ) . Convexity of Y(Y) follows by contradiction. Suppose for some Y the set Y(Y) is not convex. Then in view of the expansion (2.4) for the max function, there would exist, for an arbitrary small h, a point x + 6 x = x + Y h such that the answering set Y ( x + 6 x ) would not be convex. This contradicts the assumption in the lemma and hence Y(Y) is convex for

all YeF (x) .

Proposition 2 . There exists a Y*eF (x) and y*eY(x) such that

W ( y ,Y*)< U)(y*,Y*) < w ( y * , Y ) (A. 2)

holds for all xeX, yeY(x) and Yef (x) .

Proof: Because U)(y,Y) is continuous in the variables, because the sets Y(x) and F (x) are closed and bounded and the answering sets Y(Y) and T(y) are convex for all Yef (x) and y e Y ( x ) , respectively, the result in

proposition 2 follows by Kakutani's or Ky Fan's saddle point theoriem (lemma 6 in the p a p e r ) .

Now, the definition of (Jtf(y,Y) and the result in proposition 2 imply that

(x,y) -Y* < (x,y*) -Y* < (x,y*) *Y (A.3)

holds for all xeX, yeY(x) and Y e f(x)

(y *

By the necessary condition for the minimax solution (lemma 3 in the paper) for

Y

o < (X*,y)-Y*. (A.5)

Combining (A.4) and (A. 5) shows that

0 < (x*,y*) *Y (A.6)

must hold for all

Y ^ r

Y e F

T

x*

admissible unit direction vectors at x * . On the other hand, since y * e Y ( x * ) , y* is a global maximum of J(x*,y) and hence (x*,y*) is a local saddle point. This completes the proof of lemma 5.

The proof of lemma 7 consists in showing that any point dropped in the contraction of to X ^ by way of (2.14) cannot be the minimax solution.

Denote the complement of H (y ) by H (y ) and let xeH (y ) D X . Then, x

r n w n

J

cannot be the minimax solution since from (2.13) J(x,y ) > J(x ,y ) and

v , y n/ v n ,yn therefore

F(k) = max J(x,y)>J(x,y ) > J(x ,y ) > min max J(x,y). (A.7)

y«Y n n n xeX yey

The proof of lemma 8 consists in showing that any point dropped in the contraction of X^ to by way of (2.17) cannot be the minimax

solution, and proceeds in a similar fashion. Denote, analogously, the complement of H (y.) by H (y. ) and take xeH (y ) PI X .

UK.

h k

n K

n

the minimax solution since J(x,y.) > J(x n ,y ,) and therefore ,yk n - l ^ n - l

F(x) = max J(x,y) > J(x,yk> >

J (xn_i>yn.;|)

max J(x,y)

yeY xe X yeY

ACKNOWLEDGMENT

The author wishes to acknowledge helpful and stimulating discussions with Professors P. Kokotovic, J. B. Cruz, Jr., and W. R. Perkins, as well as his collegues in the Control Systems Group of the Coordinated Science

Laboratory. The author is indebted in particular to J. Heller for programming the algorithm on the CDC 1604 at CSL.

FS-1953

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Za. R E P O R T S E C U R I T Y C L A S S I F I C A T I O N Unclassified

2 b . G R O U P

3. R E P O R T T I T L E

ON SOME THEORETICAL AND COMPUTIONAL ASPECTS OF THE MINIMAX PROBLEM

4. D E S C R I P T I V E N O T E S ( T y p e o f r e p o r t a n d i n c l u s i v e d a t e s ) 5 . A U T H O R ( S ) ( F i r s t n a m e , m i d d l e i n i t i a l , l a s t n a m e ) MEDANIC, Juraj R E P O R T D A T E July, 1969 7 a . T O T A L N O . O F P A G E S 32__________ 7 b . N O . O F R E F S 8a. C O N T R A C T O R G R A N T N O .

DAAB 07-67-C-0199; also in part AF-AFOSR b. p r o j e c t N O . 68-1579; NSF Grant GK-3893. 9a. O R I G I N A T O R ' S R E P O R T N U M B E R ( S )

R-423

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11. S U P P L E M E N T A R Y N O T E S 12. S P O N S O R I N G Ml L I T A R Y A C T I V I T Y

Joint Services Electronics Program thru U.S. Army Electronics Command Fort Monmouth, New Jersey 07703 13. A B S T R A C T

This report contains both a brief summary of the pertinent theory of the minimax problem in finite dimensional spaces and the development of an elimination type

algorithm for the practical computation of the minimax soluti