United States - DTICprogram written and applied to an anti-submarine warfare force allocation...
Transcript of United States - DTICprogram written and applied to an anti-submarine warfare force allocation...
United StatesNaval Postgraduate School
THESIS
"AN ALGORITHM FOR THE SOLUTIONOF CONCAVE-CONVLX GAMIES
by
Lee Fredric Gunn
and
Peter Tocha
Thesis Advisor: J. M. Danskin, Jr.
September 1971
Appuoved 4c,. pubti.c tceez. e; ditAZbutZon wuai.i.ted.
An Algorithm for the Solution
of Concave-Convex Games
by
Lee Fredric GunnLieutenant, United States Navy
B.A., University of California, Los Angeles, 1965
and
Peter TochaKapitanleutnant, German Navy
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN OPERATIONS RESEARCH
from the
NAVAL POSTGRADUATE SCHOOL
Authors V' ' " '
Approved by: jk .' Thesis Advisor
Ckirman, opatment of Operataons Researc
'- "t Academic Dean
UNCLASSI FIEfD-- , Secut~t v ClaiRft¢'atItnr"o uc- " ' ' "
DOCUMENT CONTROL DATA R & D(Security classification of title, body of ab.ftarf and indezing annotation m sUif be entered when the overall report is classified)
I ORIGINATING ACTIVITY (Cororot* eauthor) 2a. REPORT SECURITY CLASSIFICATION
Naval Postgraduate School UNCLASSIFIEDMonterey, California 93940 ab. GROUP
R REPORT TITLE
AN ALGORITHM FOR THE SOLUTION OF CONCAVE-CONVEX GAMES
4. oESCRIPTIVE NOTES (Type of report andincluaive datee)
Master's Thesis; September 1971"S. AU THORIS) (First name. middle initial. laost name)
Lee F. GunnPeter Tocha
S. RE PORT OATE 7a. TOTAL NO. OF PAGES 1 76. NO. Oý ma,
September 1971 78 211a. CONTRACT OR GRANT NO. 0a. ORIGINATOR*S REPORT NUMMSER(S|
b. PROJECT NO.
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Approved for public release; distribution unlimited.
11. SUPPLEETARY NOTES V;I SP"ON-SORIN4 MILIlANY ACTIVITY
I Naval Postgraduate SchoolMI onterey, California 93940
Il. ABSTRACT
A master's thesis whi-eh discusses the solution of con-cave-convex games. An algorithm is developed, a computerprogram written and applied to an anti-submarine warfareforce allocation problem as an illustration. Techniquesfor handling concave-convex problems in high dimensionsare included.
leeu.ssed bvNATIONAL TECHNICALINFORMATION SERVICE
81110A Va. 231$1F O R ( P A G E 14 3)I II II
DD ".'..o 1473 P-UNCLASSI FI EDs/N o0o1--0o--811 77 'liyCta.aiflcaion -
UNCLASST FIEDAecurltv CIS-sifjcMiii.n
14 LINK A LINK 0 LINK C
KEY WOROS
NOLE WT ROLE WT ROLE IN T
Concave-Convex Games
Games
ASW Forces
ASW Force Allocations
Game Theory
Resource Allocations
Allocations
Allocation Problems
Concave-Convex Functions
Derivative Games
Brown-Robinson Process
MaxMin
Directional Derivative
Algorithms
DD ,o.. e 473 (BACK) IINCLASSI F'ED)', (do,*IO7.o.,, 78 Security Classification
r
ABSTRACT
A map'•-cr's thesis which discusses the solution of con-
cave-convex games. An algorithm is developed, a computer
program written and applied to an anti-submarine warfare
force allocation problem as an illustration. Techniques
for handling concave-convex problems in high dimensions
are included.
2
TABLE OF CONTENTS
I. INTRODUCTION ----------------------------------- 5
A. A GAME THEORY APPROACH TO RESOURCEALLOCATIONS ----------------------------------
B. CONTENT OF THE PAPER, BY CHAPTER ----------- 6
II. MATHEMATICAL CONCEPTS -------------------------- 7
A. GENERAL ------------------------------------- 7
B. THE ALGORITHM FOR THE SOLUTION OF LONCAVE-CONVEX GAMES ------------------------------- 8
1. The Derivative Game -------------------- 8
2. The Lemma of the Alternative ----------- 9
a. The Brown-Robinson Iterative Processin the "Auxiliary Game" ------------ 9
b. Statement of the Lemma ------------- 11
3. The Corollary of the Altcrnative ------- 12
4. The Algorithm -------------------------- 13
III. PROGRAMMING ASPECTS ON AN ILLUSTRATIVE EXAMPLE - 18
A. FORMULATION OF THE PROBLEM ----------------- 18
1. The Space X ---------------------------- 20
2. The Space Y ---------------------------- 20
3. The Function F(x,y) -------------------- 21
B. BASIC COMPUTATIONS ------------------------- 22
1. Computation of 0him -------------------- 22
2. Partial Derivatives with Respect to xIi- 23
3. The Value of F(x,y) -------------------- 23
4. Partial Derivatives with Respect to yjk 23
S. Finding Directions of Increase(Decrease) ----------------------------- 23
3
C. PROGRAMMING NOTES -------------------------- 24
1. Presentation of Input - The MissionDescription ---------------------------- 24
2. The Contraction with Respect to Vhim --- 27
3. Working in Subspaces ------------------- 27
4. Machine Accuracy Problems -------------- 29
5. Testing the Program -------------------- 30
6. General Comments ----------------------- 31
a. Iaitial Allocations ---------------- 31
b. Distance Policy -------------------- 32
c. Upper and Lower Bounds ------------- 32
d. Modification of the ObjectiveFunction --------------------------- 32
e. Assigning Values V him -------------- 33
. .Cu.ig --------- -------------------------- -3z
IV. SU'.L\IARY ---------------------------------------- 35
V. SUGGESTIONS FOR FURTHER STUDY ------------------ 36
APPENDIX: PROGRAMMING FLOWCHART ---------------------- 38
COMPUTER PROGRAM LISTING ------------------------------ 51
BIBLIOGRAPHY ------------------------------------------ 74
INITIAL DISTRIBUTION LIST ----------------------------- 76
FORMNI DD 1473 ------------------------------------------- 77
4
I. INTRODUCTION
A. A GAME THEORY APPROACH TO RESOURCE ALLOCATIONS
Problems in the allocation of resources can be divided
into two descriptive categories: Single-agent problems and
adversary problems. Single-agent problems have only one
participant optimizing without intelligent opposition. In
the solution of adversary problems, opponents work at cross
purposes. Each choice of an allocation of resources by one
participant must be made in light of those of his opponent(s).
Of course, games are adversary problems.
The great interest in game theory as a technique of
modelling which followed von Neumann's statement of the
AuT.damcLt-i concepts in 1927 ,.WJ cczLinues today. Thc
fascination of the game as a model for conflicts of almost
any sort is enhanced by the iact that the solution to a
game is entirely independent of assumptions regarding the
actual behavior of the antagonist(s).
Matrix games and differential games are extensively
treated in the literature; some references are noted here.
Matrix games or games over the square are discussed in
basic form by Williams [21] and a thorough study is done by
Karlin [8]. For a first essay in the subject of differen-
tial games, see Isaacs [6]. Taylor [16,17] gives additional
examples of the modelling of combat operations including
search using differential games.
The development of the derivative game is due to
Danskin [4]. A very large class of concave-convex problems
yield to solution when the algorithm based on this game is4V
applied. It is with the derivative game and the algorithm
evolved from it that this paper is primarily concerned.
B. THE CONTENT OF THE PAPER, BY CHAPTER
Chapter II surveys some of the more important mathe-
matical ideas necessary to the development of the algo-
rithm for the solution of concave-convex games.
Chapter III is concerned with the programming of an
anti-submarine warfare force allocation example. The ex-
ample serves to illustrate both the use of the algorithm
and the theory on which it is founded. The problem is
formulated and the basic computational steps toward the
solution are enumerated. If any fears stem from the
formidable appearance of the matrices used in the example,
they hopefully will be dispelled by the description of the
form for input in the programming notes. Technical matters
regarding programming arc considered in some detail.
Chapter IV draws together this presentation with some
concluding remarks.
Chapter V contains suggestions for further study.
The Appendix consists of the programming flowchart.
The Computer Program Listing is included as well.
6
II. MATHEMATICAL CONCEPTS
A. GENERAL
The fundamental theorem of the theory of games in the
form appropriate here is the following:
Suppose F(x,y) is a continuous function on X x Y,
where X and Y are compact and convex. Suppose that the set
of points X(y) yielding the maximum to F for fixed y is
convex for each such y, and that the set of points Y(x)
yielding the minimum to F for fixed x is convex for each
such x.
Then there exist pure strategy solutions x* and y*
satisfyingF(x",y) > FHx-,y-t),V
(1)
F(x,y 0 ) . F(xy 0 ), tV xEX.
A complete proof of the theorem in this form can be found
in [7]. Assuming that the conditions for the existence of
a solution satisfying (1) are met, the problem can be stated
in the form
Max Min F(x,y),x y
which is equivalent to
Max O(x)x
where *(x) =_ Min F(x,y).y
The theorem applies to two-person zero-sum games. Thus,
Max Min F(x,y) - Min Max F(x,y).x y y x
7
One important difficulty arises from the fact that,
although F may be smooth, *(x) is not in general differ-
entiable in the ordinary sense. Danskin, in [3], has shown
that under general conditions on X and Y, there exists a
directional derivative in every direction. It is this fact
that has provided the key to the solution of problems of
the type described.
B. THE ALGORITHM FOR THE SOLUTION OF CONCAVE-CONVEX GAMES
The algorithm to solve games concave in the maximizing
player and convex in the minimizing player is developed and
presented in great detail in [4]. The following gives a
brief survey of those results which are most important for
the design of the algorithm; to fill the apparent gaps, a
thorough reading of [4] remains necessary.
1. The Derivative Game
Suppose x°cX. Associate with x* a non-empty set
of admissible directions y, r(x°). IV is defined as the
convex hull (e.g., see [19]) of the s,.-t of points
W(y) = {Fx1 (x°,y), .. , Fyk (x,y)},
where ycY(x°).
The derivative game then isA A%
H(y,w) - yewA
defined over r(x°) x W. The maximizing player maximizes H
by choice of yer(x°), the minimizing player minimizes H by
choice of wcW.
THEOREM I
A necessary and sufficient condition for the exis-
tence of a direction of increase for O(x) is that
the value of the derivative game defined by H be
positive at x°. The yo which yields the value of
the derivative game is a pure strategy.
If the value is positive one can find and use this direction.
If the value is non-positive, a direction of increase does
not exist, i.e., the solution has been reached.
The application of the derivative game in practice is greatly
complicated by the necessity for approximations.
2. The Lemma of the Alternative
The Lemma takes into exact account the approxima-I
tions invoivO in the uplpiitaLiua uf the dernvative g•UL,.
It states that a certain process (to be explained below)
must either yield a sufficient increase to O(x) at a point
x0 or determine that the point x° is nearly optimal. Be-
fore the lemma can be formulated in mathematical terms,
some further difficulties and the tools with which to over-
come them have to be outlined.
a. The Brown-Robinson Iterative Process in the"Auxiliary Game"
The Brown-Robinson (B-R) process employs the
following idea: Let G be the pay-off function. At stage
N-O both players choose arbitrary strategies x° and y°. At
stage N-i the maximizer chooses x1 such that G is maximized
against y°; then the minimizer chooses yl to minimize G
9
against x', and so forth. At stage N the maximizer chooses
xN as if the minimizer's strategy were an evenly weighted
N-1mixture of strategies y*, . y ; the minimizer chooses
yN as if the maximizer's strategy were an evenly weighted
Nmixture of strategies x*, .* ..
For matrix games, Julia Robinson [12] proved
that [
lim sup G(x,y G 0.
Danskin (1] has generalized the proof to hold
for two-person zero-sum games with continuous pay-off de-
fined over X x Y, X and Y arbitrary compact spaces. It
should be noted here that the B-R process is very slow in
convergence when applied directly to finding an apprc'ima-
tion to the value of the game defined by (1). However, it
is not applied to the basic game in this algorithm but rather
to an "auxiliary game" for which an accurate solution is not
required.
The derivative game mentioned above cannot be
solved directly because the set Y(x°) is not known. All one
has is a single element ycY which approximately minimizes
F(x*,y). The place of the derivative game, therefore, is
taken by the "auxiliary game" employing a modified version
of the B-R process described below. This process makes it
possible to keep track of the approximations involved and
their consequences. The "auxiliary game" is defined as
follows:
10
For any c>0, denote by Y (x) the set of yeY
such that F(xy) = O(x) + e. Let Y ) C::) Yk(x), ycr(x0 )xc::
ycY(=).F(x 0 +doy,y) -(°Then H(y,y) E do , for d > 0,
d 0 0
a minimum step size, is a game over r(x 0 ) x Y (E) with y
the maximizing and y the minimizing player. This game has
optimal mixed strategies for both players. Applying the
idea of approximate optimization to the convergence proof in
[1] leads to
THEORE. II
N 1 N-1Let y be chosen such that g n_0 h(y,y ) is
maximized to accuracy ý, and yN be chosen such
that 1 n H(yn,y) is minimized to accuracy n.
Then .i olira sup g EH(y,yn) - ) N2(+n).
N-co I n=H0=
This holds for continuous II.
Let 0 be the maximum oscillation of VF(x,y) over a distance
d Then
F(x*+d° yNyn) - F(xoyn) > yN F(xoyn) n 0d0
The lemma of the alternative now can be formulated.
b. Statement of the Lemma
Suppose O<a<c , ycY C(x 0 ).
Then the generalized B-R process will, at some stage N, de-
termine that one of the two following statements is true:
1. The maximum over r(x°) of the directional
derivative does not exceed S.
11
N
2. The point x xO + do y- where
Ny-N =. n•= n
n- 1-NN _ N whrYN.
and the point y cyC(x )•whereN minimizes
F(x ,y) to accuracy c, satisfy
F(xN,yN) - F(xo,yo) > -seo- 3
o 0
3. The Corollary of the Alternative
Suppose that F(x,y) is concave in x and convex in
y. Then the modified B-R process applied at x* will, at
some stage N, determine that one of the two following state-
ments is true:
-N1. The pair x = x*, y , whereN
-N 1y. 1_+ nýO yn
are approximate optimal strategies for the game
defined by F.
2. The point x-NX yields an increase to 0(x) by at
least a specified amount.
For details and proof see [4], pp. 36 ff.
Reference [4] continues with a detailed discussion
of delicate problems which can only be listed here: The
choice of the minimal stop size d0 ; the problem of acces-
sibility of a point : from a point x*; the problem of ob-
struction; the choice of a, 0, E, their interaction with each
other, and the choice of p where p is the accuracy to which
Max O(x) is to approximate the value of the game defined by
12
(1). It must be noted that the conditions derived for the
selection of these parameters are sufficient.
4. The Algorithm
The algorithm as a consequence of the foregoing
mathematical considerations is presented in section 10 and
11 of [4] and will not be reproduced here in detail. A ver-
bal description of its basic structure - depicted in Figure
1 -, however, may be useful:
The maximizing player, called Max, having arrived
at a point xx, has a direction of maximal increase y, ob-
tained either from the derivative game D y(xx) or from
the B-R process in the auxiliary game, and a distance d > do.
The minimizing player, called Min, is at a point yy. F(xx,yy)
iskno;n. ".lax makes a proposal to nowvu Lo a point x = xx + dy.
Of course, F(x,yy) > F(xx,yy). Min accepts Max's proposal
and starts minimizing against x, looking for a direction
of maximal decrease g. If there is none, yy is a minimum
against x as well as against xx in which case Max will move
to the point x. If there is a direction of decrease Min
forms a point y - yy + Dg such that F(x,y) < F(x,yy). A test
is performed to determine whether Min has already "beaten"
Max: if F(xx,yy) > F(x,y) Min stops the minimization pro-
cess, and Max discards his proposal x because moving to x
will not increase O(x). Max halves the distance d and, with
the same y, forms a new trial point x. If F(xx,yy) < F(x,y)
Min continues to minimize until either F(xx,yy) . F(x,y) or
Min can no longer find a direction of decrease. If now
13
Figure 1. Flowchart of Algorithm.
TRIAL X AIP.1 0 YOvE
IýG THER A OP
DECREASET FLOW CRIALTOF b~ SUCCESS
R1
__ __U_ __ _ __C
C_
_ __ _
F(xxyy) < F(x,y), indicating that Min, even after a complete
minimization, was not able to "beat" Max, Max moves to the
"proposed point x realizing a gain for O(x). Max then looks
for a new direction of increase. The process terminates when
such a direction does not exist.
The situation that leads into the Brown-Robinson
process is the following one: The proposed points x = xx+dy
have been "beaten" by Min until d gets cut down to do, in
spite of the fact that y, obtained from the derivative game,
is an apparently good direction. If the trial point x = xx
+ d0y does not result in a move for Max, the B-R process,
Figure 2, is used.
Denote the present y - the one that so far has
lead Lo a failur-e for i'lax by y', and the assuciate VF x
by Z;. Minimize F completely against x = xx + doy. The
resulting y then leads to a new VFx which is averaged with
the previous i°, giving 7w. Suppose that a as input to
the derivative game D y(x) produces a new yo such that the
value of the derivative game is positive as required. (If
such a yo does not exist the problem is solved). This y,
averaged with T', gives 72 which in turn creates a new
2xj2 -Ntrial point xx + do y . x , or X in general, then is
exposed to Min's reaction as described previously. Once Max
finds a direction and an associated trial point that cannot
be "beaten" by Min, Max moves and leaves the B-R routine.
The y-strategies are also averaged and saved although their
average is never used during the computation. In case the
is
Figure 2. The Brown-Robinson Process.
CONDITIONS Fort tM111ING ThIt~BI
ATP 0ust.WS IETO 'OAPARN INRAE HWVR fiUtA1* dy HVE()
16
game defined by F(x,y) is terminated while in the B-R pro-
cess, this average of the y-strategies represents the op-
timal solution for the minimizing player.
17
III. PROGRAMMING ASPECTS OF AN ILLUSTRATIVE EXAMPLE
A. FORMULATION OF THE PROBLEM
An algorithm for the solution of a wide class of con-
cave-convex games over polyhedra has been presented in
abbreviated form above. In this chapter that algorithm is
applied to a particular game, an anti-submarine warfare
force allocation problem.
For this problem, five indices are employed: h, i, j,
k, and m. In a meaningful example, the maximum numbers
corresponding to these indices might be, respectively:
5, 25, 10, 500, and 5. The indices have the following
meanings:
li: Suabn•,'ine t.vu
i: Submarine mission
j: Type of antisubmarine weapon (or vehicle)
k: Place in which submarine and weapon encounter one
another
m: Stage of the submarine mission.
An additional index is used. L(k) is the "kind of place."
A "kind of place" might be defined by a particular set of
weapon employment parameters. These include the tactical
and the natural environment. The natural environment con-
sists of oceanographic and meteorological conditions. Ex-
amples of the tactical environment are destroyers in an ASW
screen and patrol aircraft in barrier patrol. The "kind of
place" in which an encounter occurs impacts on the outcome
18
of an encounter between submarine and weapon. A reasonable
number of "kinds of places" in the present context might be
A submarine mission is described by a matrix IIEhijkml.
This matrix has as its elements real numbers denoting the
extent to which a submarine of type h at stage m of mission
i is exposed to a weapon of type j at the kth place. The
effects of these weapons and thus of the encounters are
characterized by a "technical" matrix I IChjX(k) I I in the fol-
lowing meaning: exp[-Chji(k) Yjk] is the probability that
a submarine of type h survives one exposure to y units of
weapons of type j at a "kind of place" L(k). These en-
counters are assumed to be mutually independent. Note that
the prob~abil ity of .urviwvi to thc mth mission stage iS
conditioned on the completion of the previous stages. Sup-
pose that there are Yjk units of force of type j at the kth
place. Then the probability of a submarine's completing
stage m of mission i is
m' exp['EhijkmI Chjk(k) Yjk],m' <m3
which, due to independence of the events, equals exp [- 0 him1 ,
where
ehim = k Ehijkm' ChjI(k) Yjk"
mI<m
Now, by carrying out a premultiplication,
Ahijkm = Ehijkm ChjR(k)I
19
L
the exponent becomes
ehim A- •;I hijkm, Yjk"J km'<m
In this example, the vast majority of the Ehijkm, and there-
fore of the Ahijkm, are zero. If 3000 non-zero Ahijkm are
allowed, each type of submarine can be employed on ten dif-
ferent missions and undergo up to 60 encounters with anti-
submarine weapon systems.
1. The Space X
Let Xhi be the proportion of submarines of type h
assigned to the ith mission. Make X = 11xhilI satisfy the
conditions
Xhi. - I, for nvcry h, Xhi A 0,
and
hi .< Xhi <8hi, for every pair h,i,
where the sets {ohi}, {0hi} are supposed to satisfy
ahi <i .ihifor every h
ahi Shi
and
0 < ahi < 0hi for e.ery pair h,i.
2. The Space Y
Let Yjk be the proportion of antisubmarine forces
of type j sent to the kth place. Make Y - Ilyjkll satisfy
the conditions
20 ,o i
jk = 1, for every j, .>k 0
k Y
and
ajk - Yjk 5 bjk for every pair j,k
where the sets {ajk}, {bjk} are supposed to satisfykjk jk bk•ak < 1, < E bk
k jk k jk
and
0 < ajk < bjk for every pair jk.
3. The Function F(x,y)
Vhim is the value of accomplishing stage m of
mission i for a submarine of type h. The character of
F(x,y) can be examined.
set
Whim = Vhim exp [I-him)
where 0him is as before. Put
Thi =E "him'm
Then m
F(xy) E .Xhi Thi.h,i
This function is linear in x and exponential in y and is
therefore a concave-convex game of the type treated in [4],
defined over X x Y. The quantity F(x,y) represents the
total expected payoff to the submarine player. Re-expressing
F(x,y) in its explicit form gives:
21
F(xy) , Xhi Z Vhim exp[-. Z Ehijkm ChjtCk)yjk].
h i m j k
The remainder of this chapter gives details of the application
of the algorithm to the game defined above.
The principal result is the flow-chart (Appendix).
Since this flow chart is constructed around the algorithm
from [4], it is helpful, though not essential, to have [4]
available. The complete program listing is included following
the Appendix. The program is written in FORTRAN IV and was
run on the IBM 360/67 computer at the Naval Postgraduate
School, Monterey, California.
B. BASIC COMPUTATIONS
, L . oiiip Lt at i o f hi
For fixed (hi), 0him is non-decreasing in the mis-
sion stage m. This reflects the trivial fact that
P[submarine survives stage m]
< P[submarine survives stage m-1].
Equality holds when the "threat" due to the encounters at
stage m is non-existent, i.e., when either no ASIV-forces
Yjk are present or their effectiveness against the submarine,
Chjt(k), is zero. Hence Ohim, for each pair (h,i), is ac-
cumulated over the mission stages as follows:
Bhim" ehi(m-1) + E Z Ahijkm Yjk' Ohio O.j k
22
2. Partial Derivatives with Respect to Xhi
Because of the linearity of F in x the partials with
respect to Xhi, Fxhi, are the coefficients of Xhi and do not
explicitly contain x. Vhim exp[-Ohim ] can be represented as
a matrix of dimension (n x m), where n is the number of pairs
(h,i). Then
Fh = Vhim exp[-6 him]
hi m
is the sum of the elements in the (h,i) row of that matrix.
3. The Value of F(x,y)
F(x,y) is obtained by pre-multiplying F by xhi andFhi Xh
summing the products
F(x,y) = x.. Fx),
4. Partial Derivatives with Respect to yjk
Because of the cumulative property of 0him' a change
in Yjk during some mission stage m' will affect the following
stages as well. For each pair (h,i), premultiply xhi by
the corresponding Ahijkm , where m' is the mission stage in
which the Yjk of interest occurs. Sum Vhim exp[ -ohim] over
the mission stages for which m'<m, multiply the result with
Xhi Ahijkml and sum the products over h and i:
"_Fjk =h E xhi Ahijkm' m mVhim' exp['0him'].
S. Finding Directions of lncrcase (Decrease)
FXhi and FYjk are inputs to the derivative games for
x and y. For x, the direction y* is sought such that D y(x)=
VFxey is maximized; for y, g* is sought such that I) Max F(xy)
23
VFy~g is minimized (equivalently, -VFy g is maximized). The
side condition is that y0 and g* be unit vectors:
'hi - k 0, hy~ g9kj 1; Vh, Vj
A method of finding such directions - called THE DIRECTION
FINDING ALGORITHM - is derived from the Kuhn-Tucker conditions
*• and the Schwartz Inequality. It is contained in [4] and has
been applied to a vector - valued function by Zmuida in. [22].
C. PROGRAMMING NOTES
The flow chart (Appendix) and the associated program may
not be optimal with respect to machine time and memory space
required, and may provide opportunities for improvement. For
computer routines like this, intended to solve high-dimensional
problems, a trade-off between time and space is generally ap-
parent. The user, considering the particular facilities
available to him, must decide on his optimal trade-off, and
modify the program accordingly. The following discusses some
of the techniques that have been implemented; it points out
the major difficulties that have been encountered and the ways
presently used to deal with these, and offers some remarks
about the impact of the underlying mathematics on the use of
the algorithm for realistic problems. The numbers referenced
are statement numbers.
1. Presentation of Input - The Mission Description
In a realistic application, most of the Ahijkm will
be zero because the effectiveness Chjt(k) contained in A will
be zero. Example: suppose k denotes a place in the western
Baltic and L(k) classifies this place as shallow with extreme-
ly poor sonar conditions; j denotes a nuclear killer-submarine,
24
h representsa conventional attack submarine. Then Chju(k)
will be zero for all practical purposes.
The method of presenting the matrix E which avoids
computing of and with A when C is zero, is one which is ex-
tremely easy for the user to understand and employ. Hle gives
his description of a submarine mission by plotting it on a
chart, marks off in order the places the submarine must go,
and notes the forces it might meet at those places. He
will give the exposure E required by the particular mission
in terms of a standard which he will have set. He will mark
off the points in the mission where the various stages of
the mission will have been accomplished. Such a description
might run as follows:
m=0
h=l i=i k=l j=, L=i.uj=5 E=1.5
k=7 j=l E=2.0j=3 E=3.0
m=l (the first stage of the first mission is completed)
k=8 j=4 E=1.2j=7 E=1.7
m-2
k-iS j=3 E=0.3j=2 E-2.0j-4 E-0.5
m- 3
The above mission (mission 1 for submarines of type
1) has three stages and nine encounters, in four different
places. Then the listing starts for the next mission with
h-1, i-2, and continues until all missions for all types of
submarines have been described in this manner. Note that
this listir.g has already taken inLo account thc classification
25
of the places k by L(k). This is done in such a way that
for each encounter on this mission listing the associated
Chj,(k) is positive. In other words, a place k, and hence
an encounter, will appear on the mission listing only if it
seems possible that the opponent will allocate forces to that
place and the corresponding effectiveness against the maxi-
mizer's submarine operating in that place is positive.
There are various possible ways of storing the
information contained in the mission description in the
machine. One way which is very economical in terms of
storage space requirements is to read the list encounter by
encounter, i.e., card by card, starting with an N-1. Then
A(N) = E.Chjt(k) and an indicator array
P(N) - I.J.K.M.(h-1) + J.K.M.(i-l) + K.M.(j-1)
+ M.(k-1) + m + 1
contain the complete information. I, J, K, M denote the
maximum values of the indices i, j, k, m. The reason for
using (m+l) is that the correct m appears after the mth
stage is completed. The disadvantage of this method is
that during computations the individual indices must be re-
computed from P(N) :
h l[ [1] means "the largesth I 1integer in ."
A * P(N) I.J.K-rl(h-l)
AA - A - J.K.M.(i-1)
26
j + AA
AAA AA - K.M(j-1)
k-i + [rA]
m * AAA - M(k-1)
In the sample program contained in this paper, the indices
in their original form are stored in vector arrays 'md are
directly accessible throughout the computations.
2. The Contraction with Respect to Vhim
This contraction takes into account the possibility
that the stage m of a mission may have been noted, but that
the associated Xiim may have been declared to be zero.
Since ir is uscicsi to calculate Pnything involving a ýIm=:),
the corresponding elements in the index arrays of the mission
are eliminated. This is done prior to the execution of the
game and may therefore be referred to as the basic contrac-
tion.
3. Working in Subspaces
It can be expected that most of the xhi and Yjk
will be on their boundaries at any stage, including the
approximate optimal solution. This suggests the idea of
eliminating computations involving variables which have
fixed values either temporarily or throughout the process.
To do this, one reduces the dimensions of the
spaces, thus saving machine time. One can redefine the
space so as to "freeze" the variables on the boundaries
leaving the remainder free to move.
27
Where in the program should redefinition of the
Y-space take place? The minimizer can hold the subspace
fixed and continue to move in that subspace until an ap-
parent minimum has been reached. At this point the al-
location corresponding to this minimum has to be checked for
optimality in the full space. This will require at least
one iteration through the minimization routine in the full
space. After a minimum valid in the full space has been
obtained, the Y-space is reduced to a new subspace. An
alternative approach is to redefine the space after each
change in the Y-allocation, thus maintaining a continously
changing subspace. The authors feel that the latter will
enable the minimizing variable to stay in the subspace
v!it-r finding directcons of decrease, before Litn neced
arises to employ the full space. This idea.was implemented
in the program.
The contractions for x (statement 1871) and y (125)
eliminate computations involving elements that are on the
lower boundaries. It must be noted that the contraction
itself and the complications caused by its use take machine
time. Worthwhile time savings in computation will not be
realized until this technique is applied to relatively large
scale problems.
INDX(N), N - 1,...,NNX, contains the indices of
the Xhi > 0, iNDY(N), N - 1,...,NNY, contains the indices of
the Yjk > 0. NNX and NNY are the dimensions of the subspaces
for K and Y. These index arrays are used to control which
28
variables are to be involved in the computations. The way
this is done in practice can be seen in the program list-
ing.
One remark concerning the DIRECTION FINDING ALGO-
RITHMS, (400) and (1400), may be in order: these algorithms
have to be applied row by row to the matrices 11XhilX and
llYjkl[. In subspaces, the number of elements belonging to
a particular row varies. The outer do-loop runs over the
number of rows. To find the numbers of elements per row
(these numbers serve as the termination values of the inner
do-loops) a test on the indices stored in INDX and INDY is
conducted (402 ff), (1402 ff). The test value, NTEST, is
the index of the last element in the rows of lixhijl and
I ,Yjk I, respectIvely !1-.- Lhe t-st passer. a11 e
in the row at hand have been collected and the algorithm
begins. Before the next row is picked, NSTART is incre-
mented by the number of elements found in the previous row
so that the search through INDX or INDY always starts in the
correct position.
4. Machine Accuracy Problems
The program calls for frequent testing of floating
point numbers. Throughout the development of the program
these tests have been sources of trouble. Testing for
equality must be strictly avoided even after - as has been
done here in various places - variables close to a fixed
quantity have been reset to that quantity (e.g., (1050 ff)
or (1060 ff)). Although the program specified double-pre-
cision, it took extensive experfnmeritation to maintain
29
feasibility, i.e., satisfy the side conditions
F Xhik Yjk"i k
to at least the 1 3 th decimal place.
Of critical importance is the accuracy in the
computation of the direction matrices y (GAX) and g (GAY).
In the neighborhood of a maximum or minimum, the partials
F and F are close to zero, as are the Lagrange multi-x y
pliers (XMU and YMU ), and the differences Fx-XMUFy-YMU
(SD). In order to determine the value of the derivative
game, the sum of the squares of these differences has to be
formed, an operation that may very likely lead to erroneous
results which, in turn, carry over into the computation of
y and g.
The countermeasure taken is to premultiply the SD
when they are small by a large number, (505 ff), (1510 ff).
5. Testing the Program
As the calling of subroutines is exceedingly time
consuming, the present program does not use them. This made
debugging tedious. The "standard" routine, i.e., the pro-
gram employing the derivative games in the original form,
was tested running a small scale example where lixil was
(2 X 2) and I yI was (2 X 4), making it possible to hand-
check the computations.
The "auxiliary game" routine using the B-R process
was debugged using the following objective function:
30
F(x,y) = 2x1 exp[-2yl-y 2 1 + 2x 2 exp[-yl-2Y2]
+ x 3 exp [-yl-y 2 -y 3 ]
Subject to: xi = y Yki k
x it Yk "0
with initial allocations x = (0,0,1), y=(0,1,0). For a
discussion of this example in light of the algorithm see
[4]. The reason that this can only be solved through the
B-R routine lies in the fact that, against the initial x,
any y represents an exact minimum, however, the value of
the derivative game for x is positive, yielded by y*(0, a12, -/T/2).
Finally, an -x.am.rle withi !O0 encountcrs w:a ri
up where lixil was (4x4), llyll was (SxlO). While this does
not yet represent a problem in high-dimensional space, the
authors are confident that this example provided a suffi-
ciently severe test to demonstrate the validity of the pro-
gram as written.
6. General Comments
The following remarks are intended to facilitate
the use and modification of the program.
a. Initial Allocation
The initial allocations are generated as corner
points in the stationary part (99). It can be expected that,
in the optimal solution, most of the variables will be on
the boundaries. An initial point on the boundaries insures
that the ntmber of "absurd" allocations is minimal.
31
If one were to use an interior point as a
starting solution, it possibly would involve large numbers
of absurd allocations (e.g., a submarine in an aircraft
barrier patrol). The machine would spend an enormous
amount of time reducing such allocations to zero.
b. Distance Policy
The initial and re-set values for the dis-
tances are determined from the dimensions of the spaces;
the user may employ his own rules. A compromise between
re-set values too large causing "overshooting" and values
to small causing "creeping" should be considered. In gen-
eral, "creeping" is much costlier in terms of machine time.
The policy for halving distances and its rationale is out-
c. Upper and Lower Bounds
Fbr simplicity, 0 and 1 have been used in the
program. Specifying individual bounds h 8hi and ask)
bjk does not introduce additional problems but increases the
storage space required considerably (e.g., for y, two addi-
tional arrays of the same dimension as y).
d. Modifying the Objective Function
The algorithm as stated is valid for concave-
convex functions under quite general conditions. Though the
present program has been designed to handle a particular lin-
car-exponential function, it is quite flexible. The objec-
tive function mentioned in subsection S may serve to illus-
trate this point:
32
F(x,y) = 2x1 exp[-2y,-y 2 ] + 2x2 exp [-yl-2y 2 ]
+ x3 exp [-yl-y2 -y 3 ]
is produced by a very simple adjustment of the mission
description:
h-i i-il k=1 j-1 E=2.0k=2 j=i E=1.0k=3 j=l E=0.0 END OF MISSION 1
m=1
h-l i=2 k=l j=l E=1.0k=2 j=l E-2.0k=3 j=l E=0.0 END OF MISSION 2
m= 1
h=l i=3 k=l j-l E=1.0k=2 j=2 E=1.0k-3 j=3 E=1.0 END OF MISSION 3
m-=I
The a•.•_ýc itqtci - .v ..... 2.0
V1 2 1 - 2.0
V1 3 1 f 1.0.
The associated Chji(k) are: C ll2 ' CII3 ' 1.0
e. Assigning Values Vhim
The values (of completing stage m of mission i
for submarine of type h) are relative measures. When as-
signing Vhim, the user should consider that a submarine may
have spent part of its weapons (resources) during stage m-l.
This may reduce its operational capabilities for stage m,
and the value for stage m should reflect this fact.
f. Choosing p (RhiO)
The parameter p, mentioned in Chapter II,
section B.2.b, has to be chosen by the user. It specifics
33
the accuracy to which Max O(x) is to approximate the value
of the game F(x,y), V. p affects the y-minimization by
controlling the accuracy, c(d), to which the derivative game
D Max F(x,y) = -VF -g has to approximate its mathematical9x
value. The y-minimization is the most time consuming por-
t ion of the process.
L(d) = P s d
where 6 is the diameter of the X-space. Considering the
"worst" case, when d = d0 = p/36.L, (L is the maximum os-
cillation of Fx), it becomes apparent that
c(do) =
is of order of magnitude p2.10-.
Recall that the conditions established for
the valididty of the algorithm are intended to guarantee
that, at the approximate optimal solution, I0(x)-VIýP.Test runs of the program seem to indicate that the accuracy
actually obtained is much higher.
Although generally valid conclusions cannot be
derived from this observation the user must be aware of this
feature because he will pay heavily in terms of machine time
when p is unreasonably small.
34
IV. SUMMARY
The devlopment of the mathematical theory underlying
the derivative game and the concave-convex game algorithm
has only been sketched out in this paper. Here, that al-
gorithm has been employed in the formation of a potentially
useful example. With programming techniques designed to
provide economical running in high dimensions, large-scale
problems should be amenable to the application of the con-
cave-convex game algorithm.
35
V. SUGGESTIONS FOR FURTHER STUDY
A. MATHEMATICS
In Section III.C.6.e., the question of the proper choice
of p was addressed. Recall that
I Max O(x) - vj. p.
x
Even when the user has based his choice of a value for p
on extensive experimentation with a program, he still will
not know exactly how close Max O(x) is to the value of thex
game. The desirable state of affairs would be
I Max O(x) - V -= p
x
This wouid require the formulation of necessary conditions
on the functions a 0 and c.
B. PROGRAMMING
The fact that, for the class of games at issue, MaxMinFx y
= MinMaxF can be exploited to arrive at the neighborhood ofy x
the optimal solution faster than the present program will:
Start. the problem as MinMaxF. The course of action then is
reversed with considerable advantages. The maximizer sees
the present y-allocation; the maximization is trivial, as-
signing as much weight as possible to the Xhi with the
largest coefficients (the Fx (x,y*)), which results in a
corner solution for x. Then the B-R technique is employed
directly to F(x,y) which will bring x off the boundaries
36
again, and, after a pre-set number of iterations, will pro-
duce an allocation not far from the solution. At this stage
the problem is reinterpreted as MaxMinF and solved in the
manner of the present program.
Another feasible refinement particularly useful in the
application of the algorithm to objective functions linear
in both x and y, i.e., F x xa iyj, is the idea ofij
"doubling", outlined in [4]. The problem is treated as
MaxMin and MinMax at the same time. Here the value of the
game is approached from below and above simultaneously which
provides a stopping rule when the difference MaxF(x,y)-O(x)x
arrives at a pre-specified value.
37
APPENDIX: FLOWCHART OF THE PROGRAMMED EXAMPLE
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COMPUTER PROGRAM LISTING
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