Kleine Mitteilungen ____. ~__.____. ~~

KLEINE MITTEILUNGEN

ZAMM . Z . Angcw. Math. 11. Rfrcli. G4 (1984) 8, 363 -364

S. SCHAIBLE

Simultaneous Optimization of Absolute and Relative Terms

The optimization of a weighted sum of an absolute and a relative term is considered. Conditions for the functions defining these goals are derived that ensure one of the following properties of these nonconvex programs: a local maximum is a global one or an optimal solution is attained a t a n extreme point.

I n many applications of operations research relative rather than absolute terms are to be optimized giving rise to a fractional program [9, 16, 21, 22, 23, 241. Recently some authors have discussed the simultaneous optimization of two or more relative goals. The objective function then becomes a weighted sum of ratios [2,4,8,27]. Of particular interest is the special case where a compromise between an absolute and a relative goal is sought [l, 5-7, 10-15, 17, 18, 20, 25, 261. This may give rise to one of the following models :

X€S

or

X € S

Here the functions fi, ge may represent cost, revenue, profit, capital, risk, time etc. [21, 241. The real numbers 1 and p are parameters that express the relative importance of the absolute versus the relative goal.

An e x a m p l e of m o d e l (1) would be t.he simultaneous maximization of profit and return on investment (profit per unit capital). Model (2) arises for example when a weighted sum of risk and expected return/risk is to be maximized (p < 0).

Without loss of generality we can assume that the optimiza- tion problems are maximiza.tion problems and that the denomi- natorsf, and g1 are positive on S. Let us suppose that S is a con- vex set in Rn. We recall that if the objective function F (or G) is semistrictly quasiconcave I ) on S then a local maximum in (1) (or (2)) is a global one [3]. Also, if F (or G) is quasiconvex on S an optimal solution of (1) (or (2)) is attained a t an extreme point of S if S is compact. These features of an optimal solution me very valuable when a solution is to be calculated.

I n this note we derive conditions on fi, g i that ensure semi- strict quasiconcavity or quasiconvexity of F , G .

For the sake of convenience we IISC the following abbrevia- tions :

qcv = semistrictly quasiconcavc qcz = semistrictly quasiconvex.

For definitions and characterizations of generalized conca.ve func- tions the reader is refered to [3].

The concavity and/or convexity properties of f i , gr below ensure continuity on the relative interior of S. I n the following we assume that these functions are continuous on all of 8.

First we focus on the objective function P in (1) . We can write F as a composite function F ( z ; A) = h( fl(r), f2(z) ; 1) where

M ~ X ~ ( r ; 1) = ~ ( x ) + .f,(x)/m) ,

Max ‘3%; i d ) = / v l ( 4 + & 9 / g l ( 4 ,

3, z 0, (1)

11 f 0 . (2)

h(y,, ~ 2 ; -A) + Y~/Y, 9 ~1 6 R 9 YZ > 0 *

Furt,hormore

N Y ~ , Y ~ ; -A) = Yl/k(YZ; 1) where

k(y,;l) = y2/(1y2 + 1) for y2 > 0 , . y2 # -1p. Here k(yz;-A) is either concave or convex depending on the signof-Aandwhethery,< -l/Aor yz> -l/ilincase1<O.Henceh is the ratio of a linear function and a concave or convex function. Such a ratio is either qcz or qcv; see for example [19J As a result we find:

1) According to [3] a function f is semistrictlv quasiconcave in S if for all x , FE Ssuch that f(z) # f(z weliave f ( t i + (I - t ) ; ) > Min(f(%), f(=Z))for all 1 E (0,l).

363 ~ ~ ~- ~~

for -A > 0 h is qcx if y, 2 0 and qcv if y1 5 0, and for-A<0 h i s q c v i f y , 2 0 a n d y 2 S - - 1 / 1 o r y , ~ O

and y2 2 -111, and h is qcx if yl 2 0 and y2 2 -111 or y1 5 0 and ye 5 -111.

From a criterion for the generalized concavity of monotone compositions of convex and/or concave functions [19] we find for F(x; A ) the following properties that are summarized in the tables 1, 2 below:

Table 1 a > o

f, > O , concave fi 2 0 P qex convex 5 0 P qcv conrave

Table 2 R < O

- f z 5 -ua, 2 -I/&

fi convex concave

2 0 qm F qcx concave

5 0 F qex B qcu convex

I n [20] such properties were discovered for the special case of linear functions j , , .f,. The foregoing shows that under rather general assumptions on f t the objective function F in (1) still has valuable generalized concavity properties.

R e m a r k : Accidently in [20] the bound --A instead of -1/-A appears in case 1 < 0.

Let us now turn to the function G in (2). We can write

G(x; P ) = m(g,(x), g,M; cl)

where

m(Y19 Y2; = PYI + Y ~ / Y I 1 YI > 0 9 E R .

m(y1, Y2;

Then

= (clY;” + Y2)/Yl

which is qcx if p > 0 it‘nd qcv if p < 0. Furthermore, m is increasing in yl if y, 5 py: and decreasing in y1 if y, 2 py?, and m is increasing in y2. Applying a criterion for the generalized concavity of composite functions [19] we find that G ( r ; p) has the properties summarized in the tables 3,4:

Tab le 3

P > O

T a b l e 4 l r < O

g, convex 9 s concave 91 91

147: L 9, G qcx P!?? 2 9. G qcv convex concave

.s; 5 8% a qcx Pg: 5 9 2 G qcv concave convex

____-

For linear functions gl, g2 these properties were shown in [ZO]. There the condition pgi: 2 g, or pg; 5 g, is not needed. This is still true if only g1 is linear. Indeed we have in this case:

for p > 0 G i s qcx if g2 is convex, and for p < 0 G is qcv if g, is concave.

Like F the function G still possesses valuable generalized concavity properties under rather general assumptions. These will be useful when an optimal solution is calculated.

R e f e r e n c e s

1 AUQARWAL, S. P. ; SAXENA, P. C., The deconiposition method for linear pro- gramming problems with linear and fractional target fnnctions, Przeglad Statyst ‘LS, 211 -219 (1976).

2 ALMOUY, Y.; LEVIN. O., A class offractional programming problems, Opera- tions Research 19, 57 -67 (1971).

3 AVRIEL, M.; DIEWERT, W. E.; SCHAIBLE, S.; ZIEMBA, W. T., Introduction to concave and generalized concave fnnctions. in SCHAIBLE. 5. and W. T. ZIEMBA (eds.),-Generalized Concavity in Optimization and Ehonomics, Aca- demic Press, New York 1981, 21 -50.

4 CABOT, A. V., Maximizing the sum of certain quasiconcave functions using generalized Benders decomposition, Naval Res. Logist. Quart. 26, 473 - 482 1191Qj \_”. -,.

5 CHADHA, S. S., Duality theorem for a generalized linear and linear fractional program, Cahiers Centre d’Etud. Rech. OpBrat. 16, 167 -173, (1873).

6 CPADHA, 9.5.; GUPTA, J. M., Sensitivity analysis of the solution of a genera- lized linear and piecewise linear program, Cahiers Centre d’Etud. Rech. OpBrat. 18, 309-321 (1976).

7 CHADHA, 9. 9.; SHIVPWI, S., Parametrization of a generalized linear and piece-wise linear programming problem. Trabajos Estadist., Investigacion Operat. 28, 151-160 (1877).

___ -__- ~ _ _ _ 364 Kleine Mitteilungen

8 CHno, E. TJ., Multicriteria Linear Fractional l'rogramming, Ph. 1). Thesis,

9 CRAVEN, B. D., Mathematical Programming and Control Theory, (!Iiapmnn and Hall, London 1978.

1 0 HIRCHE, J., Zur Extremwertannahme und Dualitnt hei Optimiernngnpro- blemen mit linearem und gehrochen-Hnearem Zielfunktionsanteil, ZAMM 65, - -L 184 - 185 (1975).

1 1 HIXCHE, J . ; TAN, H. K., l!ber eine Xlasse nichtkonvexer Opt,imiernngspro- bleme, ZAMM 6'7, 247-253 (1977).

12 KANCHAN, P. R., Upper bounds in linear and piecewise linear programniing, Acta Ciencia Indica 5, 357-360 (1977).

niques in linear-plus-fractioiial programming, Cithicrs Cent,re cl'Etnii. JLecli. OpBrat. 25, No. 2 (1981).

1 4 LIJPBA, L., Sur l'nllure d'une classe d e fonctlons hyperboliques, Studia llniv. Babeg-Bolyiti, Matliematica 2% No. 2, 86-72 (1978).

15 MIBRA, S.; DAB, c, , The sum of a linear and linear fractjional function and a three dimensional transportation problem, Opaearch 18, 139-157 (1981).

16 MJELUE, K. M., Methods of the Allocation of Limited Resources. J . Wiley and Sons, Chichester 1983.

17 RITTER, K., A parametric method for solving certain nonconcave maximiza- tion problems, J. of Computer and System Sciences 1, 4 4 - 5 4 (1967).

18 SAXENA, Y. C.; PATKAR, V.; PARKABH, O., A note on an algorithm for integer solution to linear and piecewise linear programs, Pure and Appl. Math. Soi. 9, 31-36: (1979). n n

19 SCHAIHLE, S., Quasironvex optimization in general real linenr spacw, %. filr Operations llesearclr 16, 205 -213 (1972). i - 1 j = 1 20 SCHAIBLE, S., A note on the sum of a linear and linear fractional fnnrtion, Naval Res. Logist. Quart. 94, 691 -693 (1977).

21 SCHAIRLE, S., Analyse und Anwendnngen von Quot.ientenprograinriiell, Hain-Verlag, Meinenheim 1978.

22 SCHAIHLE, S., A survey of fractional programming, in SCHAIHLE, S. and W. I!. ZIEMBA (eds.), Generalized Concavity in Optimization and Economics, Acade- mic Press, New York 1981, 417-440. C i j = z Z j , i,j=l, ..., n , 23 SCHATBLR, S., Bihliography in fractional programming, Z. fiir Operations Iles. 26, 211-241 (1982).

European J . of Operational Reiiearch 12, 325 - 338 (1983).

gramming, Acta Clencia Indica 4, 199-201 (1978).

ing, (Russian), Ekon. i. Mat. Net. 6 , 440--447 (1969); lhglish translation 5.. , J - l , - Z [ ; j = l , ( q . , . > O , i . j = l , ..., n,, (10)

[rniversity of British Columbia 1981.

wohei Y = (y,, y,, ... , ?/k) wiedcr spaltenorthogonal niit 11!/,11 2 lk/zll 2 ... k 11?/h'l1 ist. Wir norltlierell die ?/I, ... , It Ulld Cr-

ganzen dicse zu einer orthogonalen Matrix 1Jniversit.y of British Colnmbia (1980).

I k , xk:+i , ... , xn E R9t.n . (6) 1 ' - ( ~ ~ ~ 1 ~ ~ ' * * ' ' 1l!/k11

Dann 1iflt sic11 1imgpl<ehrt 1.' in ,Ier Form 1 3 KANCBAN, P. R.; HOLLAND, A. 8. B.; SAHNEY, B. N., Transportation tech- (Y, 0 ) = Zdl/? (7)

mit A = Diag (&,8,, ... , 8,)

= I l ~ i l l ~ = llzi1I2, i = 1, ... , k , i = k + 1, ... , n , b

darstellen. Es gilt dnnn auch 8, 2 8,z ... 2 6,. Damit l&Bt sich (5) nlit

s p S T A ~ ~ = R p ~ T A 1' = sp d1/2%TA%,l1/2 = 2 2' Ai+ij . (8)

Diesen Ausdriwk wollen mir iiber alle orthogonalen Matxizen Z nach oben scharf absrhatzcn h w . maximicren. Hirrzu fiihren wir die Variablen

= ( z i j ) i , j=l, 2, ,.,, writcr umrc.ctlncn,

(9) ein, welche linter al1einigr.r Bcriicksiclit'igung iler Zeilen- und Spaltennormicrung der ~rt~hogonnlen Matrix Z den Rcst,riktio- nen n

24 SL'HIIHLE, s,; IsARAKI, T,, Fractional programmi,lR (Invited review),

25 SHUKLA, D. P.; KANCHAN, P. K., Sum of linear and linear fractional pro-

26 TETRREV, A. G., On a generalization of linear and piecewi*e linear programm-

27 WARBURTON, A. R., Topics in Molticritrria Optimization, Pli. I). Tlieniq,

n

in Matekon, 246-258 (1970). i z. 1 j==1

geniigen. Wir maxiniieren nun (8) statt iiber alle orthogonalrn Matrizcn Z = (zij) iiber die [ij rnit den schwacheren Kestriktio- nen (10) und hoffen, dal3 sich dabei derselbe Maximalwert und damit eine scharfe Abschitzung fur (8) ergibt.

Wir betrachten also dns folgcndr %~~ordn~rngnprohlam mi t biproportionalr?n 'Tosten,

Received Sept,ember 22, 1982, revised version March 26, 1983

Address: Professor Dr. SIEGFRIED SCHAIBLE, Faculty of Business Administration and Commerce, University of n n

i = l j - 1 Alberta, Edmonton, Alberta T6G 2G1, Canada F = 2 2 L,Bj( i j = Max! (11)

fiber n n

i - 1 j- I

mit

z [ i j = l , X [ i j = l , [ijzo, i , . j = = l , ..., n ,

ZAMRI. Z. Angcw. Mntli. u. Mcch. 64 (1984) 8 ,361 - 3 6 5

J. FOCKE A, 2 a, 2 ... 2 A,, 8, 2 8, 2 ... 2 8,.

Diescs Problcm kann man iibcr die allgemeine Theorie dcs Zu- ordnungsproblcnis bzw. der linenren Optimierung hehandeln; es l i B t sich aber such als ein spezielles Transportproblem mit biproportionalen Kost,en element,ar liisen 131. Als Mnximalwcrt

>:he Abschltzung fur quadratisehe Formen Fiir quadratische Formen gilt die bekannte Abschitzung zTAx 1, l l ~ 1 ) ~ fur alle x E Rn (1) ergibt sich dnnn

dnrch den groaten Eigenwert 1, von A und die eiiklidische Norm Max F = 1.8, 4- -1- ... -C I.+,& . (12) I & , a * I , ,* ,. \ - - I yon x . Weniger bekannt, aber in den Anwendungen sehr niitz- ]ich, ist die in foIgelldem L~~~~ angegebene vera1lgemeinerung dieser Abschatzung, welche auch eine Verallgemeinerung der

Wir erlialten daniit fiir (8) unter Benrhtung von (7) die Ab- schatzung

Ungleichung von KY FAN [I] (vergl. auch [2],%. 77) darstullt. Sp X T A X 5 1161 + A,8, + ... 4- -1- &+I . 0 + ... + 1, . 0 5

X = (Pi, 22, ... 9 Xk) E Rn'k 9 llQlll 2 llzzll 2 ... 2 l l p k l l > 0 9

(2)

(3)

die scharfe Abschutzung

SpXTAs ~ ~ i ~ / ~ l ~ ~ z +&11x~11~ + .*. + & l I z k l / ' -

Das Gleichheitszeichen wird angenommen, wenn jedes xi ein zu Li zugekiiriger Eigenvektor von A ist.

Beweis: Wir transformieren A mittels der Matrix Q = = (q,, q,, ... , 9%) der orthonormierten Eigenvektoren auf Hauptachsengestalt *

Dann transformiert sich die Spur in (3).gemlB Q T A Q = A = Diag (AL, A,, ... ,A,) rnit Q T Q = I . (4)

Sp XTAX = Sp YTAY rnit Y = QTX, 16)

A n w e n d u n g auf e i n A n s g l e i c h s p r o b l e m

I n der Statistik, insbesondere bei der sog. Hauptkomponenten- methode der Faktoranalyse, spielt, das folgcnde Ansgleichspro- blem eine wichtige Rolle,

IIA - XXT112 = Miri! , X E Rn,fi, (14) wobei A eine symmetrischc n . n-Matrix und I I .. .I I die Quadrat- summen-Matrixnorm sind. Wir wollen dieses Problcm durch scharfes Abschatzen mittels unseres Lemmas losen, wobei sich zugleich die Existenz des Minimums ergibt. Da X nur iiber das Produkt X X T in die Zielfnnktion eingeht,, ist die Minimallosung nur bis auf Rotation im R k bestimnit, und wir kBnnen uns bei der Minimierung von (14) auf spaltenorthogonale X mit gemal3 (2) geordneten Spaltenvektoren z.~ beschrinken,