Schaible JOTA 1976

8/12/2019 Schaible JOTA 1976

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348 JOTA: VOL. 19 NO. 2 JU NE 1976

ra t io cos t /v o lum e has to be min imized . Type (d ) i s enco unte red in min imiz-ing the ra t io revenue/p rof i t , i f cos ts are g iven by a convex funct ional andreven ue b y a concave funct ional , as o f ten assum ed .

2 Background

Le t L be a rea l norm ed l inear space , S a no nem pty comp act convex se tin L . Let f l , f2 S-~ R be posi tive and con t inu ous (m any of the fo llowingresu lt s are t rue a lso und er m ore genera l assumpt ions) . Consider the p rob le m

wh ere e i th e r

o r

min f x) = fl x)/fz X), 1)

l is conv ex, f2 is con cave , (2)

o r

f~ is con cav e, f2 is con vex , (3)

o r

fl is conv ex, f2 is con vex , (4)

fl is conc ave, f2 is conc ave, (5)

Obviously, (2 ) - (5 ) cover a lso the corresponding maximizat ion p rob lems,since

m ax f l / f 2) = 1/ m in f 2/ O.

Ac cor ding to R ef. 5 , a funct ional g : S -+ R is cal ledquasi-convex, if, foral l , ~ & a ~(0 , 1),

g( a + (1 - a) ~ ) ~< max (&(), g(Y)).

The functional g is said to bequasi-concave i f - g i s quas i -convex .Rece n t ly, the fo l lowing two c lasses % of mod els were inves t iga ted

(Refs. 11-13):~l-Modet: minimizat ion o f a con t inuous s t r ic t ly quas i -convex func-

t ional over a com pact convex se t ;c~2-Model: minim izat ion o f a con t inuous quas i -concave funct ional over

a com pact convex se t .T he class c~1 genera l izes the c lass o f convex p rograms. M an y theore t ica l

results , as well as algori thm s of convex programm ing~ st il l apply to ~1. In a


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JOT A: VOL. 19, NO. 2 , JU NE 1976 349

certain sen se, ~1 is the largest class of mo dels whe re each local mi nim um is a

global minimum (Refs. 11, 12). For models of class ~2, i t was recentlyprove d by Bere an u (Ref. 13) that a global min imu m is at ta ined at an extre mepoint and that there is no classessentially larger tha n q~2 with the sam eproperty of an optimal solution (for more particular results, see Refs. 11,12). M ajthay a nd W hinston (Ref. 14) proposed an algorithm which calcu-lates an opt imal extreme point of a ~2-model with a polyhedral feasiblesolution set S. A m or e particular me th od was given by Martos (Ref. 11).

3. Rat io Opt imizat ion Problems as ~l .Models or ~z Models

Recently, i t was proved that

f x) =f l x) /fz x)

in (2) is strictly quasi-co nvex (Ref. 6). Th ere for e,problem 2) is a call-model.From this, i t follows that problem 3) is a c2-model,since the object ivefunctionals in (2) an d (3) are reciprocal and th e reciprocal of a quasi-con vexfunction al is quasi-co ncave (Ref. 6).

Ho we ve r, (4) an d (5) are neit her ~l -m od el s nor c2-models, in general,as simple exa mp les show. Conside r, for instance, the following pro blem (4):

= X 1 + 2X2 + X3)/ X1q 1, 2,ix{f(x) 2 2 2 2 2 2 ~>, 3}.x2+x3)lxj j=

f x) s not quasi-conv ex. To see this, let

i = (1, 1/2, 1), ; = (1/2, 1/2, 4/3) .

On e has

Since

f() = 910/819, f ( ; ) = 910/820.

f ( (1/2)(~ + ; ) ) = 34 9/313 > m ax(f(~), f ( ; ) ) = f (g) ,

f is not quasi-convex; therefore, (4) is not always a ~ l-m od el . On the othe rhand, also 1/ [ can be conside red as an object ive funct ional in a model oftype (4). Since f is not quasi-convex,1/ f is not quasi-concave (Ref. 6).Th ere for e, (4) is not always a c2-model. In th e sam e way, it can be sho wn bysimple ex am ples tha t (5) is ne ith er a qCl-model no r a (2-model in gene ral.Therefore, (4) and (5) have none of the nice features of a cl-model orqC2-model, sin ce c~ aremaximal with respect to their corresp onding prop-erty of a n opt imal solut ion. H ow ever, in th e fol lowing it will be shown that(4) and (5) can be rela ted to an eq uivale nt problem , which is a (a-model.


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350 JOTA: VOL. 19, NO. 2, JUNE 1976

Notice , at first, that (5) can be consid ered as a particular case of (4). Forthis , observe the identi ty

f(x ) = fl( x)ff 2(x) = [ l ff2(x)] /[1/fl( x)].

Here , 1/fi is convex (Ref. 6), iffi is concave. Thus, only

min[fl(x)/f2(x)], wh ere/ 1 , f2 are convex, (6)

has to be invest igated further. B y the one- to-o ne variable t ransformation

y = x/ fl (x ), t = 1/f l(x ), x ~ S, (7)

prob lem (4) becomes

rain [1 / t f 2 (y / t ) ]T= { (~ )~ Lx R I t > O , (y / t )~ S , t f l ( y / t )= l }. (8),y)~r

Considering the definition of a convex functional (Ref. 2), i t can easily beverified that convexity of fi yields the convexity of tfi(y/t) (for twice

differentiable functionals, see Ref. 15). Since then tf2(y/t) is also quasi-convex, the reciprocal functional 1/tf2(y/t) in (8) is quasi-concave (notconc ave in general ). Thus, in (8), a quasi -conc ave functional has to beminimized over T. Now, let the equali ty

tfl(y/t) = 1

in (8) be repl aced by

tfx(y/t)


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o vious y oneha

1/ff2(~/~) < 1/t f2(~/ t) = 1/t f2(y /t ).

(, )he refo re, is no t optimal in (9). He nce , (9) and (8) have the sam e set oft

optimal solutions. Thus, not only (8), but also (9) is equivalent to (4). Asalrea dy shown, (9) is a ~2 -mod el. This proves the assertion that (4) [and also(5)] is eq uiv ale nt to a q~i-model.

Re m ark 3.1. Because of the compactness of S and continui ty off1, thefeasible region T of (8) obtained by (7) is compact. The set T will not becha nge d if, in (8), the inequa lity t > 0 is repla ced b y

t~ to = l / k for some k~m ax~l (x ) [x 6S}.

Th e calculat ion of k ma y cause some addit ional computat ional effort. Th efeasible region T1 of (9) is not compact. However, since each optimalsolution of (9) is optima l in (8), in TI the ine quality t > 0 can be re place d by

t ~> to with out cutting an optimal solution o f (9) from T1. Th e rem ainin g partof 7 1 is the n com pact. Thus, the feasible region in the equiv alent prob lem (9)of (4) can be considered as compact .

Summ arizing the p receding results, one has th e fol lowing statement:(2) is a c~l-model, (3) is a c2-model, and (4) and (5) are equivalent to ac2-model.

Th e cOl-model (2) can be solved by a convex program ming pro ced ure(Refs. 2-4). A global opt imum of the ~2-model (3) is obtainable by analgori thm of M ajtha y and W hinston (Ref. 14) provided S is polyhedral . Fo r

the solution of the c2-model (9) equivalent to (4) [and (5)], on e n eeds a nalgori thm that calculates an opt imal extrem e point of a conv ex set, whichgen erally is not a po lyhe dro n. If, for instance,

then

S = { x~ R n lA x < -b } , A = m x n m a t r i x , b c R m,

i .e. , the feasible region o f the equiv alent c2-model (9) is an inte rsection of aconvex polyhedro n and a no npolyhedra l convex se t, def ined by one ine-qual ity. Emphasis should be pu t on the dev elopm ent of an algori thm whichcom putes an opt imal ex trem e point of such a set . For this, the proce dure ofM ajthay and W hinston might be useful .


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e fe rences

1. SCHAIBLE S. Fractiona l Prog ram min g: Transformations, Duality, andAlgorithmic Aspects,Stanford University De part me nt of Ope rations ResearchTechnical Re por t No. 73-9 1973.

2. LUENBERGER D. G. Optimization by Vector Space Methods,John Wiley andSons New York New York 1969.

3. LUENBERGER D. G. Introduction to Linear and Nonlinear Programming,Ad diso n-W esle y Publishing Co mp any Massachusetts 1973.

4. DA NIEL J. W. he Approximate Minimization of Functionals,Prentice-HallEng lewoo d Cliffs New Jersey 1971.

5. ARROW K. J. and ENTHOV EN A. C. Quasi-Convex Programming,Econ ome trica V ol. 29 pp. 779-80 0 1961.

6. SCHAIBLE S. Quasi-Convex Optimization in General Real Linear Spaces,Zeitschrift ffir Ope rations Research Vol. 16 pp. 205-21 3 1972.

7. SCHAIBLE S. Quasi-Concavity and Pseudo-Concavity of Cubic Functions,Math ematical Programm ing V ol. 5 pp. 243-247 1973.

8. FERLAND J. m. Ma ximaID oma insofQu asi-Con vexity andPseudo-Convexityfor Quadratic Functions, Mathem atical Programming Vol. 3 pp. 178-1921972.

9. SCHAIBLE S. Dual Fractional Programs, Stanford University Dep artme nt ofOpera tions Research Technical Repo rt No. 75-5 1975.

10. RANI O. and KAUL R. N. Duality Theorems for a Cla ss of NonconvexProgramm ing Problems,Journal of Optimization The ory and Applications Vol.11 pp. 305 -308 1973.

11. MA RTO S B. The Direct Power of Adja cent Vertex Programming Methods,Ma nage me nt Science Vol. 12 pp. 241-252 1965.

12. MARTOS B. Quasi-Convexity and Quasi-Monoticity in NonlinearProgramming, Studia Scientarium Mathe maticarum Hungarica Vol. 2 pp.265-273 1967.

13. BEREANU B. On the Global Minim um of a Q uasi- Concave Functional,Archivder Ma them atik Vol. 25 pp. 391-393 1974.14. MAJTHAY A. and WHINSTON A. Quasi-Concave Minimization Subject to

Linear Constraints,Disc rete Mathem atics Vol. 9 pp. 35-59 1974.15. MANAS M. O n Trans[ormations of Quasi-Convex Programming Problems,

Eckono micko-M atematicky Obzor Cescoslovenska Aka dem ie VE D Vol. 7pp. 93-99 1968.

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