Duality for generalized fractional programs involving n-set functions

JOLJKNAL

OF MATHEMATICAL

Duality

ANALYSIS

AND

APPLICATIONS

149, 339-350 ( 1990)

for Generalized Fractional Programs Involving n-Set Functions G. J. ZALMAI* Washington State Umcersrty. P.O. L3o.x2548 CS, Pullmrm, Washmgtott 9916.5 Submttted by V. Lak.shmkantham Recetved March 28. 1988

Making use of a strong duality theorem for nonhnear programs mvolvmg rr-set functtons. we first estabhsh a Gordan-type transposttton theorem which IS then utrhzed to construct a dual problem for a class of generdhzed fractional programmmg problems wtth n-set functtons. As special cases of the main result. we also obtain dual problems for discrete minmax and fracttonal programmmg problems 1 1990 Academic

Press, Inc

1. INTRODUCTION

AND PRELIMINARIES

In this paper, we shall develop a Lagrangian duality theory for the following discrete minmax programming problem involving n-set functions: (Pl)

F,(S,. .... S,,) inf max Islst G,(S I,..., S,,) subject to H,(S, . .... S,,) 6 0,

jE ?I.

(S, ) . . . . S,,) E d”,

where .n/” is the n-fold product of a a-algebra .d of subsets of a given set X, ~z=(1,2 ,..., nri, F,, G,. iE 1, and H,, ~CZrzz, are real-valued functions defined on .d”, and for each i~_v, G,(S,, .... S,,) >O for all (S,, .... S,,)E ~1”. Optimization problems of this type in which the functions F,, G,, iE1. and H,. .j E m, are defined on a subset of R” (n-dimensional Euclidean space) are known in the literature as generaked fractional programming problems. These problems have arisen in multiobjective programming [ 1, 61, approximation theory [2, 31, goal programming [29], and economics c401.

The notion of duality for generalized linear fractional programs was initially considered by von Neumann [40] in the context of an economic equilibrium problem. More recently, various duality results for generalized linear and nonlinear fractional programs have appeared in [S, 15, 16, 25. * Current address: Department of Mathematics University. Marquette. MI 49855.

and Computer Sctence. Northern

Mtchtgan

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340

G. J. ZALMAI

28, 351, and some results pertaining to their computational aspects have been reported in [4, 17, 18, 21-23, 271. We observe that (Pl) contains as special casesthe discrete minmax and fractional programming problems (P2) inf max E(S,, .... S,) l=Sf
subject to H,(S,, .... S,)
.iEm,

(S 1) ...) S,) E JZP; (P3) inf

4 (S, 9.... S,) G, (S,, ...>S,)

subject to H, (S,, .... S,) < 0,

.iEm,

(S 1, ..., S,) E d”.

The significance of discrete (and continuous) minmax models and methods is well known in many areas of the decision sciences,engineering design, and applied mathematics. A fairly extensive treatment of minmax theory and applications is given in [ 191; however, no duality concepts are discussed in this monograph. Similarly, fractional programs have been used with increasing frequency as realistic models in a variety of decisionmaking situations and consequently have received considerable attention during the last three decades.Recent surveys of this area have appeared in [36, 381 and a comprehensive bibliography in [37]. Although it appears that problems like (Pl )-(P3) have not been investigated in the related literature, other types of optimization problems containing set functions have been studied in [7-12, 26, 30, 33, 34, 393. In particular, optimality conditions for nonlinear programs with set functions are discussed in [S, 30, 33, 341 and for problems with n-set functions in [ 10, 111, some duality results for nonlinear programs with set functions are obtained in [9, 30, 31, 343 and for problems with n-set functions in [lo, 123, and multiobjective problems involving set functions are studied in [7, 26, 393. In this paper, we shall construct dual problems for (Pl)-(P3) and establish appropriate duality results. This will be accomplished by resorting to a Gordan-type transposition theorem, which will be proved in the next section, and by following an approach similar to that employed in [28]. In the remainder of this section, we shall introduce some notation and recall some auxiliary results. Let (X, &, p) be a finite atomless measure space with L, (X, &, p) separable, and let d be the pseudometric on &” defined by d((R,, .... R,), (S,, ....

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341

where A denotes symmetric difference. Thus (.cu”‘, d) is a pseudometric space. A function F: d” + R is said to be c0nt‘e.y if for each i E [0, 11 and (R,, .... R,). (S,, .... S,,)EJP, lim

,I+SUP 7

F(R;

u S;

u

(R, n S, ), .... R;, u S;; u (R,, A S,,))

d j>F(R, , .... R,)+(l-I)F(S

,...., S,,)

for any sequences of sets R; c R, \, S, and Si c S, \, R, , v = 1, 2, .... satisfying I,; -+ w* ;lIRk ,s,, and I,; + w* (1 - i)Z, KI, k E _n, where A \ B denotes the complement of the set B relative to A, 1,~ L., (X. -o/, p) denotes the indicator (characteristic) function of SE .4/. and +%* denotes weak* convergence of elements in L,(X, xl, p). The intuition behind this definition of the notion of convexity for set functions and some properties of convex functions and their use in optimality and duality aspects of nonlinear programs involving set functions are discussed in [34]. The n-set counterparts of these results are given in [lo]. Consider the two problems (PI

inf f(S,, .... S,) subject to g, (S, , .... S,,) d 0. (S, , .... S,) E d”;

(D)

SUP $Ot’)

subject to w E Ry, where $(bc) = inf f(S,, i

.... S,) + f )1’,g,(S, >‘..3S,) : (S,, ...) S,,) E .cdn , I /=I

and IF?; denotes the nonnegative orthant of W”. The following strong duality result, which will be needed in the next section, was proved in [lo] as part of a Lagrangian duality theory for (P) and (D). THEOREM 1.1 [lo]. Let f and g,? j E y, be contiex and assumethat there exists (3,. .... !?,)E,c~” such that g,(S,, .... s,l)
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G. J. ZALMAI

2. A GORDAN-TYPE

TRANSPOSITION

THEOREM

Making use of Theorem 1.1, in this section we shall establish a Gordantype transposition theorem (theorem of the alternative) for convex n-set functions, which will be utilized in Section 3 for constructing dual problems for (Pl)-(P3). A transposition theorem is a statement about the solvability of two systems, say I and II, of equalities and/or inequalities which can be expressed in the following form: Either system I has a solution or system II has a solution, but never both. Theorems of this type have been used frequently in the field of optimization theory and particularly in the area of mathematical programming for obtaining optimality criteria and duality relations for various classesof problems. For detailed discussions of transposition theorems, the reader is referred to [32] for the finite-dimensional case and to [ 13, 141 for the infinite-dimensional case. Our approach to proving the desired transposition theorem is similar to that employed in [24] for point functions, and is based on Theorem 1.1. We shall begin by examining an implication of the inconsistency of a certain system of convex inequalities. This will lead to an n-set function version of a result given in [20]. THEOREM

2.1. Let h, , .... h, be real-valued convex functions defined on

d”. If the system

h,(S,,.... S,)< 0, jem,

(2.1)

has no solution (S,, .... S,) E ,clz”, then there exists w E Ry \ (0) such that f w,h,(S, ,..., S,)>O ,=I Prooj:

for all

(S ,,..., S,)E&“.

Consider the problem (E)

inf 2 subject to h,(S,, .... S,) < ;1,

.iEm,

IER.

It is clear that one can find ((9,) .... SU),2) E &” x R such that h, 6% , .... 3,) - j< 0 for all j E m. Because (2.1) is inconsistent, it follows that for any feasible solution ((S,, .... S,), 1) E &” x R of (E), the corresponding value of (E) will be finite and nonnegative. Therefore, (E) fulfills

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PROGRAMS

all the requirements of Theorem 1.1, and hence its dual problem. which takes the form ,,*

inf i+ c w,[h,(S ,,...,s,,,-j.] sup ,, ERy i.71,, .%)t .d” ,= I i.E R.

)

has an optimal solution w E R’t with nonnegative value. Hence it follows that

However, for this inequality to hold, we must have 1 -C:,l’, W,=O. because otherwise the second term can be - ~8. Thus we conclude that x71_, II‘, = I and f

,= I

w,h,(S 1. .... S,)>,O

for all

(S, . .... S,,) E .w’“,

as was to be shown. 1 This result can be recast in the form of a transposition theorem as follows. THEOREM 2.2. Let h , , .... h,, he real-valued convex -functions dcfi’ned on .d’“. Then either

I

h,(S,. .... S,)
jE p, has a solution (S, , .... S,, ) E .cJ”.

or II

?>1 c w,h,(S,, /=I

for all

. ..) S,) 3 0

(S,, .... S,)E&~

and.for some \cEIW’~‘,,,lOi,

but never both. Proof. Suppose I has a solution (S,, .... S,) E ,ca/“. Then for any w E rW; ‘\,(O}, we have that I,“= 1 u;h,(S,, .... S,) < 0, and hence II cannot have a solution. On the other hand, if I has no solution (S,, .... S,) E .d”. then by Theorem 2.1, there exists u’ E IL!; \ (0) satisfying the requirements specified in II. 1

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G.J. ZALMAI

3. DUALITY

In this section, we shall make use of Theorem 2.2 to identify a dual problem for (Pl). We begin by listing our assumptions. In addition to our standing hypotheses concerning the data of (Pl ), we shall require that the following assumptions hold throughout the present section : (Al ) The functions F,, - G,, i ET, and H,, j E g, are convex. (A21 F,(S, >.... S,) B 0 for all (S,, .... S,) E d”, i Er. (A3) There exists (si, .... 9,) E d” such that H, (3,) .... 9,) < 0, je m. In preparation for the application of Theorem 2.2, we shall first transform (Pl) into a more convenient equivalent form. We begin by observing that (Pl ) can be expressed as

Since G,(S,, .... S,) > 0 for all (S,, .... S,) E &“, i ~1, the above representation can be written in the form inf{lER:

F,(S,, .... S,)-AG,(S1,

.... S,)
iE1, H,(S, ,..., S,)QO, jem}.

Now, if for fixed 1 we define Ai.= {(Sl, ..., S,)E~“:I;(S

,,..., S,)-AG,(S,

,..., S,)QO, i~r,

H,(S,, .... S,)
then because of (A3) it follows that for sufficiently large il, the set A, is nonempty, and furthermore it is clear that the value u(P1) of (Pl) is given by o(Pl)=inf{LER:A,#@}.

(3.1)

Since F,(S, , .... S,)aO and G,(S,, .... S,)>O for all (S,, .... Sn)edpp”, z’E~,it is clear that u(P1) > 0 and hence it is sufficient to consider only ;1> 0 in (3.1). Next, we shall examine the equivalent statement for Al # 0; this will lead to the identification of a dual problem for (Pl).

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By Theorem 2.2, A, # @ if and only if the set B, =

(u, U)E UT+ x Rm, : (u, ~1)#O, i: u,CF,(S , 1 .... S,) - E,G,(S,, .... S,,)] t=I

+ 5 u,H,(S, ,..., S,,)>O for all (S ,,..., S,)E.~” J=I

is empty. Since r(Pl)=inf{~ER+ implies that B, = @ for all p > 2, it follows that

i =inf{E.ER+

:Aj,#@

tl(Pl)=sup(~~R+

i : B,=@

: B,#@Zlj + :thereexistuE[W’,,

Z’ER;, (u,u)+o,

such that 1 u,[F,(S,, .... s,)-AG,(S,, ,=I

.... S,)]

+ f u,H,(S,, .... S,) >O for all (S,. .... S,,)E.~/” ,= I We note that u # 0; otherwise, because of Theorem 2.2, (A3) will be contradicted. Now if we consider the problem (dl)

sup /IER + :thereexist u~R~\j0}. i

v~jWm+suchthat

i u,E;(S,, ...) S,)+ 2“’ u,H,(S,, .... S,) ,=I ,=I >I” i

u,G,(S ,,..., S,)forall

(S, . .... S,)E&”

.

I=1

then it has been shown that o(P1) = u(Dl), and hence (Bl) can be considered as a candidate for a dual problem for (Pl ). Since u # 0, u can be normalized so that xi=, U,= 1, and thus (dl) can be reformulated as

subject to UE IF!;, c U,= 1, ~1E R; . ,=,

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G. J. ZALMAI

The relationship obtained above between (Pl) and (Dl) is, of course, a strong duality result, and constitutes only half of the requirement for duality; the other half is a weak duality relation which should be established between (PI) and (Dl) before (Dl) can be declared a dual problem for (Pl). To show that a weak duality relation does exist between (Pl) and (Dl), let (sr, .... S,,) and (U, 0) be arbitrary feasible solutions of (Pl) and (Dl ), respectively. Then

i ii,G,(&, .... 3,)

,, .... s,)+ 5 V,H,(s,,.... 3,)

t-1

,=l

< i l&l;;@,, ...) S,) i %G,& I 1=l 1=I

.... %I

(since O,H,(S, ,..., S,)dO,jE&) d max i;I(sr, .... S,)/G,(S,, .... s,). I
.... S,)/G,(S,, .... S,),

and hence

Therefore, it follows that

Ma c ii,e(s,, .... S,) c ii,G,(S,, .... s,,). r=l

r=l

Summarizing the above conclusions, we obtain the following duality theorems linking (Pl) and (Dl). THEOREM

3.1 (Weak Duality).

Let F,, -G,,

iE[,

and H,, jem,

be

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convex fi’ith c(S,, .... S,) 20 for all (S,, .... S,,)E .d”, tee, and let (S,, .... S,,) and (u, 6) be arbitrary feasible solutions of‘ (Pl ) and (Dl ). respectively. Then the value of the objective .function qf (PI ) at (S, , .... S,,) is greater than or equal to the value of the objective function of (Dl ) ut (ii, L’). THEOREM 3.2 (Strong Duality). Let E;;, -G,, ier. and H,, JEW. hc US rn Theorem 3.1, and assunze that there exists (S,, .... S,,) E .d” such that H,(S,, .... Sn)
Comparing (Pl) and (Dl), we observe that these problems have essentially different forms in that (Pl ) is a discrete inf sup problem whereas (DI ) is a continuous supinf problem. Evidently, (Dl ) also contains dual problems for (P2) and (P3) which wc shall identify next. If we let G, = 1. I ET, in (Dl ), we obtain the dual problem for (P?) (D2)

sup

ISI.

inf i u,I;I(S ,,..., S,,)+ f v,H,(S ,..,., S,,) s,,I t 71”, = ] i-1

subject to u E Rr+ , 2 u, = 1, 1’E rWT ,=I

If we let r = 1 in (D 1). we obtain the dual problem for (P3 ) (D3)

sup

inf (S,. .sninlt4”

F,(S,, .... S,)+x::“=, v,H,(S ,r...r S,) G,(S,, .... S,)

subject to L’E iw:. If we let r = 1 and G, = 1, then (Dl) reduces to the Lagrangian dual problem treated in [12]. Obviously, with appropriate specialization, Theorems 3.1 and 3.2 can be restated for the pairs (P2)-(D2) and (P3)-(D3 ). We close with a simple example. Consider the problem (P4)

a,(js,aI dp,.... js, a, 4)

inf max 1<1
b, &

.... j.s,, h,,&)

subject to 11,(c,, c, p, .... j-s,, (,,I &)

GO,

JE!%

348

G. J. ZALMAI

where LX,,-B, : R” +[w, i~[, and y,: k!“+[w, Jim, are convex, ak, h,, ck E ,5,(X, JZZ’, p), keg, the numerators of the objective function are nonnegative, and its denominators are positive for all (S,, .... S,) E SZ’“. If we further assume the existence of (3,) .... 9,) E Sen such that

then it is easily seen that (P4) fulfills all the requirements of Theorems 3.1 and 3.2, and hence according to (Dl), the following is a dual problem for (P4):

inf sup (S,...S,)t.d”

CI=, u,a,(S,, al 4, .... Is, a, &I + C:I=, u,Y,&, ~1&, ..., js, cn &I C:=, dL(~,, b, 4, ...f fs, hn &I

1

subject to u E R’+ , c U,= 1, u E Ry. t=l Optimality conditions and duality relations under generalized p-convexity assumptions for generalized fractional programming problems involving differentiable n-set functions are investigated by different methods in [41].

REFERENCES 1. D. J. ASHTON AND D. R. ATKINS, Multicrtteria programming for financial plannmg, J. Oper. Rex Sot. 30 (1979), 259-270. 2. I. BARRODALE, Best rational approximation and strict quasiconvexity, SIAM J. Numer. Anal. 10 (1973), S-12. 3. I. BARRODALE, M. J. D. POWELL, AND F. D. K. ROBERTS, The differential correction algorithm for rational /,-approximation, SIAM J. Numer. Anal. 9 (1972), 493-504. 4. J. BORDE AND J.-P. CROUZEIX, Convergence of a Dinkelbach-type algorithm in generalized fractional programming, Z. Oper. Res. 31 (1987), 31-54. 5. S. CHANDRA, B. D. CRAVEN, AND B. MOND, Generahzed fractional programming duality: A ratio game approach, J. Ausfral. Math. Sot. Ser. B 28 (1986). 170-180. 6. A. CHARNES AND W. W. COOPER, Goal programming and multiobjective optimization, Eur. J. Oper. Rex 1 (1977), 39-54. 7. J. H. CHOW, W. S. HSIA, AND T. Y. LEE, On multiple objective programmmg problems with set functions, J. Mafh. Anal. Appl. 105 (1985), 383-394. 8. J. H. CHOU, W. S. HSIA, AND T. Y. LEE, Second order opttmality conditions for mathematical programming with set functions, J. Austral. Math. Sot Ser. B 26 (1985) 284292. 9. J. H. CHOU, W. S. HSIA, AND T. Y. LEE, Epigraphs of convex set functions, J. Math. Anal. Appl. 118 (1986), 247-254. 10. H. W. CORLEY, Optimization theory for n-set functions, J. Math. Anal. Appl. 127 (1987), 193-205.

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FRACTIONAL

PROGRAMS

349

Il. H. W. COKLEY AND S D. ROBERTS.A partitiomng problem with apphcatlons in regional design, Oper. Res 20 (1972), 101&1019 12. H. W CORLEY AND S. D. ROBERTS, Duahty relatIonships for a partitlomng problem. SIAM J. Appl. Muth. 23 (1972), 490-494. 13 B D. CRAVEN, “Mathematical Programming and Control Theory,” Chapman & Hall. London, 1978. 14. B. D. CRAVEN AND B. MOND. TrampositIon theorems for cone-convex functtona. S/:1 21 J. Appl. M&h. 24 (1973). 603-612 15. J.-P. CROUZEIX, A duality framework in quasiconvex programmmg. 111“Generalized Concavity in Optimization and Economics,” (S. Schalble and W T Zlemba, Eds I. pp. 207-225. Academic Press, San Diego, CA/New York. 1981 16 J.-P CROUZEIX, J. A. FERLAND, AND S. SCHAIBLE. Duality m generalized lmear fractional programmmg, Math. Prog. 27 (1983), 342-354. 17. J.-P. CROLIZEIX. J. A. FERLAND, AND S SCHAIBLE, An algorithm for generalized fractlonal programs, J. Optim. Theory Appl. 47 (1985). 3549. IX. J -P CROUZEIX, J A FERLAND, AND S SCHAIBLE, A note on an algorithm for generahzed fractional programs. J Opttm. Theory Appl. 50 (1986). 183-187 19. V F DEMYAN~V AND V. N. MALOZEMOV. “Introduction to Mmlmax.“ Wiley. New Yorh. 1974 20. K. FAN, J CLICKSBERG, AND A. J HOFFMAN, Systems of mequalities mvolvmg conbe\ functions, Proc. Amer. Math. Sot. 8 (19571, 617-622. 71 J A. FERLAND AND J. Y POTVIN. Generahzed fractional programming. Algorithms and numerical experimentation. Eur. J. Oper. Res. 20 (1985), 92-101 22. J FLACHS, Global saddlepomt duality for quaslconcave programs. II hlarh Pro,~. 24 (1982). 326-345. 23. J. FLACHS, Generalized Cheney-Loeb-Dmkelbach-type algorithms. Math. Oper Re.t IO (1985 ), 674-687 24. A M. GEOFFRION. Duality m nonlinear programming. A slmphfied apphcatlon oriented development, SIAM Rec. 13 (1971). 1-37. 75 E G GOL’STEIN, “Theory of Convex Programmmg,” Translations of Mathematical Monographs, Vol. 36, Amer. Math. Sot , Providence, RI. 1972. 26. W. S. HSIA ANU T. Y. LEE, Proper D-solutions of multIobJective programmmg problem> with set functions, J. Optim. Theory Appl. 53 (1987). 241-258 27 R JAGANNATHAN, An algorithm for a class of nonconvex programmmg problems with nonlinear fractIona objectives, Management Scz. 31 ( 1985 1, 847~.85I 28. R JAGANNATHAN AND S SCHAIBLE, Duality m generalized fractional programming WI Farkas’ lemma, J. Oprim. Theory Appl. 41 ( 1983). 417424. 29. J. S. H KORNBLUTH, A survey of goal programming, Omega 1 (1973). 193--205 30. H C LAG AND S. S. YANG, Saddlepomt and duahty m the optimlzatlon theory of convey set functions, J. Austral. Math. Sot. Ser. B 24 (1982). 13s-137 31. H C LAI. S S YANG, AND G. R. HWANG, Duality m mathematical programmmg of se1 functions. On Fenchel duality theorem, J. Math. And. Appl. 95 (1983 ), 223-234. 32. 0 L. MANGASARIAN, “Nonlinear Programmmg.” McGraw-Hill, New York. 1969 33. P. MAZZOLENI, On constrained optlmlzatlon for convex set functions. in “Survey of Mathematical Programmmg” (A. Prkkopa, Ed.), Vol 1. pp. 273-290. North-Holland. New York, 1979. 34. R J. T. MORRIS, Optimal constramed selection of a measurable subset, J. Murh. .-fnul Appl. 70 (1979). 546-562 35 G. S. RUBINSHTEIN, Duahty in mathematical programming and some problems of convet analysis. Russian Muth. Suroers 25 (1970), 171-200 36 S. SCHAIBLF. A survey of fractional programmmg, m “Generahzed ConcavIty m OptlmlLa-

350

37. 38. 39. 40. 41.

G. J. ZALMAI

tron and Economrcs” (S. Scharble and W. T. Ziemba, Eds ), pp. 417440. Academtc Press. San Diego, CA/New York, 1981. S. SCHAIBLE, Bibliography in fractional programmmg, Z. Oper. Rex 26 (1982). 211-241. S. SCHAIBLE AND T. IBARAKI, Fractional programming, Eur. J. Oper. Rex 12 (1983), 325-338. K. TANAKA AND Y. MARUYAMA. The multtobjective optimization problem of set functton, J. Inform. Optim. Sci. 5 (1984), 293-306. J. VON NEUMANN, A model of general economic equilibrium. Rev. Econ. Stud. 13 (1945) l-9. G. J. ZALMAI, Optimahty conditions and duality for constrained measurable subset selectton problems wtth minmax objective functions. Oplimiaztron 20 (1989) 377-395.