Kleine Mitteilungen 495

KLEINE MITTEILUNGEN C. 1%. BECTOR

Some Aspects of Quasi-Convex Programming * I n t r o d u c t i o n

Recently much attention has been paid to the class of pro- gramming problems which are more general than convex programming. ARROW and ENTHOVEN [l] and ARROW, HURWICZ and UZAWA [2], discussed certain properties of quasi-convex and quasi-concave functions with special reference to applications to mathematical programming problems and economics. MANGASARIAN [12] introduced the idea of pseudo-convex and pseudo-concave functions and investigated some of their properties in context of mathe- matical programming problems. BELA MARTOS [13] in- vestigated the necessary and sufficient conditions in terms of quasi-convex (quasi-concave) and explicit quasi-convex (explicit quasi-concave) functions under which a mathema- tical programming problem with linear constraints can be solved with the help of a technique similar to 'simplex method'. The main purpose of the present paper is to extend certain results of fundamental importance similar to those proved for convex programming by HADLEY [lo], for non-linear indefinite programming by BECTOR [4], [B], for non-linear fractional programming by BECTOR [B], for a quasi-convex programming problem in which the objective function to be optimized is quasi-convex or explicit quasi- convex and the constraint set is an appropriately restricted convex set.

P r e 1 i mi n a r i e s Let il: be an n vector of variables or points in Rn, a Euclidean space of dimension n. Let S be a non-empty closed convex sub-set in Rn and f be a function defined on S, such that f: S + R, the set of reals. We now define the function f over the convex set S to be,

(i) Explicit Quasi-Convex (EQX), if for all xl, zz satifsying f("J +f(xz)

fP "1 + (1 - 1) "21 < Max EP(xJ, f(zz)l Y v 1 E (0,1) *

fCA "I + (1 - 4 %I 5 Max [f(Zl)t f(xz:2)1 *

(ii) Quasi-Convex (QX), if for all zl, x2 E S and V 1 E (0, 1) ,

Furthermore, the function f on a convex set S is defined to be Explicit Quasi-Concave (EQV) or Quasi-Coneave (QV) according as, -f, is EQX or QX respectively.

Below we now give two results, in the form of lemmas, u hich we shall use in the analysis to follow.

Lemma 1: E'uery compact non-empty convex set admits at least one extreme point [3].

L e in in a 2 : Let Ill% - 1 be a supporting hyperplane of a non- empty compact convex set S, c Rn. Then the set S n - 1 = = H n - 1 n S n is a cowLpact non-empty convex set in (n - 1)- dimensional space [3].

Quasi - Convex P rogram mi n g P r o b 1 em The quasi-convex programming problem considered here is stated as:

(1) where the function p is continuous and is QX or EQX (as the case may be) over S. We now establish the following fundamental results for the Problem (1).

BELA MARTOS [13], among other results proved that, "If a function which is continuous on a non-empty compact convex polyhedral set L c Rn be EQX on L, then every local minimum of p over L is global minimum also". For pro- blem (1) this result can be improved as follows:

Theorem 1: If pl be EQX on S then every local minimum o j p over S is a global minimuvn also.

Optimize (Maximize or Minimize) pl(x) for x E S ,

* (i) At present visiting Asst. Professor at Purdue Univ. Lafayette, Indiana. (U. S. A,) (ii) From July 1, Associatc Professor, University of Manitoba. Winnipeg, Canada.

ZAMM 60, 495 -502 (1970)

Proof : If possible let the assertion of the theorem be false, such that if zo E S is a local minimum of pl and x*( =!= x0) is a global minimum of pl over S, we have

pl(X0) > +*) * (2)

Consider

z,l = A x* + (1 - A) xo , v 1 E (0, 1) . (3)

Since S is convex, therefore for all 1 E (0, l), x~ E S. Using the explicit quasi-convexity of pl over S, we get

But xo E S is assumed to be a local minimum of pl over S, and pl is continuous over S, therefore, there exists an eneigh- bourhood d8(z,,) of zo such that for an appropriate value 2 of 1 E (0, 1) in (3) i t is always possible for us to choose xi in such a way that x;z E Jv,(z0) n S and

d"i) 1 P("0). (6)

From ( 5 ) and (6) we see that we have a contradiction. Thus the result follows.

For a convex programming problem (see HADLEY [lo, page 901) i t is proved that the set of those points in a set, a t which the convex objective function takes on its global minima, is a convex set. For problem (1) this result can be proved as in Theorem 2 below.

Theorem 2: If p be QX on S, then the set of those points of S at which pl takes on its global minimum over S , is a convex set.

Proof : Let M denote the set of those points of S a t which pl takes on its global minimum. If M is empty or a singleton, then M is trivially a convex set. So, let US assume now that zl, 2,; x1 =k x2 are in M , so that

(7) Since pl is QX on S, therefore, for all A E [0, 11, (7) implies that

(8) plu XI + (1 - 1) "21 5 94x1) = 94%) * Since xl, 3cz are points of a global minima, therefore, in (8) pl[A x1 + (1 - A ) z2] < pl(zl) = pl(ac2) is not possible. Hence for all 1 E [0, 11 we have

v[J-=1+ (1 - 4 "21 = p l ( 4 = pl(=z) *

This implies that A z1 + (1 - A) zz E M for all 1 E [0, 11. Hence M is a convex set.

We now prove the following more results for problem (1).

Theorem 3: If Q, be EQX over the set S and $ it attains its global maximum at an interior point of 8, then it is constant in S.

Proof : Let pl be not constant in S. Let x* be an interior point of S such that

This assumption that pl is not constant in S implies that i t is possible for us to find a point z1 (different from o*) in S such that

pl(z1) < d"*) - (10)

and pl(x1) * d x z ) and xa = 1 zl + ( 1 - A) z2

xa* = x* for some value A, of 1 E ( 0 , l ) .

Let x2 be another point in S such that xz + xl, x, =k x*

for all 1 E (0, 1) , (11)

(12)

such that

Since x2 E S, therefore, either pl(x2) = pl(a*) or pl(x2)< pl(x*). In the former case we have on making use of (10)

d " 1 ) < pl(x2)

32 *

496 Kieine Mitteilunneli -

whereas in the later case we have either

pl(21) < d Z 2 )

P@l) > P(%) . or

Thus in all we have any one of the following two mutually exclusive possibilities: p-1. } (13)

y(z1) < y ( z 2 ) when either (i) pl(z2) < y ( z * ) 01 (ii) d X 2 ) = P@*)

p-2. dz1) > v(z2) when p l ( 4 < pl(Z*) (14) Assuming that P-1 exists and making use of the fact

that y is EQX over 8, we obtain ~ ( m ) < y ( q ) for all A E (0, 1) , or

y ( z ” ) =- pl(mn*) < y ( z 2 )

in particular. Which is a contradiction to (13).

we obtain Similarly when we assume that the possibility P-2 exists,

which again IS a contradiction to (10). Here the result follows.

The or c m 4 : If y be QX over S, then it cannot have a strong relative maximum in the interior of S.

Proof : If possible let Z, belonging to the interior of S be a point of strong relative maximum of pl. This means, there exist points

xl, z2 E J ~ / E ( ~ ~ ) n S , x1 * x 2 ,

pl(Z1) < & o ) , d Z 2 ) < dr,) (15)

such that

and ack = Ax1 + (1 - A ) x2 for all A E (0, I), such that X A ~ = Z, for some value I , of A E (0, 1). Now either

+l) f p l ( 4 01 pl(;cz) d P@l) *

P ( X 1 ) 4 pl(Z2)

P(4 = Max [v(=1)3 pl(%)l .

To be specific let us assume that

I. e.

(16) Using the quasi-convexity of pl over S and making use of (16) we obtain y ( q ) 5 ~ ( r . ) for all A E (0, 1) , or

V(“x0) = P(%) s (Y(z2)

P(%) 5 P(S.2)

i.e.

which is a contradiction. Hence the result. Corollary: If pl be a QX function on S, then i t cannot

have a strong global maximum in the interior of 8. Theorem 5: If the set S be bounded also and the function y

be EQX on 8, then its global maximum will be taken on at one or more extreme points of S.

Proof : Since the set X is a non-empty compact convex set, therefore, by Lemma 1, i t admits a t least one extreme point._,Now when pl attains its global maximum in the interior of the set S, then by Theorem 3, i t is constant over S and, therefore, takes on its global maximum a t an extreme point of X.

Now let us assume that pl takes on its global maximum over S a t a boundary point x* E S and not a t an interior point of S. In case x* is already an extreme point of S, we have proved the result. I f not, then we consider the set Sn - 1 = HYt - 1 n 8, where HE - 1 is a supporting hyperplane of S a t point x* E S. Since the set S is a non-empty compact convex set with S c Rn, therefore, by Lemma 2, Sn - 1 is also a non-empty compact convex set lying in a (n - 1)-dimen- sional space. Thus Lemma 1 implies that Sn-1 admits

at least one extreme point. Now if x* happens to be an interior point of S~Z- 1 with respect to the space of dimensions (n - l), then we immediately conclude that pl will be con- stant over S n - 1 and therefore i t attains its global maximum a t every point of S n - 1 and hence a t an extreme point of Sn-1, which will also be an extreme point of 8. If z* is on the boundary, we obtain another non-empty compact convex set Sn-2 = H z - 2 S n - 1 with 8 % - 2 lying in a (n - 2)-dimensional space, where H i - 2 is a supporting hyperplane of the set S n - 1 a t point x*, and repeat the above argument. This process must terminate in a finite number of stages because So will be a non-empty convex set lying in a 0-dimensional space and hence will contain only the single point x * , which by Lemma 1 must be the extreme point of So and therefore of 8. Hence the result.

R e m a r k 1: A result parallel to Theorem 5, for the case when function pl is QX can be stated as follows. “If S be bounded also and the function pl be QX on S, then its strong relative (and hence strong global) maximum will be taken on a t one or more extreme points of S”. The proof of this result makes use of Theorem 4 (Corollary to Theorem 4) and follows exactly on the lines of Theorem 5.

R e m a r k 2: I n view of Remark 1 and the results proved above we can prove the following more general result parallel to Theorem 5. “If X be a non-empty compact convex set and a function p to be optimized over S be such that i t cannot have its global optimum in the interior of 8, then pl will take on its global optimum a t one or more of the extreme points of 8”.

R e m a r k 3: If the function pl in problem (1) be QV or EQV as the case may be, then we can easily prove results ana- logous to those proved above.

R e m a r k 4: Although, in this paper, w-e have established that when pl is EQX (QX) over a non-empty compact convex set 8, then i t will take on its global maximum (strong global maximum) a t one or more extreme points of S, yet i t is not possible to exploit this fact to develop a com- putational technique typical of “Simplex Method” to find the global maximum of pl, since the well known ‘Adjacent Extreme Point Methods’, do not necessarily yield the global maximum and in our case it is possible that p may take on its local maximum, different from global maximum, a t an extreme point of the constraint set. However in such a case, to obtain a local maximum of pl over 8, a ‘Simplex- Like Technique’ can be developed. In case the function q, in addition, be EQV also over 8, then i t may be possible to obtain the global optimum of cp over X, with the help of a ‘Simplex-Like Technique’.

We now consider the following quasi-convex programming problem. Minimize pl(r) for z E I’ (17)

P = {z!; g@!) 5 0 , i E 44r = (1,2,. . . , ,WL), z 2 0, 0 E S} . where, I’ c S is the set of feasible solutions given by

KUNZI, KRJLLE and OETTLI [11], assuming the functions q~ and gi, i E M to be convex consider the following simplified programming problem

where Minimize pl(z) for c E P a (18)

Pa = {z ; gi(z) 5 0 for i E Na, Z+ 2 O for j E Ia}, N a = { i ; g i ( Z ) = O , EM},

32 = ( X l , x2, . . . , X d ’ , Ia = {j; X I = 0

and proved that if x* is an optimal solution of (17) then i t is also an optimal solution of problem (18). Here we show that the result still holds if we assume that the function pl is EQX, and all the functions gi, i E M are QX.

Proof : Assume that X* is not an optimal solution of (18), then there exists a point z with ”1 2 0, j E I=, g&) 5 0 for i E Na and p ( x ) < p(x*). Then for sufficiently small 3, > 0, we have gi[A x + (1 - A ) x * ] 5 0 i E 11.1 , A xf -1- (1 -A) x,? 2 0 for all j E I

for j E I = (1, 2,. , . , n ) )

for all

Kleine Mitteilungen 497

and because of the explicit quasi-convexity of y

vll z + (1 - 4 %I < dz*) which is a contradiction to the fact that z* is a global minimum of (17). Hence the result follows.

Remark 5: The converse of the above result is obviously not true. A solution of (18) can violate some of the dropped constraints of (17).

Acknowledgement I whish to express my gratefulness to Professor J. N. KAPUR, Head of the Department of Mathematics, Indian Institute of Technology, Kanpur, for his encouragement during the preparation of this paper.

References 1 K. J. ARROW and A. C. ENTHOVEB, Quasi-Concave Programming, Econo-

metrica 29 No. 4, pp. 779-800 (1961). 2 K. J. A R R ~ W , L. HURWICZ and H. UZAWA, Constraint Qualifications in

Maximization Problems., Nav. Res. Log. Quart. 8, pp. 175-191 (1961). 3 C. BERRBE, Topological Spaces, New York 1963, The Macmillan Company. 4 C. R. BECTOR, Non-Linear Indefinite Functional Programming with

Non-linear Constraints, Cahiers du Centre d’Etudes de Recherche Operationnelle 9, No. 4 (1967).

5 C. R. BECTOR, Non-linear Fractional Functional Programming with Non-linear Constraints, ZAMM 48, No. 6 (1968).

6 C. R. BECTOR, Some Aspects of Non-Linear Indefinite-Fractional Functional Programming, Accepted for publication in Cahiers du Centre d’Etudes de Becherche Operationnelle (1967).

7 C. R. BECTOR, and S. K. GUPTA, Nature of Quotients, products and rational powers of convex (concave)-like functions, The Mathematics Student, Jour. of Indian Math. SOC. (India) 36, (1968).

8 C. R. BECTOR, On Nature of Products, Quotients and Rational Powers of Convex-like Functions, Presented at Symposium of Operational Rcsearch Society of India, December 1968. To be published in the Bulletin of 0. R. SOC. of India December 1968.

9 C. R. BEOTOR, Mathematical Analysis of Some Non-Linear Programming Problems, Ph. D. Dissertation, Department of Mathematics, Indian Institute of Technology, Kanpur (India) 1968).

10 G. HADLEP, Non-linear and Dynamic Programming Reading Mass, 1964 Addisson Wesley.

11 H. P. E ~ ~ N z I , W. KRELLE and W. OETPLI, Non-linear Programming, Blaisdell Publishing Company (Translated by Frank Levin), Massa- chusetts Toronto, London 1966, Waltham.

1 z n T, M~NOASARIAN. Pseudo-Convex Functions, J. SIAM Control, _ _ - _ . ~ Ser. A 5(2), 1965, pp. 281 -290.

Methods, Management Science 12(3), pp. 241 -252 (1965). 13 MARTds BnLa, The Direct Power of Adjacent Vertex Programming

Anschrijt: C. R. BECTOR, Department of Mathematics, Indian Institute of Technology - Kanpur (India)

By 11. RASMUSSEN

Note on the equation y”’ + g y” - A2 y” = 0

1. In t roduc t ion The boundary value problem y”’ + y y” + i‘l y’2 = 0 (1)

y(0) = a , y’(0) = b , y‘ --f 0 as x --f 00 (2) and

where p, a , b are constants and y’ is the derivative of y with respect to x, appears in several problems in fluid mechanics. These problems include the viscous flow due to an infinite disk oscillating in its own plane, BENNEY [l], and the viscous flow between two oscillating disks, SMITH 12). When p = 2 and a = b = 0, (1) and (2) give the similarity solution for a well jet, GLAUERT [3].

In general it is not possible to obtain an explicit solution to (1) and (2); hence it is important to show that a solution does exist. KRAVCHEBEO and YABLONSKII [4] have shown that equation (1) has a t least one solution whose first derivative tends to zero as x + 00 when p satisfies the inequality - 3 5 p 5 1 .

In Section 2 we extend this result by proving the existence of a solution to (1) for all finite negative values of p satis- fying (2) with a 2 0 and b > 0. The analysis in this section is similar to the one given by COPPEL [6] in his treatment of the FALKNER-SKAN equation. A method for obtaining approximate solutions is discussed in Section 3.

2. Exis tence proof Since we are only considering negative values of p, we shall replace p by -A2 in equation (l), and the boundary value problem becomes y”’ + y y” - 1 2 y’2 == 0 (3)

y(0) = a , y‘(0) = b , y’(x) 4 0 as x + 0 0 . (4)

and

In order to prove the existence of a solution to this boundary value problem we need the following result.

Lemma: Let y = y ( x ) be a solution of (3). If y‘ converges to a finite limit as x --f 00, then this limit must be zero unless I2 = 0.

Proof: Integration of (3) by parts between the limits 0 and x gives

X Y Y X ) - Y v ) + y ( x ) y w - y(0) YW = (1 + 12) J ~ ’ W dz.

0 If y‘ + a, then for large x

Hence for large x y”(x) = A2 a2 x + o(x) . Thus y’ can only approach a finite limit if either A2 = 0 or a = 0.

The following existence theorem is now proved. Theorem: The boundary value problem (3) and (4) has at

least one solution for a 2 0 and b > 0 when A2 * 0. Proof: If we let y1 = y,,yz = y’, y3 = y” equation (3)

can be expressed in the equivalent form

Y ; = Yn 2 Y; = Y3 9 y; = Y; - Y1 Y3 * ( 5 ) Since this system is autonomous, we can represent its

solutions by curves or trajectories in the three dimensional phase space (yl , yz , y3). The method for proving the existence of a solution to (3) and (4) is to show that there exists a trajectory in the space (y l , y2, y3) which passes through the point (a , b, -c) and tends to some point on the line

The curve parameter for the curves in the phase space is the independent variable x , and it is known that two solutions which differ only by a change in the position of the origin of x are represented by the same trajectory. By the standard theorems on the solutions of ordinary differential equations one and only one trajectory passes through each regular point of the space. Let

y1> 0, y2 = y3 = 0.

l ( X ) = (Y l (4 t YZ(43 Y3(X)) be a solution of (5) which satisfies the initial condition g(0) = (y,!O), yz (0) , ~ ~ ( 0 ) ) . Then it is known that ~ ( x ) is either defmed for all x 2 0, or if g(x) is defined only for x < xl, then

lim { lYll + lYel + I Y S l } = i 0O * x+xl

It is also known that a trajectory depends continuously on its initial condition.

Let us now consider the trajectory T which passes through the point (a, b, -c) in the phase space. Here a, b, c are constants such that a 2 0, b > 0, c > 0. Motion along T is taken to be in the sense of x increasing starting from the point (a, b, - c ) .

From the equation (5) we see that T passes into the domain D given by the inequalities

In D there is y ; > 0, y ; < 0, y ; > 0. A trajectory in D cannot become infinite a t a finite value of x , since y2 and y3 are bounded and thus y1 is bounded over any finite interval of x. Since the positive y1 axis is a singular line of ( 5 ) , a trajectory cannot cross this line at a finite value of x , and by the lemma it cannot tend to (00,a,O), a some positive constant, as z + 00. Hence T must either tend to the line 0 < y1 < 00, yz = y3 = 0 or leave D through one of the faces yz = 0 and y3 = 0.