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(Ф1,Ф2)convexityRita. Pini a & Chanchal. Singh ba Istituto di Metodi Quantitativi per le Science Economiche e Aziendali , Universitià degliStudi di Milano , Via Sigieri 6, Milano, 20135, Italy E-mail:b Department of Mathematics , St. Lawrence University , Canton, New York, 13617, U.S.A.E-mail:Published online: 20 Mar 2007.

To cite this article: Rita. Pini & Chanchal. Singh (1997) (Ф1,Ф2)convexity, Optimization: A Journal of MathematicalProgramming and Operations Research, 40:2, 103-120, DOI: 10.1080/02331939708844303

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Ol~ t lmlZ( l r l~n , 1997. Val. 40. pp. 103-120 Reprmts a ia~lable d~rectly from the publ~sher Photocop)ing perm~tted by hcense on14

3; 1997 OPA (Overseas Pubhshrrs Assoaat~on) Amsterdam B.V. Pubhshed by The Netherlands under

license b] Gordon and Breach Saence Publishers Printed ~n Indm

RITA PINIL* and CHANCHAL SINGHb.**

dIstituto di Metodi Quantitatici per le Scienze Economiche e Aziendali, Unicersitir degli Studi di Milano, Via Sigieri 6, 20135 Milano, Italy;

bDepartnle~zt of Mathematics, St . Lawrence Unicersity, Canton, New York 1361 7 , V.S.A.

(Re tened 28 Februar) 1996, I n final form 31 J u l j 1996)

Convexity of a function and of a set are generalized. The new class being introduced, include many well known classes as its subclasses. The defining functions involved are required to satisfy certain regularity conditions. Some properties are studied with or with- out differentiability; in the differentiable case, first and second order conditions are stated.

Kej,u.of.ds: Generalized convexity; first order conditions; second order conditions; non- linear programming

Mathematics Subject Classijcation 1991: Primary: 26B25; Secondary: 90C30, 52A01, 52A40

1. INTRODUCTION

Convexity plays an extremely important role in mathematical, natural and social sciences. In an effort to extend existing results based on convexity, there has been an increasing interest during the last 35 years or so in the pursuit of generalizations. Excellent monographs and nu- merous journal articles that have appeared in the literature, testify to the popularity of this research area. In recent years, several definitions extending the concept of convexity of a set and of a function have been introduced, with the purpose of weakening the assumptions to establish some results concerning sufficiency of the Kuhn-Tucker conditions, and

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104 R. P I N AND C. SINGH

some classical duality results, of a mathematical ~rogramming prob- lem. For more recent developments on generalized convexity with reference to duality and optimality conditions an interested reader may consult Pini and Singh [9].

In this paper (@,,@,)-convexity is defined. This a very powerful new principle for characterizing generalized convexity of sets and functions from a unified point of view: the function @, describes a continuous deformation of straight line segments (generalized convex combinations of arguments). and @, determines generalized convex combinations of values. In this way, a large number of well-known and new convexity conditions can be included.

In Section 2. the definition of (a,. @,)-convex functions is given; we show that, for suitable choices of the functions @, and a,. some of the well-known classes of generalized convex functions are special cases of this new class. An example of a (@,, @,)-convex function is also given which is not a member of any of the known classes. We present some properties of nondifferentiable (a,, <D,) - convex functions. In this section we also investigate some properties of the solutions of a mathematical programming problem involving (<Dl. @,)-convex functions; moreover, we state a sensitivity result.

In Section 3, we consider the differentiable case. Here we state a natural necessary condition for differentiable (@,,@,)-convex func- tions: in particular, we provide criteria under which the differentiable and the nondifferentiable conditions are equivalent, extending a result in [ 6 ] Further, we state a second order sufficient condition for (@,, @,)-convexity.

In the sequel, the following conventions for equalities and inequali- ties will be used. If x, y € R k , then

x = j iff x , = xi, i = 1,2 ,..., k

x s y iff s , < y i , i = 1,2 ,..., k

x < j iff x , < y , , i = 1 , 2 ,..., k

x $ J' is the negation of x < y.

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2. (@,, cD2) - CONVEXITY: THE NONDIFFERENTIABLE CASE

Let F be a vector space of real valued functions defined on a set D c R". Assume that a , ,@, are two maps satisfying the following assump- tions:

(D,(x,y,O)=j, @,(x ,x , i )=x , VX,JED, ;.~[0,1] (i)

J, I., f ) 5 max (f (x), f (L.)), VX, !'ED, ;.E [O,1], f EF. (iii)

We will also assume that @, is continuous with respect to 2. We give the following

DEFINITION 2.1 A set D is 0, -convex if @ , ( x , y , i ) ~ D for all x, ED, k[O, 11. The intersection of 0, -convex sets is still cD, -convex. From now on, D will be a 0, -convex set.

DEFINITION 2.2 A function ~ E F is said to be (@,, @,)-convex (con- cave) if

for all X,JED,I,E[O, 11. Iff = (f f2, . . . , f,): D + Rk, fit F, and f i is (a,, @,)-convex (concave) for i = 1,2,. . . ,k, then the vector valued function f is said to be (Q1, 0 2 ) - convex (concave).

DEFINITION^.^ A function f EF is @, -quasiconvex on D if for every x, y ED, % E [0, 11,

f ( @ l k 4 ' 3 4) 5 max (f ( . 4 f (2')). (2.3)

Remarks 2.1 We point out that this definition is independent on the vector or topological structure on D; actually, D could be any set.

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106 R. PIN1 A N D C. SINGH

Remark 2.2 If (Dl,@, satisfy the assumptions (2.1), then every (a , , @,) - convex function is @, - quasiconvex; moreover, the lower level sets (XED: f (.Y) 5 r } , ~ E R , are @, -convex. We give some examples below.

Exanzple 2.1 Let D be a convex subset of Rn, and define Q,(x,y, iL) =

(1 - ;.)J f i.x, @,(x, J, ?,, f ) = (1 - i ) f (J) + if (x); then a convex func- tion on D is (@,. @,) -convex.

E.uample 2.2 If 11 : Rn x Rn -+Rn, D is a preinvex set with respect to 11,

then an q - preinvex function f : D -+ R is (@,, @,)-convex with Q , ( x , ~ , i , ) = ~ + i g ( x , j ) , @ , ( x , j , i , f ) = ( l - j - ) f ( ~ ) + i f ( x ) (see [?I).

Example 2.3 Let D G M, where A4 is an Euclidean manifold and D is geodesically convex. A geodesically convex function on D is (Q,, @,)-con- vex, with @,(x,y,i) = ;*,,,(i), @, (x, 4; I., f ) = (1 - ?,) f(y) + iif(x), where y , , is the geodesic from J to x (see [lo]).

Example 2.4 Let D be a convex subset of Rn, @,(x, J, i) = (1 - 1") ). +i.s,@,(~,j~,i,f)=(l-b~(x,j,R))f(j)+b~(x,y,/~)f(x). Then every B- vex function on D (with respect to b,) is (@,.@,)-convex (see [2], [3]).

Example 2.5 Let H be a one-to-one mapping from D r Rn to Rn, and @ a strictly monotone increasing function mapping a subset C of R onto R (see [4]). A function f :D 4 C is called (H, @)- convex if, for any x, JED and ;.€LO, 11

provided that range f' c dom @. Here

Choosing @,(x, J, 2) = M,([x, !.I, i), @,(x, Y, k f ) = mo[( f (x), f'(y)), il, we see that an (H, @)-convex function is a particular (Q,, @,)-con- vex function.

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(Q1, (D2)- CONVEXITY 107

Remark 2.3 The functions @,, @, of the Examples 1-5 satisfy (2.1).

Example 2.6 A function f : R" + R' = R u { - x) is called G-convex on the convex set D if, for every x , y ~ D , x # y, k ( 0 , I),

where G(r,, r,, 4%): R' x R1 x R+ x R+ -+R1 is continuous and nondecreas- ing in (r,,r,) and 11. I1 is an arbitrary norm on R" (see 151). If we set @1(x,y,R)=(1-A)y+3u~,andcD,(x,y,lL,f)=G(f(x),f(~~),~ x - y , 4,weget that a G-convex function is an example of (@,, @,)-convex function. We give now an example of (@,,@,)-convex function, that is not in any of the classes described above.

Example 2.7 Let D c R be the set D = (- x, - l ) u ( l , + oc), and f : D -, R be the function defined as follows

Define the functions 0,: D x D x [0,1] and a,: D x D x [0,1] x F as follows:

(I - i ) y + ;.s xy > 0 0, (x, y, 2,) = L x y < 0

The vector space F is, for instance, the set of the functions defined on D. The functions 0, and 0, satisfy conditions (2.1) (i)-(iii); the set D is @, -convex, since @,(x, J, ED for every x, y ED. R E [0, I]; moreover, it can be easily checked that f is (a,, @,)-convex on D. This function cannot belong to one of the classes described above, for one of the following reasons:

i) the set D is not connected (it cannot be B-vex, geodesically convex, G - convex);

ii) there are no one-to-one correspondences H from D to R such that @l(~,jj,>w) = H-'((1 - ~ ) H ( J ) + iH (x ) ) (indeed, if xy < 0, we have,

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108 R. PIN1 AND C. SINGH

for any H: H (@,(x, y, i)) = H(J~) for every i.e[O, 11, while (1 - i ) H (y) + iH(.u) describes a segment whenever i. varies In [0, 1);

iii) the function @, is much broader that the convex combinat~on of the values f(y) and f (x), and the function f doesn't satisfy a stronger restriction (it doesn't belong, for instance, to the class of g -pre~nvex functions, for any choice of 11; indeed, since f is differentiable on D. 17 - preinvexity implies q - invexity. However, f is not 17-invex since eaery stationary point off is not a global minimum point, as it is the case for differentiable invex functions).

Here are some properties of the class of (@,.@,)-convex functions. under suitable assumptions on @, and,'or @,.

Ohsercation ( a ) . Assume that 0, is superlinear with respect to f E F , that is @, is superadditive and positively homogeneous. Then the class of (@,,@,)-convex functions is a convex cone. (Indeed. if f, g are (a,, @,)-convex, and x > 0,

(2 f)(@l(s,j3,;.)) = x(f (@,(x,j, 1")) 5 x@2(x,y.1., f ) = @,(x,y, 2, x f)).

Refnarks 2.4 In the examples 1-4, 0, is linear with respect to f . In example 5. 0, 1s superlinear if @ is. Obsercatiorz ( h ) . Assume that f : D -+R is (@,, @,)-convex. g: R -t R 1s increasing and (@,, @,)-convex, and g c f E F . Then, if

the function g sf is (@,, @,)-convex. (Indeed. we have that

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(Q1, 0,)- CONVEXITY 109

Next we consider a scalar valued optimization problem, that can be expressed as

(P) minf (x) s.t. y(x) 5 0,

where f : D -r R, y: D -, Rk. Denote the feasible set by Do, where

Then the following holds:

PROPOSITION 2.1 Suppose that (i) g = (y,, g2,. . . , yk) is ((Dl, 02) - conuex (see Dejinition (2.2));

( i i ) f i s (@,, a,) - corzuex. Then, rlze set of solutions of problem ( P I is 0, -convex.

Proof The feasible set Do is 0, -convex; indeed, if x,, x2€DO, from (i) and (2.1) (iii) we have

for any i = 1 ,2 ,..., k. Next, let min ,,,, f (x) be attained at x: and xi . By the hypothesis (ii) and (2.1) (iii),

But f (x:) = f (xi) = minXEDo f (x), hence f (@,(x';', x:, i ) ) = f (x?), which completes the proof. Let us give the following

DEFINITION 2.4 Let x ~ E D . We say that f is (@,,@,)-strictly con- vex (concave) at xo if

we say that f is weakly (@,, @,)-strictly convex (concave) at so if (2.4) holds for some Z ~ ( 0 , l ) .

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110 R. PIN1 AND C. SINGH

If (2.4) is satisfied a t any X,ED, then f is (0,. @,)-strictly convex (concave) on D. Then we have the following

PROPOSITIO~~ 2.2 Suppose that D, is a 0, -convex yet, and (i) J is (O,, Q,) -strictly conces a t ED,;

(ii) xo is a solution of prohlenz ( P ) . Tken x, is the unique solution of (P).

Proof Let x* be another solution of (P), x* # s o . Then, for all i ~ ( O . 1 )

f (@,(x*, x,, 2) ) < @,(x*, xO, L, f ) 5 max { f (x*), f (x,)} = f (x,).

which contradicts hypothesis (ii). In case of (@,,@,)-concave functions (see Definition 2.2), we have the following

THEOREM 2.1 Suppose that (i) f is (@,,@,)-strictly concore in D;

7

(ii) Q x o ~ i n t ( D , ) 3 s , j ~ D O , ~ # j , ~ ~ ( 0 , 1 ] s u c l z that @,(.u,j,x)=x,; (iii) Do is Q, - concex, ( i t) O,(X, j., jL, f ) 2 min (f (x), f (y)) for ecerj3 x, ~ S E D,, 0 5 2 5 1. Tken there are no interior points of D, which are solutio~z of (P), i.e. if Y, is a sol~~tion o j (P), then x, is a boundary point of Do.

Proof If the solution set of (P) is empty, or int (Do) is empty, there is nothing to prove. Assume that x, is a solution of (P), and x , ~ int (D,). Then by hypothesis (ii) there exist x , y ~ D , , x # J.. and ~ E ( o , 11 such that x, = @,(x,j-. 7)). Now. by (i), we have that

f (x,) = f (O1(x, y, 1)) > @,(x, 2., 7, f ) 2 min { f (x). f ().)I 2 f (x,).

This contradiction leads us to conclude that x, is not a solution of (P). Let %,(x,) denote a neighborhood of xo of radius 6.

THEOREM 2.2 Sinppose that (i) f is (Dl, 0,) - strictlj convex;

( i i ) x,ED, is a local minimum of (P) , (iii) 6, > 0, a~zd YXED,, 3 7 ~ ( 0 , I] suck that @,(x,, x , I)E ladl (Y,); (ir) Do is 0, - convex.

Then x, is a strict global minimum of (P) .

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((Dl, 0,)- CONVEXITY 111

Proof By hypothesis (iv), for every XED,, and for every k [ O , 11, @,(x,, x, ),)ED,. Since x , is a local minimum of (P), there exists %&,) such that for every x E %&,) n Do, f (x,) 5 f (x). Now, let X E Do, x # x,. Then, by hypothesis (ii) and (iii), with 6, = 6, we have that f (x,) 5 f (@,(x,, x, I)) for some ~ E ( o , 11. Therefore, using (i) and (2.1) (iii), we have

Obviously, max{f (x,), f ( x ) ) # f (x,) since f (x,) # f (x,). Therefore, f (x,) < f (x) . Since x is an arbitrary member of D,, the proof is complete. The following results can be derived along the lines of Theorem 2.2. We leave the details for the reader.

THEOREM 2.3 Suppose that ( i ) f is (@,, 0,) - concex;

( i i ) xOeDO is a strict local minimum of ( P ) ; i i 6 > 0 and V ~ E D , , ~ T E ( O , l ] such that @,(x,, x , ;)E

~~2c,,(~,)\C~o), . ( i c ) D, is @, -convex. ?hen x, is a strict global minimum of ( P ) .

THEOREM 2.4 Suppose that ( i ) f is f a , , 0,) - comex;

( i i ) xOeDO is a local minimum of ( P ) ; ( i i i) V 6 , > 0 , and v ~ E D , , ~ ? E ( o , I ] such that @ , ( X , , X , ~ ) E q 8 , ( x o ) ; ( in ) Do is 0, -concex; ( c ) QZ(x0, x, i, f ) < max { f (x), f (x , ) ) for euery X E D , with f ( x ) f

f (x,), and for all &(O, I ). Then x , is a global miniinum of ( P ) . We now study a regularity property of the product of (a,, @,)-con- vex functions (i = 2,3). First of all, we state the following

LEMMA 2.1 Suppose that f , g are real calued functions defined on D, and satisfying the conditions

Then for ecery x , y ~ D , either

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112 R PIN1 AND C. SINGH

Proof Since, by (ii).

( f ( s ) - f ( j ~ ) ) ( g ( s ) - y ( ~ ) ) 2 0 ' d s . y ~ D ,

which further implies (in view of (i)) that either

PROPOSITION 2.3 Suppose that ( i ) f , y are ~zonnegntice f~mctions dejned on D and sutisfiiny the inequalit].

( i i ) f is (a,, 0,) -conces, g is (@,, D,) -concex. The11 f 8 is 0, - quasiconces.

Proof For any X , ED and ;.€LO, 11,

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(al; Q2) - CONVEXITY 113

Now, max{f ( x ) , f ( y ) ) . maxi&), g(y)), in view of lemma, is less than or equal to max{f ( x ) g(x), f (y)g(y)) ; hence, it follows that

Therefore f g is 0, - quasiconvex. Let us consider the following family of problems:

min f ( x ) s.t. g(x) j E ,

where f : Rn -, R, g: Rn -+ Rk, & € R k . Denote by f *(r) the function

f *: Rk -+ R, f + ( E ) = inf { f ( x ) : g(x) 5 E )

(see [5l) . Assume that f is (a,, a,)- convex, where @,(x,, x,, j", f ) = @,(f (x,) , f ( x J , 2). and the vector function g is (a,, %,)-convex, where

and @,(a,, a,, i ) is nondecreasing in (a,, a,) with respect to the com- ponentwise order (if a: j b h n d ~ $ 5 b$, Y i,j, then @,(a,, a,, i) 5 0, (b,, b,, i), for every k [ O , I]). We have the following

THEOREM 2.5 The function f * is ( 0 , , @4)-convex on R~ (i.e.

f *(@3 (81. E,, 2 ) ) 5 @4( f * f & l ) > f * ( & , h i ) ) .

Proof Notice that if g(x,) 5 E,, g(x2) 5 E,, then

in particular,

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114 R. PIN1 AND C. SINGH

Hence

3. ( a 1 : a,) - CONVEXITY: THE DIFFERENTIABLE CASE

Let us assume that @,, @, have right partial derivative with respect to 2, at i = 0. for all x , y ~ D , for all f EF. If we consider a differentiable (a,, @,)-convex function f , defined on D G Rn, taking into account (2.1), for x , ED and k ( 0 : 11 we get that

and, taking the limit of both sides for i. + 0 + (and since @,(x, y, 0 ) = y), we have

We therefore have the following

PROPOSITION 3.1 Assume that @, and hace right partial dericatice ~ ~ i t l ~ respect to j, at ;. = 0. Then a dljferentiable (@I ,@2) -co~zrex

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function f satisfies the inequality

for eGery x, yeD, where

Remarks 3.1 The same result holds in a more general setting, where D is a subset of a Riemannian manifold, and the r.h.s. of (3.1) is defined as df,.(4, (x, y)). It is easy to verify that a Dl -quasiconvex function g satisfies the condition

for every x, yeD.

DEFINITION 3.1 Let $: D x D + D. We say that $ is skew-symmetric on D x D if $(x, y) = - $(y,x) for every (x, y)eD x D.

COROLLARY 3.1 (To Proposition 3.1) Suppose that f is dzflerentiable and (Dl, a,) -convex; if 4,, 4, are related to a,, 0, as in (3.2), and skew-symmetric for any (x, y)eD x D, then Vf is 4, -monotone on D, i.e.

Proof By (3.1), we have that

and the conclusion follows from the skew-symmetry. The local condition expressed by (3.1) of Proposition 3.1 is usually not sufficient to guarantee the (a , , @,)-convexity o f f , unless we specify some more restrictive and global properties of the functions Qi and $i. Indeed, consider @,(x, y, i ) = y + iq(x, y), 02(x, y, 2, f ) = (1 - i) f (y) + >fix). In [8], Mohan and Neogy provided a counterexample, showing

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116 R. PIN1 AND C. SINGH

that the condition

does not imply in general that

f (J' +il7(?i,).)) s (1 - i.) f (4') +if (x), ViL€[O, 11.

We will assume that the function f is differentiable on D. The follow- ing results relate the necessary condition for a differentiable (a,, @,)-convex function f , and the definition of (@,, @,)-convex- ity. In the first result, we assume that a "regularity condition" is satisfied by a,, whereas @, is the usual r.h.s. of the definition of convexity, providing a slight extension of the ordinary convex case.

PROPOSITION 3.2 Assurne that (Dl is differentiable u.itlz respect to 7- iiz

[O, I ] ; if the follo\.tirzy conditions are satisfied (i) CD,(x,y,O) =4 ' , a l (x ,y ,1) =s;

for ecerj' X, !.ED, 1 1 , t, ~ E / O , 11, then a f~inction f sutisf~.ing (3.1 ) is (a,, @,) -corzces.

Proof By (3.1) and Condition (2.1) (iii), it follows that f (s) - f (1') 2 V, f (y) +,(s, J), and for every x , ~ E D , we get that the function y(s) = f (@,(s, J', s)) is convex; indeed

= y'(u) ( t - u).

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It follows that g is convex. Hence, g(A) 5 (1 - i)g(O) + i,g(l). Now by hypothesis (i) and (ii) we get that

f (@l(x,y,3")) 5 (1 - A) f (y) + i.f(x) = @,(x, y, i,, f )

(see [8], where a special case of the Proposition 3.2 is proved). More generally, the following result relating (3.1) and (a,, @,)-con- vexity holds.

THEOREM 3.1 Assume that f is a function dzjj5erentinble on D, ~chere D is a @, -comes subset of Rn. Let 4,(i = 1 , 2 ) be the function asso- ciated with 0, as in (3.2). Assume that there exists a function H : R x R x [0, I] + R, H = H(s, t, i), and the following conditions are satisfied.

( i ) H(4,(x, Dl(x, y, lL),f ), $,(J, J', jV),f 1, ;&) 5 @2( .% .Y, ;-,f) - f (O1(x, J*, i));

(ii) H is nondecreasing in (s, t), for ecerj i j ixed ( i f s , 5 s,, t , 5 t,, lte lzave that H(s,, t,, j.) 5 H (s,, t,, i));

(iii) H(Val f (Q1(xj J,, ;.)I 4,(x, @l(x, J, i ) 1 2 Valf f (x, J, 7 . ) ) 41 (y, @,(x,y,iL)),i) = 0 for euery ;.E[O, 11, f EF, x , y ~ D ;

(ic) +,(x,z, f ) 2 V, f (z) 4,(x, z ) , 'd x, z ED.

Then f is (@,, a,) - convex on D.

Proof From (iv), with z = @,(x, y, i,), we have that

Let s = 4,(x, @,(x, J, i.), f ), t = 4,(y, @,(x, y, i), f ); from (ii) and (iii), we get that

Finally, by (i), we have that

that is f is (@,,@,)-convex.

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118 R. PIN1 AND C. SINGH

Notice that Condition C in [6] is a particular case of Theorem 3.1, where @,(x, y, i ) = y + i q ( x , y). 0 2 ( x , y, i , f ) = (1 - i ) f ( y ) f i f ( x ) , and H(s, t , i,) = i s + ( 1 - i.)t.

PROPOSITION 3.3 If f : D -+ R is differentiable, and 0, sarisjes as- sumptions ( i i ) and (i i i) in Theorem 3.1, then f is @, -quasiconcex if and only if (3.3) holds.

Proof Similar to the proof given in [6]. Under suitable assumptions on @,, a differentiable (@l,@2)-convex function turns out to be invex, and we can guarantee that a stationary point is a global minimum point. Here is a sufficient condition. Assume that a, satisfies the inequality

for all x, ye D, ;.E [0, I], f E F, and for some function c = c ( x , j3, I., f ): D x D x [ O , l ] x F - t R , w i t h c ( x . ~ ~ , O , f ) = l , ( ~ c ~ S ~ ~ ) ( x , ~ ~ , > ~ , f ) ( , = , = O . Then we have the following

PROPOSITION 3.4 Let f be a differerztiahle (@,, c;D2)-cont.ex filnc- tion, \vhere 0, and 0, are diferentiable with respect to I. at >, = 0, for ecerjq X , J , E D . Assunze that condition (3 .4) holds. Then f is imex with respect to q (x , y ) = $ , ( x , y) . I n particular, ecery stationary point o f f is a global rifininzum.

Proof From (3.4), we have that

Adding and subtracting c(s , y, 0, f ) @,(x, y, i , f ) to the right hand side of the above inequality and then dividing both sides by i and taking the limit i, + O', we get

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(Q1, D2)- CONVEXITY 119

Since, by Proposition 3.1, 4,(x, y, f 1 2 V , f (y)dl(x, J), we have that

This proves that f is d,(x,y)-invex and hence every stationary point is a global minimum point. Assume now that f : D -+ R and @, : D x D x [O,1] -t D satisfy the as- sumptions

6) f gC2(D); (ii) @,(x, y,.kC2(C0, 11). Then we have the following sufficient condition for (@,,@,)-convex- ity:

PROPOSITION 3.5 In the assumptions above, f is (@,, a,) -convex for every @,(x, J, t, f ) = & g(x, y, s) ds + f (y), where g is arzy solution of the dijjerential inequality

(H,, denotes the Hessian of the function Dl, and (B@l/2t)T the trans- pose of 8Dl/2t).

Proof Consider, for every x, y E D,

where @,(x, y, t, f ) = S,g(x, y , s)ds + f (y), and g satisfies (3.5). We prove that h(t) 5 0 for every te[O, 11. We have that

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120 R. PIN1 AND C. SINGH

Therefore, h(t) 5 0 for every t~ [0, 11. and f (@,(x, y, t)) 5 a,(?;, y, t. f ) for every x, J 'E D, t~ [O. 11. The authors have developed duality theory and optimality conditions based on (@,,@,)-convexity and are about to submit for possible publication.

References

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[Z] Bector C. R. and Singh, C. (1991). B-Vex Functions, J. 0. 7: A.. 71, 237-253. [3] Bector, C. R.. Singh, C. and Suneja. S. K. (1993). Generalization of Preinvex and

B-Vex Functions. J. 0. 7: 4., 76. 577-587. [4] Ben-Tal, A. (1977). On Generalized Means and Generalized Convex Functions,

J . 0. 7: A,, 21, 1-13. [5] Fiacco, A. V. and Kyparisis. J. (19871. Generalized Convexity and Concavity of the

Optimal Value Function in Nonlinear Programming, .2lath. Progr.. 39. 285-304. [6] Mohan S. R. and Neogy. S. ti. (1995). Note: On I n ~ e x Sets and Preinr-ex Func-

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ation, J. I l a t h . Anal. Appl.. 136, 29-38. [8] Pini. R. (1994). Convexity along Curves and Inr-exit);. Optimization. 29. 301-309. [9] Pini. R. and Singh. C. (1996). A Survey of Recent (1985-1995). Advances in Gene-

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[lo] Rapcsak. T. (1987). Geodesic Convexity in Nonlinear Optimization. J. 0. Z A,, 69. 69-183.

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