Generalized Convex Functions

Transcription

1 Chater Generalized Convex Functions Convexity is one of the most frequently used hyotheses in otimization theory. It is usually introduced to give global validity to roositions otherwise only locally true, for instance, a local minimum is also a global minimum for a convex function. Moreover, convexity is also used to obtain sufficiency for conditions that are only necessary, as with the classical Fermat theorem or with Kuhn-Tucker conditions in nonlinear rogramming. In microeconomics, convexity lays a fundamental role in general equilibrium theory in duality theory. For more alications historical reference, see, Arrow Intriligator (1981), Guerraggio Molho (004), Islam Craven (005). The convexity of sets the convexity concavity of functions have been the object of many studies during the ast one hundred years. Early contributions to convex analysis were made by Holder (1889), Jensen (1906), Minkowski (1910, 1911). The imortance of convex functions is well known in otimization roblems. Convex functions come u in many mathematical models used in economics, engineering, etc. More often, convexity does not aear as a natural roerty of the various functions domain encountered in such models. The roerty of convexity is invariant with resect to certain oerations transformations. However, for many roblems encountered in economics engineering the notion of convexity does no longer suffice. Hence, it is necessary to extend the notion of convexity to the notions of seudo-convexity, quasi-convexity, etc. We should mention the early work by de de Finetti (1949), Fenchel (1953), Arrow Enthoven (1961), Mangasarian (1965), Ponstein (1967), Karamardian (1967). In the recent years, several extensions have been considered for the classical convexity. A significant generalization of convex functions is that of invex functions introduced by Hanson (1981). Hanson s initial result insired a great deal of subsequent work which has greatly exed the role alications of invexity in nonlinear otimization other branches of ure alied sciences. In this chater, we shall discuss about various concets of generalized convex functions introduced in the literature in last thirty years for the urose of weakening the limitations of convexity in mathematical rogramming. Hanson (1981) introduced the concet of invexity as a generalization of convexity for scalar constrained otimization roblems, he showed that weak duality sufficiency of S.K. Mishra et al., Generalized Convexity Vector Otimization, Nonconvex Otimization Its Alications. c Sringer-Verlag Berlin Heidelberg 009 7

2 8 Generalized Convex Functions the Kuhn-Tucker otimality conditions hold when invexity is required instead of the usual requirement of convexity of the functions involved in the roblem..1 Convex Generalized Convex Functions Definition.1.1. A subset X of R n is convex if for every x 1,x X 0 < λ < 1, we have λ x 1 +(1 λ)x X. Definition.1.. A function f : X R defined on a convex subset X of R n is convex if for any x 1,x X 0 < λ < 1, we have f (λ x 1 +(1 λ)x ) λ f (x 1 )+(1 λ) f (x ). If we have strict inequality for all x 1 x in the above definition, the function is said to be strictly convex. Historically the first tye of generalized convex function was considered by de Finetti (1949) who first introduced the quasiconvex functions (a name given by Fenchel (1953)) after 6 years. Definition.1.3. A function f : X R is quasiconvex on X if f (x) f (y) f (λ x +(1 λ)y) f (y), x, y X, λ [0, 1] or, equivalently, in non-euclidean form f (λ x +(1 λ)y) max{ f (x), f (y)}, x, y X, λ [0, 1]. For further study characterization of quasiconvex functions, one can see Giorgi et al. (004). In the differentiable case, we have the following definition given in Avriel et al. (1988): Definition.1.4. A function f : X R is said to be quasiconvex on X if f (x) f (y) (x y) f (y) 0, x, y X. An imortant roerty of a differentiable convex function is that any stationary oint is also a global minimum oint; however, this useful roerty is not restricted to differentiable convex functions only. The family of seudoconvex functions introduced by Mangasarian (1965) under the name of semiconvex functions by Tuy (1964), strictly includes the family of differentiable convex functions has the above mentioned roerty as well.

3 .1 Convex Generalized Convex Functions 9 Definition.1.5. Let f : X R be differentiable on the oen set X R n ;then f is seudoconvex on X if: f (x) < f (y) (x y) f (y) < 0, x, y X or equivalently if (x y) f (y) 0 f (x) f (y), x, y X. From this definition it aears obvious that, if f is seudoconvex f (y)=0, then y is a global minimum of f over X. Pseudoconvexity lays a key role in obtaining sufficient otimality conditions for a nonlinear rogramming roblem as, if a differentiable objective function can be shown or assumed to be seudoconvex, then the usual first-order stationary conditions are able to roduce a global minimum. The function f : X R is called seudoconcave if f is seudoconvex. Functions that are both seudoconvex seudoconcave are called seudolinear. Pseudolinear functions are articularly imortant in certain otimization roblems, both in scalar vector cases; see Chew Choo (1984), Komlosi (1993), Racsak (1991), Kaul et al. (1993), Mishra (1995). The following result due to Chew Choo (1984) characterizes the class of seudolinear functions. Theorem.1.1. Let f : X R, where X R n is an oen convex set. Then the following statements are equivalent: (i) f is seudolinear. (ii) For any x,y X, it is (x y) f (y)=0 if only if f (x)= f (y). (iii) There exists a function : X X R + such that f (x)= f (y)+(x, y) (x y) f (y). The class of seudolinear functions includes many classes of functions useful for alications, e.g., the class of linear fractional functions (see, e.g., Chew Choo (1984)). An examle of a seudolinear function is given by f (x)=x + x 3, x R. More generally, Kortanek Evans (1967) observed that if f is seudolinear on the convex set X R n, then the function F = f (x)+[f (x)] 3 is also seudolinear on X. For characterization of the solution set of a seudolinear rogram, one can see Jeyakumar Yang (1995). Ponstein (1967) introduced the concet of strictly seudoconvex functions for differentiable functions. Definition.1.6. A function f : X R, differentiable on the oen set X R n,is strictly seudoconvex on X if f (x) f (y) (x y) f (y) < 0, x, y X, x y,

4 10 Generalized Convex Functions or equivalently if (x y) f (y) 0 f (x) > f (y), x, y X, x y. The comarison of the definitions of seudoconvexity strict seudoconvexity shows that strict seudoconvexity imlies seudoconvexity. Ponstein (1967) showed that seudoconvexity lus strict quasiconvexity imlies strict seudoconvexity that strict seudoconvexity imlies strict quasiconvexity. In a minimization roblem, if the strict seudoconvexity of the objective function can be shown or assumed, then the solution to the first-order otimality conditions is a unique global minimum. Many other characterization of strict seudoconvex functions are given by Diewert et al. (1981). Convex functions lay an imortant role in otimization theory. The otimization roblem: minimize f (x) for x X R n, subject to g(x) 0, is called a convex rogram if the functions involved are convex on some subset X of R n. Convex rograms have many useful roerties: 1. The set of all feasible solutions is convex.. Any local minimum is a global minimum. 3. The Karush Kuhn Tucker otimality conditions are sufficient for a minimum. 4. Duality relations hold between the roblem its dual. 5. A minimum is unique if the objective function is strictly convex. However, for many roblems encountered in economics engineering the notion of convexity does no longer suffice. To meet this dem the convexity requirement to rove sufficient otimality conditions for a differentiable mathematical rogramming roblem, the notion of invexity was introduced by Hanson (1981) by substituting the linear term (x y), aearing in the definition of differentiable convex, seudoconvex quasiconvex functions, with an arbitrary vector-valued function.. Invex Generalized Invex Functions Definition..1. A function f : X R, X oen subset of R n,issaidtobeinvexon X with resect to η if there exists vector-valued function η : X X R n such that f (x) f (y) η T (x,y) f (y), x, y X. The name invex was given by Craven (1981) sts for invariant convex. Similarly f is said to be seudoinvex on X with resect to η if there exists vectorvalued function η : X X R n such that η T (x,y) f (y) 0 f (x) f (y), x, y X.

5 . Invex Generalized Invex Functions 11 The function f : X R, X oen subset of R n,issaidtobequasiinvex on X with resect to η if there exists vector-valued function η : X X R n such that f (x) f (y) η T (x,y) f (y) 0, x, y X. Craven (1981) gave necessary sufficient conditions for function f to be invex assuming that the functions f η are twice continuously differentiable. Ben-Israel Mond (1986) Kaul Kaur (1985) also studied some relationshis among the various classes of (generalized) invex functions (generalized) convex functions. Let us list their results for the sake comletion: (1) A differentiable convex function is also invex, but not conversely, see examle, Kaul Kaur (1985). () A differentiable seudo-convex function is also seudo-invex, but not conversely, see examle, Kaul Kaur (1985). (3) A differentiable quasi-convex function is also quasi-invex, but not conversely, see examle, Kaul Kaur (1985). (4) Any invex function is also seudo-invex for the same function η (x, x), but not conversely, see examle, Kaul Kaur (1985). (5) Any seudo-invex function is also quasi-invex, but not conversely. Further insights on these relationshis can be deduced by means of the following characterizations of invex functions: Theorem..1. (Ben-Israel Mond (1986)) Let f : X R be differentiable on the oen set X R n ; then f is invex if only if every stationary oint of f is a global minimum of f over X. It is adequate, in order to aly invexity to the study of otimality duality conditions, to know that a function is invex without identifying an aroriate function η (x, x). However, Theorem..1 allows us to find a function η (x, x),when f(x) is known to be invex; viz. [ f (x) f ( x)] f ( x), if f ( x) 0 η (x, x)= f ( x) f ( x) 0, if f ( x)=0. Remark..1. If we consider an invex function f on set X 0 X, with X 0 not oen, it is not true that any local minimum of f on X 0 is also a global minimum. Let us consider the following examle. Examle..1. Let f (x,y) =y ( x 1 ) X0 = {(x,y) : (x,y) R,x 1/, y 1}. Every stationary oint of f on X 0 is a global minimum of f on X 0, therefore f is invex on X 0. The oint ( 1/,1) is a local minimum oint of f on X 0, with f ( 1/,1)=9/16, but the global minimum is f (1,y)= f ( 1,y)=0. In order to consider some tye of invexity for nondifferentiable functions, Ben- Israel Mond (1986) Weir Mond (1988) introduced the following function:

6 1 Generalized Convex Functions Definition... A function f : X R is said to be re-invex on X if there exists a vector function η : X X R n such that (y + λη(x, y)) X, λ [0, 1], x, y X f (y + λη(x, y)) λ f (x)+(1 λ) f (y), λ [0, 1], x, y X. Weir Mond (1988a) gave the following examle of a re-invex function which is not convex. Examle... f (x)= x, x R. Then f is a re-invex function with η given as follows: x y, if y 0 x 0 x y, if y 0 x 0 η (x, y)= y x, if y > 0 x < 0 y x, if y < 0 x > 0. As for convex functions, any local minimum of a re-invex function is a global minimum nonnegative linear combinations of re-invex functions are re-invex. Pre-invex functions were utilized by Weir Mond (1988a) to establish roer efficiency results in multile objective otimization roblems..3 Tye I Related Functions Subsequently, Hanson Mond (198) introduced two new classes of functions which are not only sufficient but are also necessary for otimality in rimal dual roblems, resectively. Let P = {x : x X,g(x) 0} D = {x : (x,y) Y }, where Y = {(x,y) : x X,y R m, x f (x)+y T x g(x)=0;y 0}. Hanson Mond (198) defined: Definition.3.1. f (x) g(x) as Tye I objective constraint functions, resectively, with resect to η (x) at x if there exists an n-dimensional vector function η (x) defined for all x P such that f (x) f ( x) [ x f ( x)] T η (x, x) g( x) [ x g( x)] T η (x, x), the objective constraint functions f (x) g(x) are calledstrictly Tye I if we have strict inequalities in the above definition.

7 .3 Tye I Related Functions 13 Definition.3.. f (x) g(x) as Tye II objective constraint functions, resectively, with resect to η (x) at x if there exists an n-dimensional vector function η (x) defined for all x P such that f ( x) f (x) [ x f (x)] T η (x, x) g(x) [ x g(x)] T η (x, x). the objective constraint functions f (x) g(x) are called strictly Tye II if we have strict inequalities in the above definition. Rueda Hanson (1988) established the following relations: 1. If f (x) g(x) are convex objective constraint functions, resectively, then f (x) g(x) are Tye I, but the converse is not necessarily true, as can be seen from the following examle. Examle.3.1. The functions f : ( 0, π ) ( ) R g : 0, π R defined by f ( (x)= ) x + sinx g(x) = sinx are Tye I functions with resect to η(x) = 3 ( ) sinx 1 at x = π/6, but f (x) g(x) are not convex with resect to the same η(x)=( 3 ) (sinx 1 ) as can be seen by taking x = π/4 x = π/6.. If f (x) g(x) are convex objective constraint functions, resectively, then f (x) g(x) are Tye II, but the converse is not necessarily true, as can be seen from the following examle. Examle.3.. The functions f : ( 0, π ) ( ) R g : 0, π R defined by f (x)= x + sinx g(x)= sinx are Tye II functions with resect to η (x)= ( 1 sinx) cosx at x = π / 6, but f (x)g(x) are not convex with resect to the same η (x)= ( 1 sinx) cosx at x = π / If f (x) g(x) are strictly convex objective constraint functions, resectively, then f (x) g(x) are strictly Tye I, but the converse is not necessarily true, as can be seen from the following examle. Examle.3.3. The functions f : ( 0, π ) ( ) R g : 0, π R defined by f (x)= x + ( cosx g(x)= cosx are strictly Tye I functions with resect to η (x)= 1 )cosx at x = π / 4, but f (x)g(x) are not strictly convex with resect to ( the same η (x)=1 )cosx. 4. If f (x) g(x) are strictly convex objective constraint functions, resectively, then f (x) g(x) are strictly Tye II, but the converse is not necessarily true, as can be seen from the following examle.

8 14 Generalized Convex Functions Examle.3.4. The functions f : ( 0, π ) ( ) R g : 0, π R defined by f (x)= x + cosx g(x)= cosx are strictly Tye II functions with resect to η (x)= ( cosx ) sinx at x = π / ( 4, but f (x) g(x) are not strictly convex with resect to the same η (x)=1 )cosx. Rueda Hanson (1988) defined: Definition.3.3. f (x) g(x) as seudo-tye I objective constraint functions, resectively, with resect to η (x) at x if there exists an n-dimensional vector function η (x) defined for all x P such that [ x f (x)] T η (x, x) 0 f ( x) f (x) 0 [ x g(x)] T η (x, x) 0 g(x) 0. Definition.3.4. f (x) g(x) as quasi-tye I objective constraint functions, resectively, with resect to η (x) at x if there exists an n-dimensional vector function η (x) defined for all x P such that f (x) f ( x) 0 [ x f ( x)] T η (x, x) 0. g(x) 0 [ x g(x)] T η (x, x) 0. Pseudo-Tye II quasi-tye II objective constraint functions are defined similarly. It was shown by Rueda Hanson (1988) that: 1. Tye I objective constraint functions seudo-tye I objective constraint functions, but the converse is not necessarily true, as can be seen from the following examle. Examle.3.5. The functions f : ( π, π ) ( R g : π, π ) R defined by f (x) = cos ( x g(x) = cosx are seudo-tye I functions with resect to ) η (x) = 1 / + cosx at x = π / 4, but f (x) g(x) are not Tye I with ( ) resect to the same η (x)= 1 / + cosx as can be seen by taking x = 0.. Tye II objective constraint functions seudo-tye II objective constraint functions, but the converse is not necessarily true, as can be seen from the following. Examle.3.6. The functions f : ( π, π ) ( R g : π, π ) R defined by f (x) = cos ( x g(x) = cosx are seudo-tye II functions with resect to η (x)=sin x cosx / ) at x = π / 4, but f (x) g(x) are not Tye II with ( resect to the same η (x)=sinx cosx / ) as can be seen by taking x = π / 3.

9 .3 Tye I Related Functions Tye I objective constraint functions quasi-tye I objective constraint functions, but the converse is not necessarily true, as can be seen from the following examle. Examle.3.7. The functions f : (0,π) R g : (0,π) R defined by f (x)= sin 3 x g(x) = cosx are quasi-tye I functions with resect to η (x)= 1 at x = π /, but f (x) g(x) are not Tye I with resect to the same η (x)= 1 as can be seen by taking x = π / Tye II objective constraint functions quasi-tye II objective constraint functions, but the converse is not necessarily true, as can be seen from the following examle. Examle.3.8. The functions f : (0, ) R g : (0, ) R defined by f (x)= 1 x g(x) =1 x are quasi-tye II functions with resect to η (x) =1 x at x = 1, but f (x) g(x) are not Tye II with resect to the same η (x)=1 x as can be seen by taking x =. 5. Strictly Tye I objective constraint functions Tye I objective constraint functions, but the converse is not necessarily true, as can be seen from the following examle. Examle.3.9. The functions f : ( π, π ) R g : ( π, π ) R defined by f (x)= sinx g(x)= cosx are Tye I functions with resect to η (x)=sinx at x = 0, but f (x) g(x) are not strictly Tye I with resect to the same η (x)=sinx at x = Strictly Tye II objective constraint functions Tye II objective constraint functions, but the converse is not necessarily true, as can be seen from the following examle. Examle The functions f : ( 0, π ) R g : ( 0, π ) R defined by f (x)= sinx g(x)= e x are Tye II functions with resect to η (x)=1at x = 0, but f (x)g(x) are not strictly Tye I with resect to the same η (x) at x = 0. Kaul et al. (1994) further extended the concets of Rueda Hanson (1988) to seudo-quasi Tye I, quasi-seudo Tye I objective constraint functions as follows. Definition.3.5. f (x) g(x) as quasi-seudo-tye I objective constraint functions, resectively, with resect to η (x) at x if there exists an n-dimensional vector function η (x) defined for all x P such that f (x) f ( x) 0 [ x f ( x)] T η (x, x) 0 [ x g(x)] T η (x, x) 0 g(x) 0.

10 16 Generalized Convex Functions Definition.3.6. f (x) g(x) as seudo-quasi-tye I objective constraint functions, resectively,with resect to η (x) at x if there exists an n-dimensional vector function η (x) defined for all x P such that [ x f ( x)] T η (x, x) 0 f (x) f ( x) 0 g(x) 0 [ x g(x)] T η (x, x) 0..4 Univex Related Functions Let f be a differentiable function defined on a nonemty subset X of R n let φ : R R k : X X R +. For x, x X, we write k (x, x)= lim b(x, x,λ ) 0. λ 0 Bector et al. (199) defined b-invex functions as follows. Definition.4.1. The function f is said to be B-invex with resect to η k, at x if for all x X, we have k (x, x)[f (x) f ( x)] [ x f ( x)] T η (x, x). Bector et al. (199) further extended this concet to univex functions as follows. Definition.4.. The function f is said to be univex with resect to η,φ k, at x if for all x X, we have k (x, x)φ [ f (x) f ( x)] [ x f ( x)] T η (x, x). Definition.4.3. The function f is said to be quasi-univex with resect to η,φ k, at x if for all x X, we have φ [ f (x) f ( x)] 0 k (x, x)η (x, x) T x f ( x) 0. Definition.4.4. The function f is said to be seudo-univex with resect to η,φ k, at x if for all x X, we have η (x, x) T x f ( x) 0 k (x, x)φ [ f (x) f ( x)] 0. Bector et al. (199) gave the following relations with some other generalized convex functions existing in the literature. 1. Every B-invex function is univex function with φ : R R defined as φ (a)= a, a R, but not conversely. Examle.4.1. Let f : R R be defined by f (x)=x 3,where, { x η (x, x)= + x + x x, x > x x x,x x

11 .4 Univex Related Functions 17 { x / (x x), x > x k (x, x)= 0, x x. Let φ : R R be defined by φ (a)=3a. The function f is univex but not b-invex, because for x = 1, x = 1/, k (x, x)φ [ f (x) f ( x)] < η (x, x) T x f ( x).. Every invex function is univex function with φ : R R defined as φ (a) = a, a R, k (x, x) 1, but not conversely. Examle.4.. The function considered in above examle is univex but not invex, because for x = 3, x = 1, f (x) f ( x) < η (x, x) T x f ( x). 3. Every convex function is univex function with φ : R R defined as φ (a)= a, a R,k (x, x) 1, η (x, x) x x, but not conversely. Examle.4.3. The function considered in above examle is univex but not convex, because for x =, x = 1, f (x) f ( x) < (x x) T x f ( x). 4. Every b-vex function is univex function with φ : R R defined as φ (a) = a, a R, η (x, x) x x, but not conversely. Examle.4.4. The function considered in above examle is univex but not b-vex, because for x = 1 10, x = 1 100,k (x, x)[f (x) f ( x)] < (x x)t x f ( x). Rueda et al. (1995) obtained otimality duality results for several mathematical rograms by combining the concets of tye I functions univex functions. They combined the Tye I univex functions as follows. Definition.4.5. The differentiable functions f (x) g(x) are called Tye I univex objective constraint functions, resectively with resect to η,φ 0,φ 1,b 0,b 1 at x X, if for all x X, we have b 0 (x, x)φ 0 [ f (x) f ( x)] η (x, x) T x f ( x) b 1 (x, x)φ 1 [g( x)] η (x, x) T x g( x). Rueda et al. (1995) gave examles of functions that are univex but not Tye I univex. Examle.4.5. The functions f,g : [1, ) R, definedby f (x) =x 3 g(x) = 1 x, are univex at x = 1 with resect to b 0 = b 1 = 1,η (x, x) =x x,φ 0 (a) = 3a,φ 1 (a)=1, but g does not satisfy the second inequality of the above definition at x = 1. They also ointed out that there are functions which are Tye I univex but not univex. Examle.4.6. The functions f,g : [1, ) R,definedby f (x)= 1/x g(x)= 1 x, are Tye I univex with resect to b 0 = b 1 = 1,η(x, x)= 1/(x x), φ 0 (a)=a, φ 1 (a)= a,at x = 1, but g is not univex at x = 1.

12 18 Generalized Convex Functions Following Rueda et al. (1995), Mishra (1998b) gave several sufficient otimality conditions duality results for multiobjective rogramming roblems by combining the concets of Pseudo-quasi-Tye I, quasi-seudo-tye I functions univex functions..5 V-Invex Related Functions Jeyakumar Mond (199) introduced the notion of V-invexity for a vector function f =(f 1, f,..., f ) discussed its alications to a class of constrained multiobjective otimization roblems. We now give the definitions of Jeyakumar Mond (199) as follows. Definition.5.1. A vector function f : X R is said to be V-invex if there exist functions η : X X R n α i : X X R + {0} such that for each x, x X for i = 1,,...,, f i (x) f i ( x) α i (x, x) f i ( x)η (x, x). for = 1 η (x, x) =α i (x, x)η (x, x) the above definition reduces to the usual definition of invexity given by Hanson (1981). Definition.5.. A vector function f : X R is said to be V-seudoinvex if there exist functions η : X X R n β i : X X R + {0} such that for each x, x X for i = 1,,...,, f i ( x)η (x, x) 0 β i (x, x) f i (x) β i (x, x) f i ( x). Definition.5.3. A vector function f : X R is said to be V-quasiinvex if there exist functions η : X X R n δ i : X X R + {0} such that for each x, x X for i = 1,,...,, δ i (x, x) f i (x) δ i (x, x) f i ( x) f i ( x)η (x, x) 0. It is evident that every V-invex function is both V-seudo-invex (with β i (x, x)= 1 α i (x, x) ) V-quasi-invex (with δ i (x, x)= 1 α i (x, x) ).Alsoifweset = 1, α i (x, x)=1, β i (x, x)=1, δ i (x, x)=1η (x, x) =x x, then the above definitions reduce to those of convexity, seudo-convexity quasi-convexity, resectively. Definition.5.4. A vector otimization roblem: (VP) V min( f 1, f,..., f ) subject to g(x) 0,

13 .5 V-Invex Related Functions 19 where f i : X R,i = 1,,..., g : X R m are differentiable functions on X, is said to be V-invex vector otimization roblem if each f 1, f,..., f g 1, g,...,g m is a V-invex function. Note that, invex vector otimization roblems are necessarily V-invex, but not conversely. As a simle examle, we consider following examle from Jeyakumar Mond (199). Examle.5.1. Consider ( x min 1, x ) 1 x 1,x R x x subject to 1 x 1 1,1 x 1. Then it is easy to see that this roblem is a V-invex vector otimization roblem with α 1 = x x, α = x 1 x 1, β 1 = 1 = β,η(x, x)=x x; but clearly, the roblem does not satisfy the invexity conditions with the same η. It is also worth noticing that the functions involved in the above roblem are invex, but the roblem is not necessarily invex. It is known (see Craven (1981)) that invex roblems can be constructed from convex roblems by certain nonlinear coordinate transformations. In the following, we see that V-invex functions can be formed from certain nonconvex functions (in articular from convex-concave or linear fractional functions) by coordinate transformations. Examle.5.. Consider function, h : R n R defined by h(x) =(f 1 (φ(x)),..., f (φ(x))), wheref i : R n R,i = 1,,...,, are strongly seudo-convex functions with real ositive functions α i,φ : R n R n is surjective with φ ( x) onto for each x R n. Then, the function h is V-invex. Examle.5.3. Consider the comosite vector function h(x) =(f 1 (F 1 (x)),..., f (F (x))), where for each i = 1,,...,, F i : X 0 R is continuously differentiable seudolinear with the ositive roortional function α i (, ), f i : R R is convex. Then, h(x) is V invex with η (x, y) =x y. This follows from the following convex inequality seudolinearity conditions: f i (F i (x)) f i (F i (y)) f i (F i (y))(f i (x) F i (y)) = f i (F i (y))α i (x, y)f i (y)(x y) = α i (x, y)(f i F i ) (y)(x y). For a simle examle of a comosite vector function, we consider ( h(x 1, x )= e x 1/x, x ) 1 x, where X 0 = { (x 1, x ) R : x 1 1, x 1 }. x 1 + x Examle.5.4. Consider the function H(x)=(f 1 ((g 1 ψ)(x)),..., f ((g ψ)(x))), where each f i is seudolinear on R n with roortional functions α i (x, y),ψ is a

14 0 Generalized Convex Functions differentiable maing from R n onto R n such that ψ (y) is surjective for each y R n, f i : R R is convex for each i. Then H is V-invex. Jeyakumar Mond (199) have shown that the V-invexity is reserved under a smooth convex transformation. Proosition.5.1. Let ψ : R R be differentiable convex with ositive derivative everywhere; let h : X 0 R be V-invex. Then, the function is V-invex. h ψ (x)=(ψ (h 1 (x)),...,ψ (h (x))), x X 0 The following very imortant roerty of V-invex functions was also established by Jeyakumar Mond (199). Proosition.5.. Let f : R n R be V-invex. Then y R n is a (global) weak minimum of f if only if there exists 0 τ R, τ 0, τ i f i (y)=0. By Proosition.5., one can conclude that for a V-invex vector function every critical oint (i.e., f i (y)=0, i = 1,...,) is a global weak minimum. Hanson et al. (001) extended the (scalarized) generalized tye-i invexity into a vector (V-tye-I) invexity. Definition.5.5. The vector roblem (VP) is said to be V-tye-I at x Xifthereexist ositive real-valued functions α i β j defined on X X an n-dimensional vector-valued function η : X X R n such that f i (x) f i ( x) α i (x, x) f i ( x)η (x, x) g j ( x) β j (x, x) g j ( x)η (x, x), for every x X for all i = 1,,..., j = 1,,...,m. Definition.5.6. The vector roblem (VP) is said to be quasi-v-tye-i at x Xif there exist ositive real-valued functions α i β j defined on X X an n- dimensional vector-valued function η : X X R n such that for every x X. τ i α i (x, x)[f i (x) f i ( x)] 0 m j=1 λ j β j (x, x)g j ( x) 0 m j=1 τ i η (x, x) f i ( x) 0 λ j η (x, x) g j ( x) 0,

15 .5 V-Invex Related Functions 1 Definition.5.7. The vector roblem (VP) is said to be seudo-v-tye-i at x X if there exist ositive real-valued functions α i β j defined on X X an n-dimensional vector-valued function η : X X R n such that τ i η (x, x) f i ( x) 0 m j=1 for every x X. λ j η (x, x) g j ( x) 0 τ i α i (x, x)[f i (x) f i ( x)] 0 m j=1 λ j β j (x, x)g j ( x) 0, Definition.5.8. The vector roblem (VP) is said to be quasi-seudo-v-tye-i at x X if there exist ositive real-valued functions α i β j defined on X X an n-dimensional vector-valued function η : X X R n such that τ i α i (x, x)[f i (x) f i ( x)] 0 m j=1 for every x X. λ j η (x, x) g j ( x) 0 m j=1 τ i η (x, x) f i ( x) 0 λ j β j (x, x)g j ( x) 0, Definition.5.9. The vector roblem (VP) is said to be seudo-quasi-v-tye-i at x X if there exist ositive real-valued functions α i β j defined on X X an n-dimensional vector-valued function η : X X R n such that for every x X. τ i η (x, x) f i ( x) 0 m j=1 λ j β j (x, x)g j ( x) 0 τ i α i (x, x)[f i (x) f i ( x)] 0 m j=1 λ j η (x, x) g j ( x) 0, Nevertheless the study of generalized convexity of a vector function is not yet sufficiently exlored some classes of generalized convexity have been introduced recently. Several attemts have been made by many authors to introduce ossibly a most wide class of generalized convex function, which can meet the dem of a real life situation to formulate a nonlinear rogramming roblem therefore get a best ossible solution for the same. Recently, Aghezzaf Hachimi (001) introduced a new class of functions, which we shall give in next section.

16 Generalized Convex Functions.6 Further Generalized Convex Functions Definition.6.1. f is said to be weak strictly seudoinvex with resect to η at x X if there exists a vector function η (x, x) defined on X X such that, for all x X, f (x) f ( x) f ( x)η(x, x) < 0. This definition is a slight extension of that of the seudoinvex functions. This class of functions does not contain the class of invex functions, but does contain the class of strictly seudoinvex functions. Every strictly seudoinvex function is weak strictly seudoinvex with resect to the same η. However, the converse is not necessarily true, as can be seen from the following examle. Examle.6.1. The function f =(f 1, f ) defined on X = R,by f 1 (x)=x(x + ) f (x) =x(x + ) is weak strictly seudoinvex function with resect to η (x, x) = x + at x = 0, but it s not strictly seudoinvex with resect to the same η (x, x) at x because for x =, we have f (x) f ( x) but f ( x)η (x, x)=0 0. Definition.6.. f is said to be strong seudoinvex with resect to η at x Xifthere exists a vector function η (x, x) defined on X X such that, for all x X, f (x) f ( x) f ( x)η (x, x) 0. Instead of the class of weak strictly seudoinvex, the class of strong seudoinvex functions does contain the class of invex functions. Also, every weak strictly seudoinvex function is strong seudoinvex with resect to the same η. However,the converse is not necessarily true, as can be seen from the following examle. Examle.6.. The function f =(f 1, f ) defined on X = R, by f 1 (x) =x 3 f (x) =x(x + ) is strongly seudoinvex function with resect to η (x, x) =x at x = 0, but it is not weak strictly seudoinvex with resect to the same η (x, x) at x because for x = 1 f (x) f ( x) but f ( x)η (x, x)=(0, 4) T 0, also f is not invex with resect to the same η at x, as can be seen by taking x =. There exist functions f that are seudoinvex but not strong seudoinvex with resect to the same η. Conversely, we can find functions that are strong seudoinvex, but they are not seudoinvex with resect to the same η. Examle.6.3. The function f : R R,definedby f 1 (x)=x(x ) f (x)= x(x 3), is seudoinvex with resect to η (x, x)=x x at x = 0, but it is not weak strictly seudoinvex with resect to the same η (x, x) when x =.

17 .6 Further Generalized Convex Functions 3 Examle.6.4. The function f : R R,definedby f 1 (x)=x(x ) f (x)= x (x 1), is strong seudoinvex with resect to η (x, x)=x x at x = 0, but it is not seudoinvex with resect to the same η (x, x) at that oint. Remark.6.1. If f is both seudoinvex quasiinvex with resect to η at x X, then it is strong seudoinvex function with resect to the same η at x. Definition.6.3. f is said to be weak quasiinvex with resect to η at x Xifthere exists a vector function η (x, x) defined on X X such that, for all x X, f (x) f ( x) f ( x)η (x, x) 0. Every quasiinvex function is weak quasiinvex with resect to the same η. However, the converse is not necessarily true. Examle.6.5. Define a function f : R R,by f 1 (x) =x(x ) f (x) = x (x ), then the function is weak quasiinvex with resect to η (x, x) =x x at x = 0, but it is not quasiinvex with resect to the same η (x, x) at x = 0, because f (x) f ( x) but f ( x)η (x, x) 0, for x =. Definition.6.4. f is said to be weak seudoinvex with resect to η at x Xifthere exists a vector function η (x, x) defined on X X such that, for all x X, f (x) < f ( x) f ( x)η (x, x) 0. The class of weak seudoinvex functions does contain the class of invex functions, seudoinvex functions, strong seudoinvex functions strong quasiinvex functions. Remark.6.. Notice from Examles , that the concets of weak strictly seudoinvex, strong seudoinvex, weak seudoinvex, seudoinvex vectorvalued functions are different, in general. However, they coincide in the scalarvalued case. Definition.6.5. f is said to be strong quasiinvex with resect to η at x Xifthere exists a vector function η (x, x) defined on X X such that, for all x X, f (x) f ( x) f ( x)η (x, x) 0. Every strong quasiinvex function s both quasiivex strong seudoinvex with resect to the same η. Aghezzaf Hachimi (001) introduced the class of weak requasiinex functions by generalzing the class of reinvex (Ben-Israel Mond 1986) the class requasiinvex functions (Suneja et al. 1993). Definition.6.6. We say that f is weak requasiinvex at x X with resect to η if Xisinvexat x with resect to η, for each x X, f (x) f ( x) f ( x + λη(x, x)) f ( x), 0 < λ 0.

18 4 Generalized Convex Functions Every requasiinvex function is weak requasiinvex with resect to the same η. But the converse is not true. Examle.6.6. The function f : R R,definedby f 1 (x)=x(x ) f (x)= x(x ), is weak requasiivex at x = 0 with resect to η (x, x)=x x, but it is not requasiinvex with resect to the same η (x, x) at x = 0, because f 1 (x)=x(x ) is not requasiinvex at x = 0 with resect to same η (x, x).

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