Some New Classes of Generalized Concave Vector Valued Functions. Riccardo CAMBINI

Transcription

1 UNIVERSITÀ DEGLI STUDI DI PISA DIPARTIMENTO DI STATISTICA E MATEMATICA APPLICATA ALL ECONOMIA Some New Classes of Generalized Concave Vector Valued Functions Riccardo CAMBINI This is a preliminary version of the following paper: Cambini R. : Some new classes of generalized concave vector-valued functions, in Optimization of Generalized Convex Problems in Economics, Proceedings of the Workshop Ottimizzazione di problemi convessi generalizzati interessanti l Economia, National Research Project 40%, held in Milano, Università Cattolica, March 10, 1994, edited by P. Mazzoleni, pp , Please find and cite the published final version. Via Cosimo Ridolfi, PISA Tel. (050) Centralino Segr. Amm /231 - Segr. Stud Fax Cod. Fisc Partita IVA

2 SOME NEW CLASSES OF GENERALIZED CONCAVE VECTOR VALUED FUNCTIONS RICCARDO CAMBINI Department of Statistics and Applied Mathematics, University of Pisa Via Cosimo Ridolfi, Pisa Abstract The aim of this paper is to study in a systematic way relationships and first-order characterizations among several classes of functions which are possible extensions of scalar generalized concavity. These classes are defined by using two order relations generated by a cone C or the interior of C, or the cone C without the origin. Keywords Generalized Concavity, Vector Optimization 1. Introduction In these last years several of articles dealing with scalar generalized concavity have appeared in scientific journals and numerous textbooks have specific chapters in this subject. On the contrary, vector generalized concavity is not yet sufficiently explored; only occasionally, with the aim to extend to the vector case some properties of scalar generalized concavity, some authors have considered componentwise generalized concavity for the Paretian case or have defined some specific classes of vector generalized concavity with respect to a cone [1, 2, 4-6].

3 The aim of this paper is to study in a systematic way relationships and first-order characterizations among several classes of functions which are possible extensions of scalar generalized concavity. These classes are defined by using two order relations generated by a cone C ( 1 ) or the interior C 00 of C, or the cone C without the origin, denoted by C Extension of concavity in vector optimization In this section we will extend the concept of concavity to a vector function. Definition 2.1 Consider the vector function f:s R m, where S R n is a convex set, and let be C R m a closed cone with vertex at the origin and with nonempty interior. Set C* {C,C 0,C 00 }; we will say that: f is C*-concave [C*.cv] if and only if x,y S, x y, f(x+λ(y-x))-λ(f(y)-f(x)) f(x)+c* λ (0,1); f is C*-semiconcave [C*.smcv] if and only if x,y S, x y, f(y) f(x)+c f(x+λ(y-x))-λ(f(y)-f(x)) f(x)+c* λ (0,1) Let us note that the previous definition concerns with six classes of functions. In the Paretian case C=R m + ={y R m : y> = 0}, there exists a strict relation between the C*--concavity, with C* {C,C 0,C 00 }, and the componentwise concavity of the function, as is pointed out in Property 2.1. Property 2.1 Let S R n be a convex set and consider the function f:s R m, with f(x)=(f 1 (x),,f m (x)), and the Paretian cone C=R m +. Then i), ii), iii) hold: i) f is C-concave if and only if f 1,,f m are concave functions; ii) if f 1,,f m are concave functions where at least one is strictly concave, then f is C 0 -concave; iii) f is C 00 -concave if and only if f 1,,f m are strictly concave functions. 1 A set C R m is said to be a cone with vertex at the origin if x C implies kx C k 0; C is a pointed cone if x C, x 0, implies -x C; C is a convex cone if x,y C implies x+y C.

4 Let us note that in the Paretian case C=R m +, there is no relations between the C*--semiconcavity, with C* {C,C 0,C 00 }, and the componentwise generalized concavity of the function, as is shown in the following example: Example 2.1 Consider the the cone C=R+ 3 and the function f(x)=(x 2,-x 2 -x,x), where S=R : f is C 00 -semiconcave since / x,y S, x y, such that f(y) f(x)+c, while the first component of f is strictly convex. The following picture points out the relations among the above defined classes of functions: C 00.smcv C 0.smcv C.smcv C 00.cv C 0.cv C.cv In the diagram is shown that in the scalar case (m=1 e C=R + ), the classes of the C 0 -concave, C 00 -concave, C 0 -semiconcave and C 00 -semiconcave functions collapse to the class of the strictly concave functions, while the classes of C-concave and C-semiconcave functions collapse to the class of the concave functions [3]. The following examples show that the inclusions among the classes are proper. Examples 2.2 Consider the cone C=R+ 2. i) the function f(x)=(x, e -x ), x R is C 00.smcv since / x,y R, x y, such that f(y) f(x)+c; on the other hand the second component of f is not a convave function so that for Property 2.1 f is not C.cv; ii) for Property 2.1, the function f(x)=(-x 2 +2x, 0) is C 0.cv since it first component is a strictly concave function and the second one is concave; f is not C 00.smcv, since, for x=0, y=1, we have f(y) f(x)+c but, taking into account that the second component of f is the null function, / λ (0,1) such that f(x+λ(y-x))-λ(f(y)-f(x)) f(x)+c 00 ; iii) the null vector function is C.cv but it is not C 0.smcv.

5 3. Extension of quasi-concavity in vector optimization In this section we will extend the concept of quasi-concavity to a vector function. Definition 3.1 Consider the vector function f:s R m, where S R n is a convex set, and let be C R m a closed cone with vertex at the origin and with nonempty interior. Set C* {C,C 0,C 00,0,C\C 00 } and C # {C,C 0,C 00 }. We will say that f is (C*,C # )-quasiconcave [(C*,C # ).qcv] if and only if for every x,y S, x y, f(y) f(x)+c* f(x+λ(y-x)) f(x)+c # λ (0,1) Let us note that the previous definition concerns with fifteen classes of functions, since we can choice C* and C # independently. The relationships among these classes of functions are shown in the following picture: (0,C).qcv (C\C 00,C).qcv (C,C).qcv (C 0,C).qcv (C 00,C).qcv (0,C 0 ).qcv (C\C 00,C 0 ).qcv (C,C 0 ).qcv (C 0,C 0 ).qcv (C 00,C 0 ).qcv (0,C 00 ).qcv (C\C 00,C 00 ).qcv (C,C 00 ).qcv (C 0,C 00 ).qcv (C 00,C 00 ).qcv The diagram points out that, in the scalar case, the classes of (C 00,C 0 ).qcv, (C 00,C 00 ).qcv, (C 0,C 0 ).qcv e (C 0,C 00 ).qcv functions collapse to the class of semistrictly quasi-concave functions, the classes of (C,C 0 ).qcv and (C,C 00 ).qcv functions collapse to the class of strictly quasi-concave functions, the classes of (C 00,C).qcv and (C 0,C).qcv functions collapse to the class of generalized quasi-concave functions, the classes of (0,C).qcv and (C\C 00,C).qcv functions collapse to the class of generalized quasi-concave functions, while the classes of

6 (C\C 00,C 0 ).qcv, (C\C 00,C 00 ).qcv, (0,C 0 ).qcv and (0,C 00 ).qcv, functions collapse to the class of generalized strictly quasi-concave functions [3]. Some classes of vector quasi-concave functions can be characterized as is stated in the the following theorems: Theorem 3.1 Consider the vector function f:s R m, where S R n is a convex set, and let be C R m a closed cone with vertex at the origin and with nonempty interior. Set C* {C,C 0,C 00 }. Then the following conditions are equivalentl: i) f is (C,C*)-quasiconcave; ii) f is both (C 0,C*)-quasiconcave and (0,C*)-quasiconcave. iii) f is both (C 00,C*)-quasiconcave and (C\C 00,C*)-quasiconcave. Proof. It follows by the given definitions. Theorem 3.2 Consider the vector function f:s R m, where S R n is a convex set, and let be C R m a closed cone with vertex at the origin and with nonempty interior. Then f is (C,C 0 ).qcv if and only if f is (C 0,C 0 ).qcv and x,y S, f(y)=f(x) λ (0,1) such that f(x+ λ(y-x)) f(x)+c 0. (3.1) Proof. The necessary condition is obvious; as regards to the sufficiency condition, since f is (C 0,C 0 ).qcv we must verify that x,y S, x y, with f(y)=f(x), we have f(x+λ(y-x)) f(x)+c 0 λ (0,1). Property (3.1) implies the existence of λ (0,1) such that f(x+ λ(y-x)) f(x)+c 0 ; the thesis follows from the definition of a (C 0,C 0 ).qcv function applied to the intervals of end points x, x+ λ(y-x) and x+ λ(y-x), y, respectively, taking into account that f(y)=f(x). The following example shows that the inclusions between the classes of (C\C 00,C*).qcv and (C,C*).qcv functions are proper. Example 3.1 (x,x) per x [0,2], x 1 Consider the function f(x)= (-1,1) per x=1 and the cone C=R 2 + ; f is (C\C 00,C 00 ).qcv (consequently f is (C\C 00,C 0 ).qcv and (C\C 00,C).qcv, too) since

7 / x,y S, x y, such that f(y) f(x)+c\c 00 ; on the other hand f is not (C,C).qcv (and consequently f is nor (C,C 0 ).qcv nor (C,C 00 ).qcv) since f(2) f(0)+c but f(1) f(0)+c. Under continuity assumption, the class of (C\C 00,C*).qcv functions collapse to the class of (C,C*).qcv functions; this is shown in the following theorem: Theorem 3.3 Let S R n be a convex set, let f:s R m be a vector continuous function and let C R m be a pointed ( 2 ) convex and closed cone with vertex at the origin and with non empty interior. Set C* {C,C 0,C 00 }. Then f is (C,C*)-quasiconcave if and only if f is (C\C 00,C*)-quasiconcave. Proof. necessity : it follows directly from the definitions of the classes; sufficiency: case C*=C Assume that f is (C\C 00,C).qcv but not (C,C).qcv; then there exist x,y S, λ (0,1) such that f(y) f(x)+c 00 and f(x+ λ(y-x)) f(x)+c. For the continuity of f there exists λ ~ ( λ,1) such that f(x+λ ~ (y-x)) f(x)+c\c 00 ; applying the definition of (C\C 00,C).quasiconcavity in the interval of end points x and x+λ ~ (y-x) we have (x+λ(y-x)) f(x)+c λ (0,λ ~ ), and this is absurd since λ (0,λ ~ ). case C*=C 0 Since f is (C\C 00,C 0 ).qcv, it is also (C\C 00,C).qcv and consequently, for the previous case, f is (C,C).qcv. Assume now that f is not (C,C 0 ).qcv; then there exist x,y S, λ (0,1) such that f(y) f(x)+c 00 and f(x+ λ(y-x)) f(x)+c\c 0, that is f(x+ λ(y-x))=f(x); applying the definition of (C\C 00,C 0 ).quasiconcavity in the interval of end points x and x+ λ(y-x) we have f(x+λ(y-x)) f(x)+c 0 λ (0, λ). The condition f(y) f(x)+c 00 implies, for the continuity of f, the existence of λ ~ (0, λ ) such that f(x+λ ~ (y-x)) f(x)+c 0 and f(y) f(x+λ ~ (y-x))+c 00 ; applying the definition of (C,C).quasiconcavity in the interval of end points x+λ ~ (y-x) and y we have 2 Let C R m be a convex cone; then x C 00, y C x+y C 00. Furthermore if C is pointed then x C 0, y C x+y C 0.

8 f(x+λ(y-x)) f(x+λ ~ (y-x))+c λ (λ ~,1). Since λ (λ ~,1) and f(x+λ ~ (y-x)) f(x)+c 0, taking into account that C is a pointed convex cone, we have f(x+ λ(y-x)) f(x)+c 0, and this is absurd. case C*=C 00 Since f is (C\C 00,C 00 ).qcv, it is also (C\C 00,C 0 ).qcv and consequently, for the previous case, f is (C,C 0 ).qcv. Assume now that f is not (C,C 00 ).qcv; then there exist x,y S, λ (0,1) such that f(y) f(x)+c 00 and f(x+ λ(y-x)) f(x)+c 0 \C 00. Applying the definition of (C\C 00,C 00 ).quasiconcavity in the interval of end points x, x+ λ(y-x) we have f(x+λ(y-x)) f(x)+c 00 λ (0, λ) and consequently, for the continuity of f and taking into account that f(y) f(x)+c 00, there exists λ ~ (0, λ) such that f(x+λ ~ (y-x)) f(x)+c 00 and f(y) f(x+λ ~ (y-x))+c 00. Applying the definition of (C,C 0 ).quasiconcavity in the interval of end points x+λ ~ (y-x), y we have f(x+λ(y-x)) f(x+λ ~ (y-x))+c 0 λ (λ ~,1). Since λ (λ ~,1), f(x+λ ~ (y-x)) f(x)+c 00 and C is a pointed convex cone, we have f(x+ λ(y-x)) f(x)+c 00, and this is absurd. When f is a continuous function, the relationships among the classes reduces to the ones given in the following picture: (0,C).qcv (C\C 00,C).qcv (C,C).qcv (C 0,C).qcv (C 00,C).qcv (0,C 0 ).qcv (C\C 00,C 0 ).qcv (C,C 0 ).qcv (C 0,C 0 ).qcv (C 00,C 0 ).qcv (0,C 00 ).qcv (C\C 00,C 00 ).qcv (C,C 00 ).qcv (C 0,C 00 ).qcv (C 00,C 00 ).qcv The following example show that the above inclusion are proper even if f is continuous. Examples 3.2

9 Let C be the Paretian cone R+ 2. (0,1-x) per x [0,1] i) the function f(x)= (0,x-1) per x [1,2] is (C00,C 00 ).qcv (consequently f is (C 00,C 0 ).qcv and (C 00,C).qcv, too) since / x,y S=[0,2] such that f(y) f(x)+c 00 ; on the other hand f is not (C 0,C).qcv (and consequently nor (C 0,C 0 ).qcv nor (C 0,C 00 ).qcv) since f(2) f(1/2)+c 0 but f(1) f(1/2)+c; ii) the function f(x)=(x 2 -x,-x 2 +x), with x S=R, is (C 0,C 00 ).qcv (consequently f is (C 0,C 0 ).qcv and (C 0,C).qcv, too) since / x,y S such that f(y) f(x)+c 0 ; f is not (C,C).qcv (consequently f is nor (C,C 0 ).qcv nor (C,C 00 ).qcv) since f(1) f(0)+c but f(1/2) f(0)+c; iii) the function f(x)=(x,x 2 -x), with x S=[0,1], is (0,C 00 ).qcv (consequently f is (0,C 0 ).qcv and (0,C).qcv, too) since / x,y S such that f(y)=f(x); f is not (C,C).qcv (consequently f is nor (C,C 0 ).qcv nor (C,C 00 ).qcv) since f(1) f(0)+c but f(1/2) f(0)+c; iv) the function f(x)= (x,x) per x [0,1] (2-x,2-x) per x [1,2] is (C,C).qcv in S=[0,3] (0,0) per x [2,3] (consequently f is (C 0,C).qcv, (C 00,C).qcv and (0,C).qcv, too); f is not (C 00,C 0 ).qcv (and consequently f is nor (C 0,C 0 ).qcv nor (C,C 0 ).qcv) since f(1) f(3)+c 00 but f(2) f(3)+c 0, furthermore f is not (0,C 0 ).qcv since f(3)=f(0) while f(2) f(0)+c 0 ; v) the function f(x)= (x,0) per x [0,1] (2-x,x-1) per x [1,2] is (C,C 0 ).qcv in S=[0,3] (0,3-x) per x [2,3] (consequently f is (C 0,C 0 ).qcv,(c 00,C 0 ).qcv and (0,C 0 ).qcv, too); f is not (C 00,C 00 ).qcv (consequently f is nor (C 0,C 00 ).qcv nor (C,C 00 ).qcv) since f(3/2) f(0)+c 00 but f(1) f(0)+c 00, furthermore f is not (0,C 00 ).qcv since f(3)=f(0) but f(2) f(0)+c 00. Remark 3.1 Examples 3.2 i) e v) point out that the property for which a continuous semistrictly quasiconcave scalar function is quasiconcave too, cannot be exteded to the vector case.

10 As is well known, a scalar function is quasiconcave if and only if its upper level sets are convex. A natural extension of the definition of an upper level set in vector optimization is the following one: Definition 3.2 Consider the vector function f:s R m, where S R n is a convex set, and let be C R m a closed cone with vertex at the origin and with nonempty interior. The set U(f,µ)={x S: f(x) µ+c}, µ R m, is said to be a C-upper level set of f. The following example and Theorem 3.4 show that the convexity of the C-upper level sets of a function f is a necessary but not sufficient condition for f to be (C,C)-quasiconcave. Example 3.3 Consider the cone C=R+ 2 and the following function: (x,-x) if x [-1,1], x 0, and y=0 f(x,y)= (-2,2) if x=0 and y=0 ; (-1/2,-1/2) if x [-1,1] and y<0 f is (C,C).quasiconcave in S={(x,y): x [-1,1], y 0} but, choosing µ=(-1/2,-1/2) (µ=f(x,y) y<0, x [-1,1]) the following C-upper level set is not convex: U(f,µ)={(x,y): x [-1/2,1/2]\{0}, y=0} {(x,y): x [-1,1], y<0}. Theorem 3.4 Consider the vector function f:s R m, where S R n is a convex set, and let be C R m a closed cone with vertex at the origin and with nonempty interior. If the C-upper level sets U(f,µ ) are convex,with µ f(s), then f is (C,C)-quasiconcave. Proof. Let x,y S are such that f(y) f(x)+c; setting µ=f(x) we have x,y U(f,µ) so that, for the convexity of U(f,µ), it results x+λ(y-x) U(f,µ) λ (0,1), that is f(x+λ(y-x)) f(x)+c. Remark 3.2 Let us note that in [5] the (C,C)-quasiconcavity of a function has been characterized assuming that the C-upper level sets are star-shaped in every point x such that µ=f(x).

11 4. Extension of pseudo-concavity in vector optimization Now we will extend to a vector function the concept of pseudo-concavity as given by Thompson in [8]. Definition 4.1 Consider the vector function f:s R m, where S R n is a convex set, and let be C R m a closed cone with vertex at the origin and with nonempty interior. Set C* {C,C 0,C 00 }, C # {C 0,C 00 }; we will say that : f is strictly (C*,C # )-pseudoconcave[(c*,c # ).spcv] if and only if for every x,y S, x y, f(y) f(x)+c* ξ(x,y) C # t.c. λ (0,1) f(x+λ(y-x)) f(x)+λ(1-λ)ξ(x,y)+c, where the vector ξ(x,y) depends on x and y and is indipendent from λ. The relationships among the classes of (C*,C # )-pseudoconcave functions are given in the following picture which point out that in the scalar case the classes of strictly (C,C 0 )-pseudoconcave and strictly (C,C 00 )-pseudoconcave functions collapse to the class of strictly pseudo-concave functions, while the other classes reduce to the class of pseudo-concave functions. (C,C 0 ).spcv (C 0,C 0 ).spcv (C 00,C 0 ).spcv (C,C 00 ).spcv (C 0,C 00 ).spcv (C 00,C 00 ).spcv Now we will extend to a vector differentiable function the concept of pseudoconcavity as given by Mangasarian in [7]. Definition 4.2 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a closed cone with vertex at the origin and with non empty interior. Set C* {C,C 0,C 00 }, C # {C 0,C 00 }, we will say that: f is(c*,c # )-pseudoconcave[(c*,c # ).pcv] if for every x,y S, x y, f(y) f(x)+c* J f (x)(y-x) C #.

12 Also the concept of a differentiable quasi-concave function can be extended in the same way of Definition 4.2. Definition 4.3 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a closed cone with vertex at the origin and with non empty interior. Set C* {C,C 0,C 00 }, we will say that: f is weakly (C*,C)-quasiconcave [(C*,C).wqcv] if for every x,y S, x y, f(y) f(x)+c* J f (x)(y-x) C. Remark 4.1 In the scalar case (m=1 e C=R + ) the classes of functions weakly (C*,C)-quasiconcave and (C*,C # )-pseudoconcave are the first-order characterization of the classes of quasi-concave functions and pseudo-concave or strictly pseudoconcave functions, respectively. The relationships among the above classes are given in the following picture that point out that in the scalar case the classes of functions (C,C # )-pseudoconcave, with C # {C 0,C 00 }, reduce to the class of strictly pseudo-concave functions, while the classes of (C*,C # )-pseudoconcave functions, with C* {C 0,C 00 }, C # {C 0,C 00 }, reduce to the class of pseudo-concave functions and the other classes of functions reduce to the class of quasi-concave functions [3]. (C,C).wqcv (C 0,C).wqcv (C 00,C).wqcv (C,C 0 ).pcv (C 0,C 0 ).pcv (C 00,C 0 ).pcv (C,C 00 ).pcv (C 0,C 00 ).pcv (C 00,C 00 ).pcv

13 5. Relationships among the defined classes of functions In this section we will study relationships among all classes of vector functions defined in the previous sections. The following theorems hold: Theorem 5.1 Let S R n be a convex set, let f:s R m be a vector continuous function and let C R m be a pointed convex and closed cone with vertex at the origin and with non empty interior. If f is strictly (C*,C # )-pseudoconcave, with C* {C,C 0,C 00 }, C # {C 0,C 00 }, then f is (C*,C # )-quasiconcave. Proof. Let x,y S, x y, are such that f(y) f(x)+c*. Since f is (C*,C # )-spcv there exists ξ(x,y) C # such that f(x+λ(y-x)) f(x)+λ(1-λ)ξ(x,y)+c λ (0,1) ; this last relation, taking into account that C is a pointed and convex cone, implies f(x+λ(y-x)) f(x)+c #. Theorem 5.2 Let S R n be a convex set, let f:s R m be a vector continuous function and let C R m be a pointed convex and closed cone with vertex at the origin and with non empty interior. If f is C*-semiconcave, with C* {C,C 0,C 00 }, then f is (C,C*)-quasiconcave. Proof. Let x,y S, x y, are such that f(y) f(x)+c; since f is C*-semiconcave we have f(x+λ(y-x)) f(x)+λ(f(y)-f(x))+c* for every λ (0,1); the thesis follows taking into account that C is a pointed and convex cone. Theorem 5.3 Let S R n be a convex set, let f:s R m be a vector continuous function and let C R m be a pointed convex and closed cone with vertex at the origin and with non empty interior. i) If f is C-semiconcave then f is strictly (C 00,C 00 )-pseudoconcave; ii) If f is C 0 -semiconcave then f is strictly (C 0,C 0 )-pseudoconcave; iii) If f is C 00 -semiconcave then f is strictly (C,C 00 )-pseudoconcave.

14 Proof. i) Assume that f is not (C 00,C 00 ).spcv; then there exist x,y S such that f(y) f(x)+c 00 and furthermore, ξ(x,y) C 00 there exists λ (0,1) such that f(x+λ(y-x)) f(x)+λ(1-λ)ξ(x,y)+c. Since C is a convex cone we have λ(1-λ)ξ(x,y)+c λξ(x,y)+c, and thus ξ(x,y) C 00 there exists λ (0,1) such that f(x+λ(y-x)) f(x)+λξ(x,y)+c. As a consequence, setting ξ(x,y)=f(y)-f(x) C 00 there exists λ (0,1) such that f(x+ λ(y-x)) f(x)+ λ(f(y)-f(x))+c, and this is absurd since f is C.smcv. ii) the proof is similar to i). iii) Assume that f is not strictly (C,C 00 ).pseudoconcave; then (see proof given in i)) there exist x,y S, x y, such that f(y) f(x)+c and furthermore ξ(x,y) C 00 there exists λ (0,1) such that f(x+λ(y-x)) f(x)+λξ(x,y)+c. As a consequence, for every ε C 00, setting ξ(x,y)=f(y)-f(x)+ε C 00, there exists λ (0,1) such that f(x+ λ(y-x)) f(x)+ λ(f(y)-f(x)+ε)+c; when ε 0, taking into account that C is a closed cone, we have f(x+ λ(y-x)) f(x)+ λ(f(y)-f(x))+c 00, and this is absurd since f is C 00 -semiconcave. The results obtained in theorems 5.1, 5.2, 5.3 are shown in the following picture. (C*,C # ).spcv (C*,C # ).qcv C*.smcv (C,C*).qcv (C,C 00 ).spcv (C 0,C 0 ).spcv (C 00,C 00 ).spcv C 00.smcv C 0.smcv C.smcv Now we will point out that in the differentiable case, the functions defined in definitions 4.2 and 4.3 extend the ones defined in definitions 3.1 and 4.1. Theorem 5.4 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a closed cone with vertex at the origin and with non empty interior. Set C* {C,C 0,C 00 } e C # {C,C 0,C 00 }.

15 If f is (C*,C # )-quasiconcave then f is weakly (C*,C)-quasiconcave. Proof. Since f is (C*,C # )-quasiconcave, choosing x,y S, with x y such that f(y) f(x)+c*, we have f(x+λ(y-x))-f(x) λ C # λ (0,1); the differentiability of f implies J f (x)(y-x) - y-x σ(λ,0)+c # ; the thesis follows approaching λ 0, taking into account that C is a closed cone. Theorem 5.5 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a pointed, convex and closed cone with vertex at the origin and with non empty interior. Set C* {C,C 0,C 00 } e C # {C 0,C 00 }. If f is strictly (C*,C # )-pseudoconcave then f is (C*,C # )-pseudoconcave. Proof. Since f is (C*,C # )-pseudoconcavità di f, for x,y S, with x y such that f(y) f(x)+c*, we have: f(x+λ(y-x))-f(x) λ (1-λ)ξ(x,y)+C with ξ(x,y) C # λ (0,1); the differentiability of f implies J f (x)(y-x) - y-x σ(λ,0)+(1-λ)ξ(x,y)+c, with ξ(x,y) C # ; the thesis follows approaching λ 0, taking into account that C is a pointed, convex and closed cone. The following picture points out the result given in Theorem 5.4, 5.5. (C*,C # ).qcv (C*,C).wqcv (C*,C # ).spcv (C*,C # ).pcv Example 5.1 Consider the cone C=R 3 + and the following differentiable function f:[0,3] R3 : f(x)= (-x 2 +2x)[1/2,1/2,1] T if x [0,1] [1/2,1/2,1] T +(-2x 3 +9x 2-12x+5)[1,-1,0] T if x ]1,2[ [3/2,-1/2,1] T +(x-2) 2 [-5/6,7/6,-1/3] T if x [2,3],

16 It results f(0)=[0,0,0] T, f(3)=[2/3,2/3,2/3] T, and thus f(3) f(0)+c 00 ; on the other hand f(2)=[3/2,-1/2,1] T f(0)+c, so that f is not (C 00,C).qcv (consequently f does not belong to any class of functions previously defined) but it is (C,C 00 ).pcv (consequently f verifies Definitions 4.2 e 4.3) since the condition f(y) f(x)+c holds if and only if x [0,1[, y ]x,3] and for such x,y we have J f (x)(y-x)=[1/2,1/2,1] T (y-x) C First-order characterizations of some classes of functions As is known, in the scalar case it is possible to find first-order characterization for several classes of generalized concave functions; in the vector case this is not possible since the inclusion between the classes of weakly (C*,C)- quasiconcave and (C*,C)-quasiconcave functions and the classes of (C*,C # )-pseudoconcave and stictly (C*,C # )-pseudoconcave functions, are proper. Nevertheless we can obtain the following first-order characterization for C*-concave functions: Theorem 6.1 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a pointed, convex and closed cone with vertex at the origin and with non empty interior. Set C* {C,C 0,C 00 }. Then f is C*-concave if and only if J f (x)(y-x) f(y)-f(x)+c*. Proof. case C*=C The C-concavity of the function f implies that allora risulta, x,y S x y, f(x+λ(y-x))-f(x) λ f(y)-f(x)+c λ (0,1); since f is differentiablewe obtain J f (x)(y-x) - y-x σ(λ,0)+f(y)-f(x)+c; the thesis follows approaching λ 0, taking into account that C is a closed cone. Consider now x,y S, λ (0,1). We have J f (x+λ(y-x))(y-(x+λ(y-x)))=f(y)-f(x+λ(y-x))+c 1, c 1 C, J f (x+λ(y-x))(x-(x+λ(y-x)))=f(x)-f(x+λ(y-x))+c 2, c 2 C. Adding the first equality multiplied for for λ to the second multiplied for (1-λ), we obtain f(x+λ(y-x))=f(x)+λ(f(y)-f(x))+λc 1 +(1-λ)c 2 f(x)+λ(f(y)-f(x))+c.

17 case C* {C 0,C 00 } If f is C*-concave then f is C-concave too, so that for the previous case we have J f (x)((x+λ(y-x))-x) f(x+λ(y-x))-f(x)+c; since J f (x)((x+λ(y-x))-x)=λj f (x)(y-x) and f(x+λ(y-x))-λ(f(y)-f(x)) f(x)+c* we have λj f (x)(y-x) λ(f(y)-f(x))+c*. The sufficiency condition is similar to the one given in the previous case. As regards to the class of C*-semiconcave functions, we have the following results: Theorem 6.2 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a closed cone with vertex at the origin and with non empty interior. Set C* {C,C 0,C 00 }. If f is C*-semiconcave then f is weakly (C,C)-quasiconcave. Proof. Since f is C*-semiconcave, for x,y S, x y such that f(y) f(x)+c, we f(x+λ(y-x))-f(x) have λ f(y)-f(x)+c* λ (0,1), so that the differentiability of f implies J f (x)(y-x) - y-x σ(λ,0)+f(y)-f(x)+c #. The thesis follows approaching λ 0, taking into account that C is a closed cone. Now we will give some sufficient conditions to have first-order characterization for strictly (C*,C # )-pseudoconcave and (C*,C # )-quasiconcave functions. Lemma 6.1 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a closed cone with vertex at the origin and with non empty interior. If J f (x)(y-x) C 00, with x,y S, x y, then there exists ε>0 such that f(x+λ(y-x)) f(x)+c 00 λ (0,ε). f(x+λ(y-x))-f(x) Proof. If L=J f (x)(y-x)=lim λ 0 λ C 00, there exists a neighbourhood of L contained in C 00, so that there exists ε>0 such that: f(x+λ(y-x))-f(x) λ C 00 λ (0,ε). Lemma 6.2

18 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a convex closed cone with vertex at the origin and with non empty interior. Assume that f is (C 00,C)-quasiconcave. If f is (C 00,C 00 )-pseudo-concave then f is (C 00,C 00 )-quasiconcave. Proof. Let x,y S, x y, are such that f(y) f(x)+c 00 ; then J f (x)(y-x) C 00 and, for Lemma 6.1, ε>0 such that f(x+λ(y-x)) f(x)+c 00 λ (0,ε). Since f(y) f(x)+c 00 there exists λ* (0,ε) such that f(x+λ*(y-x)) f(x)+c 00, f(y) f(x+λ*(y-x))+c 00 ; on the other hand f is (C 00,C)-qcv, and so: f(x+λ(y-x)) f(x+λ*(y-x))+c λ (λ*,1). This last relation, taking into account that f(x+λ*(y-x)) f(x)+c 00 and the convexity of C, holds for λ (0,λ*], and consequently for every λ (0,1). The following theorem points out that a (C 00,C 00 )-pseudo-concave function can be view as the characterization of a strictly (C 00,C 00 )-pseudo-concave function. Theorem 6.3 Let S R n be a convex set, let f:s R m be a vector differentiable function and let C R m be a convex closed cone with vertex at the origin and with non empty interior. Assume that f is (C 00,C)-quasiconcave. Then f is (C 00,C 00 )-pseudoconcave if and only if f is strictly (C 00,C 00 )-pseudoconcave. Proof. if. Assume that f is not (C 00,C 00 ).spcv ; then there exist x,y S, x y, such that f(y) f(x)+c 00 and furthermore ξ C 00 there exists λ ξ (0,1) such that f(x+λ f(x+λ ξ (y-x)) f(x)+λ ξ (1-λ ξ )ξ+c, that is ξ (y-x))-f(x) λ (1-λ ξ ξ )ξ+c. Setting u C 00, ξ i =u(1/i), where i is a positive integer, there exists a sequence {λ i } [0,1] such that f(x+λ i(y-x))-f(x) u(1-λ i )(1/i)+C, and a subsequence of λ i {λ i }, that without loss of generality we can suppose to be the same sequence, converging to λ* [0,1]. If λ*=0, we have f(x+λ J f (x)(y-x)= lim i (y-x))-f(x) C 00, and this is absurd since f is (C 00,C 00 ).pcv i + λ i

19 f(x+λ If λ*=1 we have f(y)-f(x)= lim i (y-x))-f(x) i + λ C 00, and this contradicts the i assumptions. Assume now that λ* (0,1); in such a case we have: f(x+λ*(y-x))-f(x) f(x+λ λ* = lim i (y-x))-f(x) i + λ C 00 i that is f(x+λ*(y-x)) f(x)+c 00, since λ*>0; this last relation implies that f is not (C 00,C 00 ).qcv, and this is absurd because of Lemma 6.2. only if. it follows from Theorem 5.5. In order to give some other characterization we need of the following: Definition 6.1 Let S R n be a convex set, let f:s R m be a vector function, let C R m be a convex closed cone with vertex at the origin and with non empty interior and let x,y S are such that f(y) f(x)+c. We will say that f(x+λ*(y-x)), λ* [0,1], is α-minimal with respect to x and y if ( 3 ): α C ++ such that α T f(x+λ*(y-x))= min λ [0,1] {αt f(x+λ(y-x))}. The following thorem holds: Theorem 6.4 Let S R n be an open convex set, let f:s R m be a vector differentiable function and let C R m be a pointed, convex, closed cone with vertex at the origin and with non empty interior. Let x,y S, x y, λ (0,1) are such that f(y) f(x)+c and f(x+ λ(y-x)) f(x)+c. Then i) and ii) hold: i) λ* (0,1) such that: a) f(x+λ*(y-x)) is α-minimal with respect to x and y; b) J f (x+λ*(y-x))(y-x) C 0 ; ii) λ # (0,1) such that J f (x+λ # (y-x))(y-x) C. Proof. 3 Let C R m be a cone. The positive polar of C is the cone C + ={α R m : α T c 0 c C}; the strict positive polar of C is the cone C ++ ={α R m : α T c>0 c C, c 0} C +. If C is a convex closed cone we have C ++ Ø if and only if C is pointed.

20 i) Since f(x+ λ(y-x))-f(x) C, applying a known separation theorem ( 4 ), there exists α C ++ such that α T [f(x+ λ(y-x))-f(x)]<0, that is α T f(x+ λ(y-x))<α T f(x). Since f(y)-f(x) C we have α T [f(y)-f(x)] 0, that is α T f(y) α T f(x); as a consequence we obtain α T f(x+ λ(y-x))<α T f(x) α T f(y). The one variable function g(λ)=α T f(x+λ(y-x)), λ [0,1] has, for Weierstrass theorem, a global minimum point λ* [0,1] and this implies that f(x+λ*(y-x)) is α-minimal with respect to x and y. Taking into account that α T f(x+ λ(y-x))<α T f(x)=g(0) α T f(y)=g(1), necessarily λ* (0,1), so that 0=g'(λ*)=α T [J f (x+λ*(y-x))(y-x)]; this last relation, taking into account that α C ++, implies J f (x+λ*(y-x))(y-x) C 0. ii) Since g(0)>g(λ*), there exists λ # (0,λ*) such that g'(λ # )<0, that is α T [J f (x+λ # (y-x))(y-x)]<0; since α C ++ C + we have J f (x+λ # (y-x))(y-x) C, and the thesis follows. Corollary 6.1 Let S R n be an open convex set, let f:s R m be a vector differentiable function and let C R m be a pointed, convex, closed cone with vertex at the origin and with non empty interior. Set C* {C,C 0,C 00 }. If for every x,y S, x y, f(y) f(x)+c* J f (x+λ(y-x))(y-x) C λ (0,1), then f is (C*,C)-quasiconcave Proof. assume that f is not (C*,C).qcv ; then there exist x,y S, x y, λ (0,1) such that f(y) f(x)+c* and f(x+ λ(y-x)) f(x)+c; from Theorem 6.4 there exists λ # (0,1) such that J f (x+λ # (y-x))(y-x) C, and this contradicts the assumption. 4 Let C R m is a convex closed pointed cone with non empty interior and let x such that x C. Then α C ++ such that α T x<0.

21 Theorem 6.5 Let S R n be an open convex set, let f:s R m be a vector differentiable function and let C R m be a pointed, convex, closed cone with vertex at the origin and with non empty interior. Assume that for every α-minimal point f(x+λ*(y-x)) with respect to x and y, there exists λ ~ (λ *,1] such that f(x+λ ~ (y-x)) f(x+λ*(y-x))+c*, with C* {C,C 0,C 00 }. If f is (C*,C 0 )-pseudo concave, then f is (C,C)-quasiconcave. Proof. Ab absurdo assume that f is not (C,C).qcv; then there exist x,y S, x y, λ (0,1) such that f(y) f(x)+c and f(x+ λ(y-x)) f(x)+c; from i) of Theorem 6.4 there exists λ* (0,1) such that f(x+λ*(y-x)) is α-minimal with respect to x and y and furthermore J f (x+λ*(y-x))(y-x) C 0. Taking into account the assumptions of the theorem, there exists λ ~ (λ*,1] such that f(x+λ ~ (y-x)) f(x+λ*(y-x))+c*, with C* {C,C 0,C 00 }, and consequently J f (x+λ*(y-x))(x+λ ~ (y-x)-x-λ*(y-x)) C 0, that is J f (x+λ*(y-x))(y-x)(λ ~ -λ*) C 0. Since λ ~ >λ* we obtain J f (x+λ*(y-x))(y-x) C 0, and this contradicts Theorem 6.4. Theorem 6.6 Let S R n be an open convex set, let f:s R m be a vector differentiable function and let C R m be a pointed, convex, closed cone with vertex at the origin and with non empty interior. Assume that for every x,y S, x y, λ # (0,1) such that f(y) f(x)+c and J f (x+λ # (y-x))(y-x) C there exists λ ~ (λ #,1] verifying the condition f(x+λ ~ (y-x)) f(x+λ # (y-x))+c*, with C* {C,C 0,C 00 }. If f is (C*,C)-pseudo concave, then f is (C,C)-quasiconcave. Proof. Assume that f is not (C,C).qcv; then there exist x,y S, x y, λ (0,1) such that f(y) f(x)+c and f(x+ λ(y-x)) f(x)+c and, from ii) of Theorem 6.4 there exists λ # (0,1) such that J f (x+λ # (y-x))(y-x) C. One of the assumptions of the theorem implies the existence of λ ~ (λ #,1] such that f(x+λ ~ (y-x)) f(x+λ # (y-x))+c*, with C* {C,C 0,C 00 }, and consequently J f (x+λ # (y-x))(x+λ ~ (y-x)-x-λ # (y-x)) C, that is J f (x+λ # (y-x))(y-x)(λ ~ -λ # ) C. Since λ ~ >λ # we have J f (x+λ # (y-x))(y-x) C, and this contradicts one of the assumptions of the theorem.

22 Corollary 6.2 Let S R n be an open convex set, let f:s R m be a vector differentiable function and let C R m be a pointed, convex, closed cone with vertex at the origin and with non empty interior. Assume that i) and ii) hold:: i) for every α-minimal point f(x+λ*(y-x)) with respect to x and y there exists ~ ~ λ (λ*,1] such that f(x+λ (y-x)) f(x+λ*(y-x))+c 00 ; ii) for every x,y S, x y, and λ # (0,1) such that f(y) f(x)+c and J f (x+λ # (y-x))(y-x) C there exists λ ~ (λ #,1] such that f(x+λ ~ (y-x)) f(x+λ # (y-x))+c 00. Then f is (C 00,C 00 )-pseudoconcave if and only if f is strictly (C 00,C 00 )- pseudoconcave. Proof. From Theorems 6.5 and 6.6 the function f is (C,C).qcv and thus (C 00,C).qcv too; the thesis follows from Theorem 6.3. References [1] A. Cambini and L. Martein, An approach to optimality conditions in vector and scalar optimization, in Mathematical Modelling in Economics, edited by Diewert, Spremann, and Stehling, Springer Verlag, (1993) [2] R. Cambini, Una nota sulle possibili estensioni a funzioni vettoriali di significative classi di funzioni concavo-generalizzate, Technical report 57, Dipartimento di Statistica e Matematica Applicata all Economia, Università di Pisa, [3] R. Cambini, Alcune nuove classi di funzioni concavo-generalizzate, Technical report 74, Department of Statistics and Applied Mathematics, University of Pisa, 1993.

23 [4] E. Castagnoli and P. Mazzoleni, Scalar and vector generalized convexity, in Nonsmooth optimization and related topics, edited by F.H. Clarke, V.F. Dem yanov, and F. Giannessi, Plenum Press, New York (1989) [5] J. Jahn, Mathematical vector optimization in partially ordered linear spaces, Springer-Verlag, Frankfurt, [6] D.T. Luc, Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems 319, Springer-Verlag, Berlin, [7] O.L. Mangasarian, Pseudo-convex functions, J. SIAM Control Ser. A 3 (1965) [8] W.A. Thompson and D.W. Parke, Some properties of generalized concave functions, Operation Research 21 (1973)