AND OTHER GENERALIZED CONVEX FUNCTIONS
GIORGIO GIORGI
We precise some known and new properties of preinvex functions, in both the non differentiable case and in the differentiable case. In particular, we study the relationships of this class of functions with other classes of generalized con- vex functions.
AMS 2000 Subject Classification: 90C29, 90C30, 90C46.
Key words: Multiobjective programming, optimization.
1. INTRODUCTION
Hanson [6] has introduced a new class of generalized convex functions, subsequently called by Craven [3] “invex functions”, with the aim to extend the validity of the sufficiency of the Kuhn-Tucker conditions. Since the pa- pers of Hanson and Craven, many authors have studied invex functions, their generalizations and related functions: see, e.g., [2,4,5,8,14] and, for what concerns Romanian mathematicians, [9,10,11,12,15].
Definition 1. A differentiable real-valued functionf defined on an open set X⊆Rn is said to be invex if there exists a vector functionη(x, y) defined on X×X such that
f(x)−f(y)≥η(x, y)∇f(y), ∀x, y∈X.
Obviously a differentiable convex function (on an open convex set X) is also invex (choose η(x, y) =x−y). A useful characterization of the class of invex functions is given in the following result (see[2]).
Theorem 2. A differentiable function is invex (with respect to the same η) if and only if every stationary point is a global minimum point.
This shows that the class of pseudoconvex functions of n variables is contained in the class of invex functions (Pini [14] shows that if n = 1 the two classes coincide), whereas there is only a partial overlap of quasiconvex functions and invex functions. Indeed, f(x) =x3, x∈R, is quasiconvex, but
MATH. REPORTS10(60),4 (2008), 317–325
not invex, since x = 0 is a stationary point which is not a minimum point.
On the other hand (see [2]), the function f(x1, x2) = x31 +x1−10x32−x2 is invex (there are no stationary points) but not quasiconvex: taking y= (0,0), x1= 2, x2 = 1,yieldsf(x)−f(y)<0 but (x−y)∇f(y)>0.
Moreover, unlike pseudoconvex and convex functions, there is no dis- tinction between pseudoinvex and invex functions (a function f is said to be pseudoinvex ifη(x, y)∇f(y)≥0⇒f(x)−f(y)≥0). Note that a pseudoinvex function may not be invex for the same η, but can be invexfor some η.
Some nice properties of convex functions are lost by invex functions.
For example, the restriction of an invex function to a non open set does not maintain the local-global property. Consider, e.g., the function f(x, y) = y(x2−1)2 defined on the set X =
(x, y)∈R2 :y >0 . It is easy to verify that the stationary points (1, y),(−1, y),y >0,are global minimum points for f, sof is invex. Now, considerf on the closed setS={(x, y)∈R2 :x≥ −1/2, y ≥1}.The point (−1/2,1) is a local minimum forf onS but it is not global since f(−1/2,1) = 9/16> f(1,1) = 0.
For the same function defined on X =
(x, y)∈R2:y >0 , the set of all minimum points is{(1, y) :y >0} ∪ {(−1, y) :y >0}which is not a convex set. So, unlike convex functions, for an invex function the set of all minimum points is not necessarily a convex set (contrary to what was asserted in [14]).
Since invexity requires the differentiability assumption, in [2,7] a new class of functions, not necessarily differentiable, has been introduced. We first need the following definition.
Definition 3. A subsetX of Rn is said to be η-invex with respect to a function η:Rn×Rn→Rn ifx, y∈X,λ∈[0,1]⇒y+λη(x, y)∈X.
It is obvious that the above definition is a generalization of that of a convex set. It is to be noted that any set in Rn is invex with respect to η(x, y) ≡ 0, ∀x, y ∈ Rn. However, the only function f :Rn → R invex with respect to η(x, y)≡0 is the constant functionf(x) =c,c∈R.
Definition 4. Let f be a real-valued function defined on an η-invex set X;f is said to be preinvex with respect toη if
(1) f[y+λη(x, y)]≤λf(x) + (1−λ)f(y), ∀x, y∈X, ∀λ∈[0,1].
2. MAIN PROPERTIES OF PREINVEX FUNCTIONS
The sum of two or more preinvex functions, with respect to the same η, is again a preinvex function with respect to η. More generally, if fi :X →R are preinvex (with respect to η),i= 1,2, . . . , k,then Pk
i=1λifi(x) is preinvex (with respect to η), where λi ≥0,i= 1,2, . . . , k (see [17]).
As for convex functions, for a preinvex function every local minimum also is global and every strict local minimum also is a strict global minimum.
Furthermore, it is possible to establish saddle points and duality theorems following the classical approach used in the convex case [18].
Another property of convex functions shared by preinvex functions is given in the following result.
Theorem 5. Let X be an η-invex set f : X → R a preinvex function with respect to η, and φ:R→Ra convex increasing function. Then φ◦f is preinvex with respect to η.
Proof. By the assumptions we have
f[y+λη(x, y)]≤λf(x) + (1−λ)f(y).
As φis increasing and convex, we get
φ(f[y+λη(x, y)])≤φ(λf(x) + (1−λ)f(y)≤λφ(f(x)) + (1−λ)φ(f(y)).
A differentiable function satisfying (1) also is invex [2] and this is the reason why functions satisfying (1) are called preinvex in [18].
However, the converse is not generally true, in the sense that iff is invex, with respect to some η, it is not necessarily preinvex with respect to η. For example, f(x) = ex, x ∈ R, is invex with respect to η = −1, but it is not preinvex with respect to the same function η. We shall come back to this question in the next section.
Of course, preinvexity is a generalization of convexity: if in (1) we choose η(x, y) =x−y, we obtain the definition of convex functions. In [18] the authors give an eample of a preinvex function that is not convex. A simple condition for the preinvexity of an invex function is given below.
Theorem 6. LetX ⊆Rnbe an open convex set. A function f :X→R invex with respect to some η and concave on X also is preinvex on X with respect to the same η.
Proof. The concavity of the differentiable function f implies (2) f[y+λη(x, y)]−f(y)≤λη(x, y)∇f(y), ∀x, y∈X, ∀λ∈[0,1]. The invexity of f implies η(x, y)∇f(y)≤f(x)−f(y).Asλ≥0, we also have λη(x, y)∇f(y)≤λ(f(x)−f(y)) and, taking (2) into account,
f[y+λη(x, y)]−f(y)≤λ(f(x)−f(y)), i.e.,
f[y+λη(x, y)]−f(y)≤λf(x) + (1−λ)f(y), ∀x, y∈X, ∀λ∈[0,1].
More general conditions, ensuring that a differentiable function invex on a η-invex set X also is preinvex onX, with respect to the same η, are given by Mohan and Neogy [13].
Definition 7 (Condition C). Let η : X ×X → Rn. We say that the function η satisfies Condition C if
η(y, y+λη(x, y)) =−λη(x, y), η(x, y+λη(x, y)) = (1−λ)η(x, y) for every x, y∈X and everyλ∈[0,1].
There are many vector functions that satisfy Condition C besides the trivial case ofη(x, y) =x−y. For example, letX =R\ {0} and
η(x, y) =
x−y ifx≥0, y≥0 x−y ifx≤0, y≤0
−y otherwise.
Then X is an invex set and η satisfies Condition C.
In [20] it is proved that Condition C holds if η(x, y) =x−y+o(kx−yk) The main result of Mohan and Neogy is as follows.
Theorem 8. Suppose that X⊆Rnis an η-invex set and that f :X→R is differentiable on an open set containing X.Further, suppose that f is invex on X with respect to η and η satisfies Condition C. Then f is preinvex with respect to η on X.
Similarly to convex functions, it is possible to characterize preinvex func- tions in terms of invexity of their epigraphs, however not with reference to the same functionη (first of all, one is ann-vector, the other is an (n+ 1)-vector).
Theorem 9. Let f :X→R, where X⊆Rn is an η-invex set. Then f is preinvex with respect to η if and only if the set
epif ={(x, α) :x∈X, α∈R,f(x)≤α}
is an invex set with respect to η1: epif×epif →Rn+1, whereη1((y, β),(x, α)) = (η(y, x), β−α) for all (x, α),(y, β)∈epif.
Proof. Necessity. Let (x, α)∈epif and (x, β)∈epif, that is f(x) ≤α and f(y)≤β. From the preinvexity of f we have
f(y+λη(x, y))≤(1−λ)f(x) +λf(y)≤(1−λ)α+λβ, λ∈[0,1] . It follows that (x+ λη(y, x),(1−λ)α +λβ) ∈ epif, λ ∈ [0,1]. That is (x, α) +λ(η(y, x), β−α) ∈epif, λ∈[0,1]. Hence epi f is an invex set with respect to
η1((y, β),(x, α)) = (η(y, x), β−α).
Sufficiency. Assume that epif is an invex set with respect to η1((y, β),(x, α)) = (η(y, x), β−α).
Let x, y ∈ X and α, β ∈ R such that f(x) ≤ α, f(y) ≤ β. Then (x, α) ∈ epif and (x, β) ∈ epif. From the invexity of the set epi f with respect to η1((y, β),(x, α)) = (η(y, x), β−α), we have
(x, α) +λη1((y, β),(x, α))∈epif, λ∈[0,1]. It follows that
(x+λη(y, x),(1−λ)α+λβ)∈epif, λ∈[0,1],
that is, f(y+λη(x, y) ≤λα+ (1−λ)β.Hence f is a preinvex function with respect to η(x, y) onX.
Another result on preinvex functions that will be used in the next sec- tion is
Theorem 10. Let f : X → R be preinvex. If f has a unique global minimizer at x∗ ∈X,then f is convex at x∗.
Proof. As f is preinvex, there exists η : X×X → Rn such that y+ λη(x, y) ∈ X and λf(x) + (1−λ)f(y) ≥ f(y +λη(x, y)) for any x, y ∈ X, λ∈[0,1].In particular, when x=x∗ and λ= 1,
f(x∗)≥f(y+η(x∗, y)), ∀y ∈X.
Since x∗ is a unique global minimizer (that is,f(x∗)< f(x),∀x∈X,x6=x∗), we have
y+η(x∗, y) =x∗, ∀y∈X, that is, η(x∗, y) =x∗−y,∀y∈X. Thus,
λf(x∗) + (1−λ)f(y)≥f(λx∗+ (1−λ)y), ∀y∈X, λ∈[0,1], hence f is convex atx∗.
In the case of vector-valued functions, f : Rn → Rm is preinvex with respect to η if each component is preinvex with respect to η. So, preinvex vector-valued functions are convexlike. We recall that f :Rn→Rm is said to be convexlike if there exists z∈Rn such that
f(z)≤λf(x) + (1−λ)f(y), ∀λ∈[0,1]. Other properties of preinvex functions are given in [19].
3. RELATIONSHIPS WITH OTHER CLASSES OF GENERALIZED CONVEX FUNCTIONS
We have seen in the previous section that an invex function with respect toηmay not be preinvex with respect to the sameη. However, Mititelu [10,11]
and the authors of the book [16] (the last chapter is specifically taken from paper [11] of Mititelu), state that ifX is an open set, the two classes coincide in case of differentiability off (obviously, with possible different functions η).
Theorem 10 can be used to demonstrate that the invex function (defined on R) in the example below is not preinvex.
Example11. Letf :R→R,f(x) = 1−e−x2;f has a unique global mini- mizer at x∗ = 0, withf0(0) = 0, and is therefore invex (f also is pseudoconvex and strictly pseudoconvex). However, f isnot convex atx∗ and therefore not preinvex. Asx∗ = 0 andf0(x∗) = 0, we haveλf(x∗)+(1−λ)f(y) = (1−λ)f(y) and f(λx∗+ (1−λ)y) =f((1−λ)y). Taking for instance y= 5, λ= 0.5, we have (1−λ)f(y)u0.5< f((1−λ)y)u0.988. Thus (1−λ)f(y)f((1−λ)y),
∀y∈R, sof is not convex atx∗.
We have remarked in Section 2 that there is a strict inclusion of the class of convex functions into the class of preinvex functions. We now investigate the relationships between preinvex functions and pseudoconvex functions.
Theorem 12. There is only a partial overlapping of the class of diffe- rentiable preinvex functions and the class of pseudoconvex functions.
Proof. We prove the theorem by means of two examples.
a) From Example 11 we know that the function f(x) = 1−e−x2 is pseudoconvex (and also strictly pseudoconvex) but not preinvex.
b) We take the opportunity to correct Example 3 of [5]. Let f(x, y) = (x2−y2) defined on X={y≥ −x∩y ≤x}.Consider two points
u= x1
y1
, v= x2
y2
.
Let us show that f is preinvex with respect to η(u, v) = −v. We begin by showing thatX is anη-invex set for thisη, i.e., for eachλ∈[0,1] and each (u, v)∈X the vector
(3) v+λ(−v) =
x2−λx2 y2−λy2
belongs to X. Indeed equation (3) can be rewritten as v+λ(−v) =λ
0 0
+ (1−λ) x2
y2
,
and it remains to note that the points (0,0),(x2, y2) ∈X while X is convex.
It follows that X isη-invex with respect to the chosenη. Now, we have f(v−λv) =f
x2−λx2
y2−λy2
=x22(1−λ)2−y22(1−λ)2
≤λf(u) + (1−λ)f(v) =λ(x21−y12) + (1−λ)(x22−y22).
We obtain
−(x22−y22)(1−λ)≤(x21−y21),
which is always verified for eachλ∈[0,1] and eachu, v∈X. So f is preinvex on X.
We now show thatf is not pseudoconvex. Consider, e.g., the points (2,1) and (2,0) for whichf(2,1) = 3< f(2,0) = 4, but (0,−1)∇f(2,0) = 0, sof is not pseudoconvex.
We also note that there is a partial overlapping of the class of differen- tiable preinvex functions and the class of differentiable quasiconvex functions.
Indeed, as already pointed out, there are (differentiable) quasiconvex functions that are not invex, and therefore not preinvex. Moreover, a differentiable func- tion, both preinvex (and therefore invex) and quasiconvex is pseudoconvex.
More precisely, an invex function is quasiconvex if and only if it also is pseu- doconvex [4].So, on account of Example b) in the proof of Theorem 12, we can assert the existence of differentiable preinvex functions (therefore invex) which are not quasiconvex.
Bector and Singh [1] introduced the class of B-vex functions, as a gener- alization of convex functions.
Definition 13. Let X ⊆ Rn be a convex set. A function f : X → R is said to beB-vex on X with respect tob(x, y, λ) if
f(λx+ (1−λ)y)≤λb(x, y, λ)f(x) + [1−λb(x, y, λ)]f(y)
for allx, y∈Xandλ∈[0,1] whereb:X×X×[0,1]→R+, withλb(x, y, λ)∈ [0,1] for all x, y∈X, λ∈[0,1].
One of the results they obtained (Theorem 2.4 of [1]) is that a B-vex function is quasiconvex, but not necessarily preinvex. Their example is not however consistent, as it refers to a function which is not preinvex with respect to the particular choice η =x−y (i.e., a function which is not convex), and not with respect toanychoice ofη. However, the assertion is true, as the class of B-vex functions coincides with the class of quasiconvex functions.
Theorem 14. The following statements are equivalent.
i)f is B-vex on a convex set X ⊆Rn with respect to a function b.
ii) f is quasiconvex on the convex set X.
Proof. i) ⇒ ii). Since f is B-vex on X, noting that λb ∈ [0,1] for any x, y∈X and λ∈[0,1], we get
f(λx+ (1−λ)y)≤λbf(x) + (1−λb)f(y)≤max{f(x), f(y)}
for any x, y∈X and λ∈[0,1]. This shows that f is quasiconvex on X.
ii) ⇒i). Define b:X×X×[0,1]→R+ as b(x, y, λ) =
( λ−1 λ∈(0,1] and f(x)≥f(y) 0 λ= 0 orf(x)< f(y).
It follows that for allx, y∈Xandλ∈[0,1], we haveλb∈[0,1], λbf(x) + (1− λb)f(y) = max{f(x), f(y)}.So, by the quasiconvexity off onX we have
f(λx+ (1−λ)y)≤λbf(x) + (1−λb)f(y) for all x, y∈X, λ∈[0,1].
On account of the remarks following Theorem 12, we can even assert that it is not true that a preinvex function is necessarily B-vex (i.e. quasiconvex).
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Received 4 January 2008 University of Pavia
Faculty of Economics Pavia, Italy
