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A critical view on invexity

The aim of this note is to emphasize the fact that in many papers on invexity published in prestigious journals there are not clear definitions, trivial or not clear statements and wrong proofs. We also point out the unprofessional way of answering readers' questions by some authors. We think that this is caused mainly by the lack of criticism of the invexity community


Introduction
There are a lot of articles written on invexity. A search in MathScieNet gives 350 articles having in their titles the words "invex, invexity, preinvex" (310 in Zentralblatt für Mathematik).
In my opinion, the number of articles devoted to invexity (invex and generalized invex functions) is too big with respect to its importance. In fact, when I read the first time about (classic!?) invex functions, in the differentiable case, I realized that this is another way of saying that any critical (or stationary) point of the function, that is, a point at which the differential is 0, is necessarily a global minimum. Then the notion was generalized to non differentiable functions, but saying the same thing: every critical point (in the sense that 0 is in a certain type of subdifferential at that point) is a global minimum. If one looks to the applications of the results in Ref. 1 we see that they state something like: every local solution is a global one.
Rapidly one had generalizations of the notion: quasiinvex functions, preinvex functions, and so on. The common feature for many papers on invexity and its generalizations is the lack of clarity of the notions and statements of the results and doubtful proofs; when the proofs are correct many of them are trivial. After the first draft of this note was written (and after the change of messages with some authors of the cited papers) I had a closer look to several reviews in Mathematical Reviews and Zentralblatt für Mathematik (referred in Section 4), some of them related to the quoted articles; it seems that the opinions of the reviewers didn't influence the authors of papers on invexity.

About statements and proofs
Let us consider the following text quoted from Ref. 1  "Definition 1.1. See Refs. 1-2. A set K ⊆ R n is said to be invex if there exists a vector function η : R n × R n → R n such that x, y ∈ K, λ ∈ [0, 1] ⇒ y + λη(x, y) ∈ K. Remark 1.1. A convex set is an invex set; i.e., take η(x, y) = x − y. But the converse does not hold." 1 Note on this remark: Of course, the converse does not hold because by Definition 1.1 in Ref. 1 quoted above, any set is invex: just take η(x, y) := 0. The honest definition is: let η : R n × R n → R n be a function. The set K ⊂ R n is said to be η-invex if ...
Or there is another formulation: One says that f is (θ, α)-d invex if there exist θ etc such that f satisfies a certain condition involving θ etc. Of course, correctly is to introduce first θ, α, d and after that to say that f is (θ, α)-d invex if ... Let us quote from Ref. 2: with respect to η and b on R n if and only if its every stationary point is a global minimum (maximum) in R n ." At least two remarks are in order with respect to this statement. The first one: If the statement is true, f is B-(0, r)-invex with respect to η and b (in the sense of Definition 1 in Ref. 2 if and only if f is invex (because, as seen above, the invexity of the differentiable function f : R n → R is equivalent to the fact that every stationary point is a global minimum); so which is the need to introduce B-(0, r)-invexity? The second one: The statement gives the impression that the functions η and b, as well as r ∈ R, are given. Consider n = 1, r = 0, b(x, u) := 1 and η(x, u) := u −1 x 2 − u 2 + sgn u for u = 0, η(x, u) := 0 for u = 0. Taking f (x) = 1 2 x 2 for x ∈ R we see that every stationary point of f (that is, u = 0) is a global minimum, but f is not B-(0, r)-invex with respect to η and b on R. Let us quote also from Definition 8 in Ref. 3: "Let S ⊂ R n be a nonempty invex set with respect to η. A function f : S → R is said to be pre-invex with respect to η if, there exists a vector-valued function η : S × S → R n such that the relation ..."; Definition 9 in Ref. 3 is obtained by changing pre-invex by invex. So, first η is given, and one line below one asks the existence of an η.
I am not used with this kind of text in mathematics. If invexity is not a domain of mathematics it is advisable to say it explicitly. Why is this important? Because we are judged in comparison with other mathematicians for getting jobs, for promotions, for getting grants. Even if one declares that invexity is not a domain of mathematics, this does not change a lot the situation because now one asks for interdisciplinarity.
Why did I ask if invexity is a topic in mathematics? Because I had the impression that the next text quoted from Ref. 4 is not a mathematical text (Ref. 4 appears as being cited 21 times in Google Scholar and 3 times in MathSciNet): "Remark 2.3. We will show that Assumption C holds if η(x, y) = x − y + o(x − y). In fact, the following two equalities hold: (i) η(y, y + λη(x, y)) "Assumption C. See Ref. 6. Let η : X × X → R n . Then, for any x, y ∈ R n and for any λ ∈ [0, 1], η(y, y + λη(x, y)) = −λη(x, y), η(x, y + λη(x, y)) = (1 − λ)η(x, y)." Somewhere it is written that X ⊂ R n ; probably for the authors it is not very important to speak about η(x, y) when x or y is not in X. Let us consider X = R n . (Also note that in To see that the condition η(x, y) = x − y + o(x − y) does not imply Condition C let us consider η : R × R → R be defined by η(x, y) = x − y + (x − y) 2 . Of course η satisfies the hypothesis of the statement in Remark 2.3 of Ref. 4 (above). For λ = 1, the first relation of Condition C is equivalent to each of the next ones: −η(x, y) + (η(x, y)) 2 = −η(x, y), (η(x, y)) 2 = 0, η(x, y) = 0, (x − y)(1 + x − y) = 0, x − y ∈ {0, −1}. So, taking x = 0, y = 2 and λ = 1 one sees that η does not verify Condition C.
In Ref. 5 one finds: Then, it is easy to verify that f is invex with respect to η on K and that f and η satisfy Assumptions A and C. However, f is not convex." The authors seem to not realize that η defined in these two examples is not a function because η(2, 0) gives 2 using the first expression and −2 using the third expression. A possible modification for η defined in Example 2.1 of Ref. 5 could be: Take x, y ∈ R with x > 0 > y and λ ∈ [0, 1] and let us look to the second relation in Condition C. We have that η(x, y) = 2 − y and y ′ : Hence, in such a situation (y < 0 and y + λ(2 − y) ≥ 0) one has η(x, y + λη(x, y)) = (1 − λ)η(x, y) if and only if x = 2. Is it possible to have y < 0, y + λ(2 − y) ≥ 0 and λ ∈ [0, 1]? The answer is YES! Just take λ = 1. Hence, for x = 1, y = −1 and λ = 1 the second relation in Condition C is not verified because In fact an adequate modification of the function η in Example 2.1 from Ref.
The function η defined in this way satisfies indeed Condition C.
Maybe (1) is true whenever Condition C holds, but some additional arguments must be provided.
I do not propose myself to mention all doubtful sentences or statements in articles about invexity, but the majority I had occasion to browse are like that.

About the triviality of results and generalizations
Another problem with invexity is given by the triviality of some results or generalizations. Let us mention some of them found in recent articles published in prestigious journals.
It is well known that for a Gâteaux differentiable function f : D → R with D an open subset of a normed vector space (but X could be a topological vector space), for any (distinct) "Theorem 14. Let S ⊂ R n be a nonempty invex set with respect to η : S × S → R n , and P ab be an arbitrary η-path contained in int S. Moreover, we assume that f : S → R is defined on S and differentiable on int S. Then, for any a, b ∈ S, there exists c ∈ P 0 ab such that the following relation They could constitute easy exercises for students following a first course in analysis.
Another example in this sense is provided by Ref. 6. As seen in the title of Ref. 6, there is some G there. What is it? It is a function defined on a certain set A ⊂ R with values in R which is increasing (s, t ∈ A, s < t ⇒ G(s) < G(t)), and moreover G is differentiable. In fact G is defined on the image of a real-valued function f defined in its turn on an η-invex set X ⊂ R n . (By the way, if A = I f (X) is {0, 1}, what does differentiability of G mean?) One defines G-invex and G-pre-invex functions. Let us quote Definition 3 in Ref. 6: "Definition 3. Let X be a nonempty invex (with respect to η) subset of R n and f : X → R be a differentiable function defined on X. Further, we assume that there exists a differentiable real-valued increasing function G : I f (X) → R. Then f is said to be (strictly) Ginvex at u ∈ X on X with respect to η if there exists a vector-valued function η : X ×X → R n such that, for all x ∈ X (x = u), G(f (x)) − G(f (u)) ≥ G ′ (f (u))∇f (u)η(x, u) (>).
(2) If (2) is satisfied for any u ∈ X then f is G-invex on X with respect to η." Taking into account that for f (Fréchet) differentiable one has ∇h(u) = G ′ (f (u))∇f (u), where h := G • f , the inequality above says that h(x) − h(u) ≥ ∇h(u)η(x, u). Having this inequality for all x, u ∈ X this means that h is invex. So, one can say simply that f is G-invex (at u) if G • f is invex (at u). This simple remark is not made in Ref. 6, but one has (quoted from Ref. 6): "We remark that the G-invexity assumption generalizes a hypothesis of Avriel et al. [6], Avriel [7], Hanson [11] and Antczak [3] for differentiable functions. Thus, the following remarks are true: Remark 5. In the case when η(x, u) = x − u, we obtain a definition of a differentiable G-convex function introduced Avriel et al. [6].
Remark 6. Every invex function with respect to η introduced by Hanson [11] is G-invex with respect to the same function η, where G : I f (X) → R is defined by G(a) ≡ a. The converse result is, in general, not true (see also Remark 13 and Example 14).
Remark 7. Every r-invex function with respect to η introduced by Antczak [1,3] is Ginvex with respect to the same function η, where G : I f (X) → R is defined by G(a) = e ra , where r is any finite real number." (However, note that for r ≤ 0 the function G defined by G(a) = e ra is not increasing.) It is suggestive to quote also the definition a G-pre-invex function (but probably the reader already guesses it): "Definition 9. Let X be a nonempty invex (with respect to η) subset of R n . A function f : X → R is said to be (strictly) G-pre-invex at u on X with respect to η if there exist a continuous real-valued increasing function G : I f (X) → R and a vector-valued function η : (2) is satisfied for any u ∈ X then f is G-pre-invex on X with respect to η." Of course, the author does not (want to) observe that this means that h := G • f is pre-invex at u. What does he obtain in Theorem 10 of Ref. 6? He obtains that f is G-invex provided f and G are differentiable and f is G-pre-invex. I quote from page 646 in Ref. "The converse result is not true in general, that is, there exist G-invex functions with respect to η which are not G-pre-invex with respect to the same function η. To prove the converse theorem the function η should satisfy the following condition C (see [16]). Condition C. ..." Of course one states Theorem 11 and one gives a detailed proof. As a conclusion for paper Ref. 6: The function f is G-"word" if G • f is "word". If an existing result holds for "word" then in Ref. 6 one has a result for G-"word" with detailed proof. And this is published in a prestigious journal. "Let K be a nonempty closed set in a real Hilbert space H. We denote by ., . and . the inner product and norm respectively. Let F : K → H and η(., .) : K × K → R be continuous functions. Let α : K × K → R \ {0} be a bifunction. First of all, we recall the following well-known results and concepts.
Definition 2.1. Let u ∈ K. Then the set K is said to be α-invex at u with respect to η(., .) and α(., .), if, for all u, v ∈ K, t ∈ [0, 1], u + tα(v, u)η(v, u) ∈ K. K is said to be an α-invex set with respect to η and α, if K is α-invex at each u ∈ K. The α-invex set K is also called αη-connected set. Note that the convex set with α(v, u) = 1 and η(v, u) = v − u is an invex set, but the converse is not true." First note that u + tα(v, u)η(v, u) above does not make sense if H = R because u ∈ H and tα(v, u)η(v, u) ∈ R; next, if η(., .) : "The function F on the α-invex set K is said to be α-preinvex with respect to α and η, that is (I say), F is η ′ -preinvex (however, one must take F : K → R as in Ref. "Definition 2.6. The differentiable function F on K is said to be an α-invex function with respect to α and η, if where F ′ (u) is the differential of F at u ∈ K. The concepts of the α-invex and α-preinvex functions have played very important role in the development of convex programming; see [6,7]. Note that for α(v, u) = 1, Definition 2.6 is mainly due to Hanson [1]".
Unfortunately not, even in this case, F is α-invex with respect to α and η iff F is η ′ -invex. What is new and surprising for me is the emphasized text above.

Conclusions
In this note we pointed out that several papers published in prestigious journals contain important drawbacks in the formulation of the notions and in the statements of the results, as well as very serious mistakes in the proofs. Also, there are many trivial generalizations of notions and results. In this sense it is useful to mention that there are several reviews in Mathematical Reviews and Zentralblatt für Mathematik which are concordant with our opinions; let us cite the reviews MR1989930 (  Hadjisavvas) where it is mentioned that "Many other notions and properties introduced in this paper can be derived in the same way from the usual generalized invexity notions that can be found in other papers in the field. When this is not the case, mistakes occur frequently". We also pointed out the unprofessional way some authors answered questions related to their papers. In conclusion we consider that there are too many papers related to invexity, much more that the domain deserves. We consider that the editors of mathematical journals have to pay much more attention when accepting to publish such papers, taking into account at least the lack of criticism in the Invexity Community.