In this section, we introduce the concept of pseudoinvexity defined by upper Dini directional derivative and give some properties for this class of pseudoinvexity.
Definition 3.1. (See [18]) The function f : Γ → R is said to be pseudoconvex on convex set Γ if
The following concept of pseudoinvexity is a natural extension for pseudoconvexity.
Definition 3.2. The function f : Γ → R is said to be pseudoinvex with respect to η on the η-invex set Γ if
By Definitions 3.1 and 3.2, it is clear that every pseudoconvex function is pseudoinvex function with respect to η(x, y) = x - y, but the converse is not true.
Examples 3.1 and 3.2 will show that a pseudoinvex function with respect to a given mapping η : Γ × Γ → R n is not necessarily pseudoconvex function.
Example 3.1. Let, and let f : Γ → R, η : Γ × Γ → R2 be functions defined by
where x = (x1, x2) ∈ Γ, y = (y1, y2) ∈ Γ. Then, f is pseudoinvex with respect to the given mapping η. But it is not pseudoconvex, because there exist, , such that, but.
Example 3.2. Let f : R → R, η : R × R → R be defined by
Then, f is pseudoinvex with respect to the given mapping η. But it is not pseudoconvex, because there exist x = -5, y = 4, such that, but f (x) = -5 < - 4 = f (y).
Definition 3.3. (See [19]) The bifunction h : Γ × Rn → R is said to be pseudomonotone on if, for any x, y ∈ Γ,
We extend this pseudomonotonicity to η-pseudomonotonicity.
Definition 3.4. Let Γ × Γ → Rn be a given mapping. The bifunction h : Γ × Rn → R ∪ {± ∞} is said to be η-pseudomonotone on Γ if, for any x, y ∈ Γ,
The above implication is equivalent to the following implication:
Next, we present some properties for pseudoinvexity.
Theorem 3.1. Let f : Γ → R be radially upper semicontinuous on the η-invex set Γ. Suppose that
-
(1)
f is a pseudoinvex function with respect to η on Γ;
-
(2)
f and η satisfy Conditions A and C, respectively;
-
(3)
is subodd in the second argument.
Then, f is a prequasiinvex function with respect to the same η on Γ.
Proof. Suppose, on the contrary, f is not prequasiinvex with respect to η on Γ. Then, ∃ x, y ∈ Γ, , such that
Without loss of generality, let f (y) ≥ f (x), then
(3.1)
From radially upper semicontinuity of f, (3.1) and Condition A, there exists λ* ∈ (0, 1) such that
i.e.,
(3.2)
Hence,
(3.3)
Since η satisfies Condition C and is subodd in the second argument, then (3.3) implies
According to the pseudoinvexity of f, we have
(3.4)
From (3.1) and (3.2), we get
which contradicts (3.4). Hence, f is a prequasiinvex function with respect to η on Γ. ■
Theorem 3.2. Let Γ be an η-invex subset of Rn, and let η : Γ × Γ → Rn be a given mapping. Suppose that
-
(1)
f : Γ → R is radially upper semicontinuous on Γ ;
-
(2)
f and η satisfy Conditions A and C, respectively;
-
(3)
is subodd in the second argument.
Then, f is pseudoinvex with respect to η on Γ if and only ifis η-pseudomonotone on Γ.
Proof. Suppose that f is pseudoinvex with respect to η on Γ. Let x, y ∈ Γ,
(3.5)
In order to show is η-pseudomonotone on Γ, we need to show . Assume, on the contrary,
(3.6)
According to the pseudoinvexity of f, from (3.5), we get f (y) ≥ f (x), from (3.6), we get f (x) ≥ f (y). Hence, f (x) = f (y). By Theorem 3.1, we know that f is also prequasiinvex with respect to the same η on Γ, which implies
Consequently,
which contradicts (3.6).
Conversely, suppose that is η-pseudomonotone on Γ. Let x, y ∈ Γ,
(3.7)
In order to show that f is a pseudoinvex function with respect to η on Γ, we need to show f (x) ≥ f (y). Assume, on the contrary,
(3.8)
According to the mean value Theorem 2.1, , such that
(3.9)
From (3.8) and (3.9), we get
(3.10)
Note that , if , from (3.10), we get , which contradicts (3.7). Hence, . Since is subodd in the second argument, (3.10) implies
(3.11)
It follows from (3.11) and that . By Condition C, we get
From the η-pseudomonotonicity of , we get . Again using the Condition C, we obtain , which contradicts (3.7). ■
Theorem 3.3. Let f : Γ → R be radially upper semicontinuous on the η-invex set Γ. Suppose that
-
(1)
f is a pseudoinvex function with respect to η on Γ;
-
(2)
f and η satisfy Conditions A and C, respectively;
-
(3)
is subodd in the second argument.
Then, f is semistrictly prequasiinvex with respect to the same η on Γ.
Proof. Suppose, on the contrary, f is not a semistrictly prequasiinvex function with respect to η on Γ, then there exist points x, y ∈ Γ with f (x) ≠ f (y) and such that
Without loss of generality, let f (y) > f (x), then
(3.12)
By the mean value Theorem 2.1, ∃λ* ∈ (0, 1), such that
(3.13)
According to Condition C and (3.13), we get . By the suboddity and positively homogeneity of in the second argument, we have
(3.14)
Note that , if λ* = 0, then . From (3.13), we get
Since is subodd and positively homogeneous in the second argument, from Condition C, we have
Consequently, . By the pseudoinvexity of f, , which contradicts (3.12). Therefore, λ* ≠ 0 and λ* ∈ (0, 1). From (3.14), , Remark 2.1 and λ* ∈ (0, 1), we get
By Theorem 3.2, we know is η-pseudomonotone on Γ. Therefore,
From Condition C, the suboddity and positively homogeneity of in the second argument, we obtain
Hence, . From the pseudoinvexity of f, we get , which contradicts (3.12). ■