A Class of Semilocal E-Preinvex Functions and Its Applications in Nonlinear Programming

Definition 2.1 (see [11].)A set K ⊂ Rⁿ is said to be E-convex if there is a map E such that

Definition 2.2 (see [11].)A function f : Rⁿ → R is said to be E-convex on a set K ⊂ Rⁿ if there is a map E such that K is an E-convex set and

Definition 2.3 (see [12].)A function f : Rⁿ → R is said to be semi-E-convex on a set K ⊂ Rⁿ if there is a map E such that K is an E-convex set and

Definition 2.4 (see [15].)A set K ⊂ Rⁿ is said to be E-invex with respect to η if

Definition 2.5 (see [15].)Let K ⊂ Rⁿ be an E-invex set with respect to η. A function f : Rⁿ → R is said to be E-preinvex on K with respect to η if

Definition 2.6 (see [16].)Let K ⊂ Rⁿ be an E-invex set with respect to η. A function f : Rⁿ → R is said to be semi-E-preinvex on K with respect to η if

Definition 2.7 (see [17].)A set K ⊂ Rⁿ is said to be local star-shaped invex with respect to η if for any x, y ∈ K, there is a maximal positive number a(x, y) ≤ 1 satisfying

Definition 2.8 (see [13].)A set K ⊂ Rⁿ is said to be local star-shaped E-convex if there is a map E such that corresponding to each pair of points x, y ∈ K, and there is a maximal positive number a(x, y) ≤ 1 satisfying

Definition 2.9 (see [13].)A function f : Rⁿ → R is said to be semilocal E-convex on a local star-shaped E-convex set K ⊂ Rⁿ if for each pair of x, y ∈ K(with a maximal positive number a(x, y) ≤ 1 satisfying (2.8)), there exists a positive number b(x, y) ≤ a(x, y) satisfying

Definition 2.10 (see [18].)A vector function f : X₀ → R^k is said to be a convex-like function if for any x, y ∈ X₀ ⊂ Rⁿ and 0 ≤ λ ≤ 1, there is z ∈ X₀ such that

where the inequalities are taken component wise.

Lemma 2.11 (see [19].)Let S be a nonempty set in Rⁿ, and let ψ : S → R^k be a convexlike function then either ψ(x) < 0 has a solution x ∈ S or λ^Tψ(x) ≥ 0 for all x ∈ S and some λ ∈ R^k, λ ≥ 0, and λ ≠ 0, but both alternatives are never true.

Definition 3.1. A set K ⊂ Rⁿ is said to be local star-shaped E-invex with respect to a given mapping η if there is a map E such that corresponding to each pair of points x, y ∈ K, and there is a maximal positive number a(x, y) ≤ 1 satisfying

Remark 3.2. Every E-convex set is a local star-shaped E-invex set with respect to η, where η(x, y) = x − y, a(x, y) = 1, for all x, y ∈ Rⁿ. Every local star-shaped E-convex set is a local star-shaped E-invex set with respect to η, where η(x, y) = x − y, for all x, y ∈ Rⁿ. Every E-invex set with respect to η is a local star-shaped E-invex set with respect to η, where a(x, y) = 1, for all x, y ∈ Rⁿ. But their converses are not necessarily true.

Example 3.3. Let K = [−4, −1)⋃[1,4],

We can testify that K is a local star-shaped E-invex set with respect to η.

However, when x₀ = 1, y₀ = 3, there exists a λ₁ ∈ [0,1] such that  λ₁E(x₀)+(1 − λ₁)E(y₀) = −1 + 2λ₁ ∉ K, namely, K is not an E-convex set.

Also, there is a λ₂ ∈ [0,1] such that E(y₀) + λ₂η(E(x₀), E(y₀)) = −1 ∉ K, that is, K is not an E-invex set with respect to η.

Similarly, for any positive number a ≤ 1, there exists a sufficiently small positive number λ₃ ≤ a satisfying λ₃E(x₀)+(1 − λ₃)E(y₀) = −1 + 2λ₃ ∉ K, that is, K is not a local star-shaped E-convex set.

Proposition 3.4. If a set K ⊂ Rⁿ is local star-shaped E-invex with respect to η, then E(K) ⊂ K.

Proof. Since K is local star-shaped E-invex, then for any x, y ∈ K, there exists a maximal positive number a(x, y) ≤ 1 satisfying E(y) + λη(E(x), E(y)) ∈ K, for all λ ∈ [0, a(x, y)].

Thus, for λ = 0, E(y) ∈ K.

Hence, E(K) ⊂ K.

Proposition 3.5. Let E(K) be local star-shaped invex with respect to η, E(K) ⊂ K, then K is local star-shaped E-invex with respect to the same η.

Proof. Assume that x, y ∈ K, then E(x), E(y) ∈ E(K). Since E(K) is local star-shaped invex with respect to η, thus for E(x), E(y) ∈ E(K), there exists a positive number a(E(x), E(y)) ≤ 1 satisfying

Hence, K is local star-shaped E-invex with respect to η.

Remark 3.6. Every local star-shaped invex set with respect to η is local star-shaped E-invex set, where E is an identity map, but its converse is not necessarily true. See the following example.

Example 3.7. Let K₁ = {(x, y) ∈ R² : (x, y) = λ₁(0,0) + λ₂(2,3) + λ₃(0,2)}, K₂ = {(x, y) ∈ R² : (x, y) = λ₁(0,0) + λ₂(−2, −3) + λ₃(0, −4)}, and K = K₁ ∪ K₂, where λ₁, λ₂, λ₃ ≥ 0 and $$. Let η : R² × R² → R² be defined as η(x, y) = x − y, and let E : R² → R² be defined as

It is not difficult to prove that K is a local star-shaped E-invex set with respect to η. However, by taking x = (2,3), y = (0, −4), we know that there exists no maximal positive number a(x, y) ≤ 1 such that y + λη(x, y) ∈ K, for all λ ∈ [0, a(x, y)].

That is, K is not a local star-shaped invex set with respect to η.

Proposition 3.8. Let K_i ⊂ Rⁿ  (i = 1,2, …, m) be a collection of local star-shaped E-invex sets with the same map η, then $$ is a local star-shaped E-invex set with respect to η.

Proof. For all$$, we have x, y ∈ K_i  (i = 1,2, …, m).

Since K_i  (i = 1,2, …, m) are all local star-shaped E-invex sets, then there exist positive numbers a_i(x, y) ≤ 1  (i = 1,2, …, m) such that

Taking a(x, y) = min a_i(x, y), i = 1,2, …, m, we can get Therefore, the proposition is proved.

Remark 3.9. Even if K₁, K₂ are all local star-shaped E-invex set with respect to η, K₁ ∪ K₂ is not necessarily a local star-shaped E-invex set. See the following example.

Example 3.10. Let the map η : R² × R² → R² be defined as η(x, y) = x − y, and the map E : R² → R² be defined as E(x, y) = (x/2, y/2). Consider the two sets

where λ_i ≥ 0, i = 1,2, 3, and $$.

We can easily prove that the two sets K₁, K₂ are all local star-shaped E-invex sets with respect to η. However, when x = (2,0), y = (0, −2), there is not a positive number a(x, y) ≤ 1 such that

Thus, K₁ ∪ K₂ is not a local star-shaped E-invex set with respect to η.

Proposition 3.11. Let K ⊂ Rⁿ be a local star-shaped E₁ and E₂-invex set with respect to the same η, then K is a local star-shaped (E₁∘E₂) and (E₂∘E₁)-invex set with respect to the same η.

Proof. By contradiction, assume that for a pair of x, y ∈ K, for all a(x, y)∈(0,1], there exists a λ₀ ∈ (0, a(x, y)] such that E₁∘E₂(y) + λ₀η((E₁∘E₂(x), E₁∘E₂(y)) ∉ K, that is, E₁(E₂y) + λ₀η((E₁(E₂x), E₁(E₂y)) ∉ K.

Since, from Proposition 3.4, E₂(x), E₂(y) ∈ K, then E₁(E₂y) + λ₀η(E₁(E₂x), E₁(E₂y)) ∉ K contradicts the local star-shaped E₁-invexity of K.

Hence, K is a local star-shaped (E₁∘E₂)-invex set.

Similarly, K is a local star-shaped (E₂∘E₁)-invex set.

Definition 4.1 (see [15].)A function f : Rⁿ → R is said to be local E-preinvex on k ⊂ Rⁿ with respect to η if for any x, y ∈ K (with a maximal positive number a(x, y) ≤ 1 satisfying (3.1)), there exists 0 < b(x, y) ≤ a(x, y) such that K is a local star-shaped E-invex set and

Definition 4.2. A function f : Rⁿ → R is said to be semilocal E-preinvex on k ⊂ Rⁿ with respect to η if for any x, y ∈ K (with a maximal positive number a(x, y) ≤ 1 satisfying (3.1)), there exists 0 < b(x, y) ≤ a(x, y) such that K is a local star-shaped E-invex set and

If the inequality sign above is strict for any x, y ∈ K and x ≠ y, then f is called a strict semilocal E-preinvex function.

A vector function f : Rⁿ → R^k is said to be semilocal E-preinvex on a local star-shaped E-invex set K ⊂ Rⁿ with respect to η if for each pair of points x, y ∈ K(with a maximal positive number a(x, y) ≤ 1 satisfying (3.1)), there exists a positive number b(x, y) ≤ a(x, y) satisfying

where the inequalities are taken component wise.

The definition of strict semilocal E-preinvex of a vector function f : Rⁿ → R^k is similar to the one for a vector semilocal E-preinvex function.

Remark 4.3. Every semilocal E-convex function on a local star-shaped set K is a semilocal E-preinvex function, where η(x, y) = x − y, for all x, y ∈ Rⁿ. Every semi-E-preinvex function with respect to η is a semilocal E-preinvex function, where a(x, y) = b(x, y) = 1, for all x, y ∈ Rⁿ. But their converses are not necessarily true.

Example 4.4. Let the map E : R → R be defined as

and the map η : R × R → R be defined as Obviously, R is a local star-shaped E-convex set and a local star-shaped E-invex set with respect to η. Let f : R → R be defined as

We can prove that f is semilocal E-preinvex on R with respect to η. However, when x₀ = 2, y₀ = 3, and for any b ∈ (0,1], there exists a sufficiently small λ₀ ∈ (0, b] satisfying

That is, f(x) is not a semilocal E-convex function on R.

Similarly, taking x₁ = 1, y₁ = 4, we have

for some λ₁ ∈ [0,1].

Thus, f(x) is not a semi-E-preinvex function on R with respect to η.

Theorem 4.5. Let f : K ⊂ Rⁿ → R be a local E-preinvex function on a local star-shaped E-invex set K with respect to η, then f is a semilocal E-preinvex function if and only if f(E(x)) ≤ f(x), for all x ∈ K.

Proof. Suppose that f is a semilocal E-preinvex function on set K with respect to η, then for each pair of points x, y ∈ K (with a maximal positive number a(x, y) ≤ 1 satisfying (3.1)), there exists a positive number b(x, y) ≤ a(x, y) satisfying

By letting λ = 0, we have f(E(x)) ≤ f(x), for all x ∈ K.

Conversely, assume that f is a local E-preinvex function on a local star-shaped E-invex set K, then for any x, y ∈ K, there exist a(x, y)∈(0,1] satisfying (3.1) and b(x, y)∈(0, a(x, y)] such that

Since f(E(x)) ≤ f(x), for all x ∈ K, then The proof is completed.

Remark 4.6. A local E-preinvex function on a local star-shaped E-invex set with respect to η is not necessarily a semilocal E-preinvex function.

Example 4.7. Let K, E, and η be the same as the ones of Example 3.3 and f : R → R be defined by f(x) = x², then f is local E-preinvex on K with respect to η.

Since f(E(2)) = 16 > f(2) = 4, from Theorem 4.5, it follows that f is not a semilocal E-preinvex function.

Theorem 4.8. Let f : Rⁿ → R be a semilocal E-preinvex function on a local star-shaped E-invex set K ⊂ Rⁿ with respect to η, and let φ : R → R be a nondecreasing and convex function, then φ(f(x)) is semilocal E-preinvex on K with respect to η.

Theorem 4.9. If the functions f_i : Rⁿ → R  (i = 1,2, …, m) are all semilocal E-preinvex on a local star-shaped E-invex set K ⊂ Rⁿ with respect to the same η, then the function $$ is semilocal E-preinvex on K with respect to η for all a_i ≥ 0, i = 1,2, …, m.

Proof. Since K is a local star-shaped E-invex set with respect to η, then for all x, y ∈ K, there exists a positive number a(x, y) ≤ 1 such that

On the other hand, f_i  (i = 1,2, …, m) are all semilocal E-preinvex on K with respect to the same η; thus, there exist positive numbers b_i(x, y) ≤ a(x, y) such that Now, letting b(x, y) = min b_i(x, y),   i = 1,2, …, m, we have That is, f(x) is semilocal E-preinvex on Kwith respect to η.

Definition 4.10. The set G = {(x, α) : x ∈ K ⊂ Rⁿ, α ∈ R} is said to be a local star-shaped E-invex set with respect to η corresponding to Rⁿ if there are two maps η, E and a maximal positive number a((x, α₁), (y, α₂)) ≤ 1, for each (x, α₁), (y, α₂) ∈ G such that

Theorem 4.11. Let K ⊂ Rⁿ be a local star-shaped E-invex set with respect to η, then f is a semilocal E-preinvex function on K with respect to η if and only if its epigraph G_f = {(x, α) : x ∈ K, f(x) ≤ α, α ∈ R} is a local star-shaped E-invex set with respect to η corresponding to Rⁿ.

Proof. Assume that f is semilocal E-preinvex on K with respect to η and (x, α₁), (y, α₂) ∈ G_f, then x, y ∈ K, and f(x) ≤ α₁, f(y) ≤ α₂. Since K is a local star-shaped E-invex set, there is a maximal positive number a(x, y) ≤ 1 such that

In addition, in view of f being a semilocal E-preinvex function on K with respect to η, there is a positive number b(x, y) ≤ a(x, y) such that That is, (E(y) + λη(E(x), E(y)), λα₁ + (1 − λ)α₂) ∈ G_f, for all λ ∈ [0, b(x, y)].

Therefore, G_f = {(x, α) : x ∈ K, f(x) ≤ α, α ∈ R} is a local star-shaped E-invex set with respect to η corresponding to Rⁿ.

Conversely, if G_f is a local star-shaped E-invex set with respect to η corresponding to Rⁿ, then for any points (x, f(x)), (y, f(y)) ∈ G_f, there is a maximal positive number a((x, f(x)), (y, f(y))) ≤ 1 such that

That is, E(y) + λη(E(x), E(y)) ∈ K, Thus, K is a local star-shaped E-invex set, and f is a semilocal E-preinvex function on K with respect to η.

Theorem 4.12. If f is a semilocal E-preinvex function on a local star-shaped E-invex set K ⊂ Rⁿ with respect to η, then the level set S_α = {x ∈ K : f(x) ≤ α} is a local star-shaped E-invex set for any α ∈ R.

Proof. For any α ∈ R and x, y ∈ S_α, then x, y ∈ K and f(x) ≤ α, f(y) ≤ α. Since K is a local star-shaped E-invex set, there is a maximal positive number a(x, y) ≤ 1 such that

In addition, due to the semilocal E-preinvexity of f, there is a positive number b(x, y) ≤ a(x, y) such that That is, E(y) + λη(E(x), E(y)) ∈ S_α, for all λ ∈ [0, b(x, y)].

Therefore, S_α is a local star-shaped E-invex set with respect to η for any α ∈ R.

Theorem 4.13. Let f be a real-valued function defined on a local star-shaped E-invex set K ⊂ Rⁿ, then f is a semilocal E-preinvex function with respect to η if and only if for each pair of points x, y ∈ K (with a maximal positive number a(x, y) ≤ 1 satisfying (3.1)), there exists a positive number b(x, y) ≤ a(x, y) such that

whenever f(x) < α, f(y) < β.

Proof. Let x, y ∈ K and α, β ∈ R such that f(x) < α, f(y) < β. Due to the local star-shaped E-invexity of K, there is a maximal positive number a(x, y) ≤ 1 such that

In addition, owing to the semilocal E-preinvexity of f, there is a positive number b(x, y) ≤ a(x, y) such that

Conversely, let (x, α) ∈ G_f, (y, β) ∈ G_f (see epigraph G_f in Theorem 4.11), then x, y ∈ K, f(x) ≤ α, and f(y) ≤ β. Hence, f(x) < α + ϵ and f(y) < β + ϵ hold for any ϵ > 0. According to the hypothesis, for x, y ∈ K (with a positive number a(x, y) ≤ 1 satisfying (3.1)), there exists a positive number b(x, y) ≤ a(x, y) such that

Let ϵ → 0⁺, then That is, (E(y) + λη(E(x), E(y)), λα + (1 − λ)β) ∈ G_f, for all λ ∈ [0, b(x, y)].

Therefore, G_f is a local star-shaped E-invex set corresponding to Rⁿ.

From Theorem 4.11, it follows that f is semilocal E-preinvex on K with respect to η.

Theorem 5.1. The following statements hold for programming (P).

• (i)

  The optimal solution set ω for (P) is a local star-shaped E-invex set with respect to η.

• (ii)

  If x₀ is a local minimum for (P) and E(x₀) = x₀, then x₀ is a global minimum for (P).

• (iii)

  If the real-valued function f is a strict semilocal E-preinvex function on K with respect to η, then the global optimal solution for (P) is unique.

Proof. (i) Assume that x, y ∈ ω, then x, y ∈ K and f(x) = f(y). On account of K being a local star-shaped E-invex set with respect to η, there is a maximal positive number a(x, y) ≤ 1 such that

Besides, due to semilocal E-preinvexity of f, there is a positive number b(x, y) ≤ a(x, y) such that From the optimality of x, we have f(E(y) + λη(E(x), E(y))) ≥ f(x).

Hence, f(E(y) + λη(E(x), E(y))) = f(x), that is, E(y) + λη(E(x), E(y)) ∈ ω, for all λ ∈ [0, b(x, y)].

This shows that ω is a local star-shaped E-invex set with respect to η.

(ii) Assume that N_ϵ(x₀) is a neighbourhood of x₀ with radius ϵ > 0, and f attains its local minimum at x₀ ∈ N_ϵ(x₀)∩K. For x, x₀ ∈ K, there is a maximal positive number a(x, x₀) ≤ 1 such that E(x₀) + λη(E(x), E(x₀)) ∈ K, for all λ ∈ [0, a(x, x₀)].

Moreover, there is a positive number b(x, x₀) ≤ a(x, x₀) such that

Owing to E(x₀) = x₀, so for sufficiently small λ > 0, Thus, f(x₀) ≤ f(E(x₀) + λη(E(x), E(x₀))) ≤ λf(x)+(1 − λ)f(x₀), namely, f(x₀) ≤ f(x).

This means that x₀ is a global minimum for (P).

(iii) By contradiction, assume that x₁, x₂ ∈ K are two global optimal solutions for (P) and x₁ ≠ x₂. For x₁, x₂ ∈ K, there is a maximal positive number a(x₁, x₂) ≤ 1 such that

Furthermore, there is a positive number b(x₁, x₂) ≤ a(x₁, x₂) such that This contradicts the fact that x₁ is a global optimal solution for (P).

Therefore, the global optimal solution for (P) is unique.

Theorem 5.2. Let u ∈ K and E(u) = u. If f is differentiable on K, then u is a minimum for programming (P) if and only if u satisfies the inequality ∇f(u) ^Tη(E(v), u) ≥ 0, for all v ∈ K.

Proof. Assume that u is a minimum for (P). Since K is a local star-shaped E-invex set with respect to η, for any v ∈ K, there is a maximal positive number a(u, v) ≤ 1 such that

From the differentiability of f and E(u) = u, we get Dividing the inequality above by λ and letting λ → 0⁺, we have

Conversely, if ∇f(u) ^Tη(E(v), u) ≥ 0, owing to semilocal E-preinvexity of f on K, there is a positive number b(u, v) ≤ a(u, v) such that

which together with E(u) = u implies Letting λ → 0⁺ in the inequality above, we obtain This follows that u is a minimum for (P).

Theorem 5.3. Assume that f, g_i  (i ∈ I) are all semilocal E-preinvex on K with respect to η, then

• (i)

  K₀ is a local star-shaped E-invex set with respect to η;

• (ii)

  the optimal solution set ω for (NP) is a local star-shaped E-invex set with respect to η;

• (iii)

  if x₀ is a local minimum for (NP) and E(x₀) = x₀, then x₀ is a global minimum for (NP);

• (iv)

  if the real-valued function f is a strict semilocal E-preinvex function on K with respect to η, then the global optimal solution for (NP) is unique.

Lemma 5.4. Let x^* be a local optimal solution for (NP). Assume that g_j is continuous at x^* for any $$, and f, $$ possess the directional derivatives at x^* along the direction η(E(x), x^*) for each x ∈ K, then the system:

has no solution in K, where f′(x^*; d) denotes the directional derivative of f at x^* along the direction d and $$.

Theorem 5.5. Let x^* be a local optimal solution for (NP). Assume that functions g_j are continuous at x^* for any $$, and f, g possess the directional derivatives with respect to η(E(x), x^*) at x^* for each x ∈ K. If $$ is a convex-like function and E(x^*) = x^*, then there are $$, $$ such that

Additionally, if g is a semilocal E-preinvex function on K with respect to η and there is $$ such that $$, then there exists λ^* ∈ R^m such that

Proof. Define vector function $$. Then ψ(x) is a convexlike function. By Lemma 5.4, the system φ(x) < 0 has no solution in K. Thus, from Lemma 2.11, there are $$ such that

Hence, by letting $$, we further have Subsequently, we testify that $$, and if this is not true, we get from above On account of g being semilocal E-preinvex on K with respect to η and E(x^*) = x^*, from Proposition 5.9(i) below, we have $$.

So $$. But this contradicts the fact that $$ and $$.

Thus, $$. Dividing (5.17) and the first equality of (5.18) by $$ and letting $$, we know that (5.19) and (5.20) hold.

Consequently, the whole proof is finished.

Definition 5.6. A real-valued function f defined on a local star-shaped E-invex set k ⊂ Rⁿ is said to be quasisemilocal E-preinvex (with respect to η) if for all x, y ∈ K (with a maximal positive number a(x, y) ≤ 1 satisfying (3.1)) satisfying f(x) ≤ f(y), there is a positive number b(x, y) ≤ a(x, y) such that

The definition of quasi-semilocal E-preinvex of a vector function f : Rⁿ → R^k is similar to the one for a vector semilocal E-preinvex function.

Definition 5.7. A real-valued function f defined on a local star-shaped E-invex set K ⊂ Rⁿ is said to be pseudosemilocal E-preinvex (with respect to η) if for all x, y ∈ K (with a maximal positive number a(x, y) ≤ 1 satisfying (3.1)) satisfying f(x) < f(y), there are a positive number b(x, y) ≤ a(x, y) and a positive number c(x, y) such that

The definition of pseudo-semilocal E-preinvex of a vector function f : Rⁿ → R^k is similar to the one for a vector semilocal E-preinvex function.

Remark 5.8. Every semilocal E-preinvex function on a local star-shaped E-invex set K with respect to η is both a quasi-semilocal E-preinvex function and a pseudo-semilocal E-preinvex function.

Proposition 5.9. Let f be a real-valued function on a local star-shaped E-invex set K ⊂ Rⁿ, and f possesses directional derivative with respect to the direction η(E(x), y) at y for all x, y ∈ K. If E(y) = y, then the following statements hold true:

• (i)

  if f is semilocal E-preinvex on K with respect to η, then f′(y; η(E(x), y)) ≤ f(x) − f(y),

• (ii)

  if f is quasi-semilocal E-preinvex on K with respect to η, then f(x) ≤ f(y) implies that f′(y; η(E(x), y)) ≤ 0,

• (iii)

  if f is pseudo-semilocal E-preinvex on K with respect to η, then f(x) < f(y) implies that f′(y; η(E(x), y)) < 0.

Theorem 5.10. Let x^* ∈ K₀ and E(x^*) = x^*. Suppose that f, g possess directional derivatives with respect to the direction η(E(x), x^*) at x^* for any x ∈ K, and assume that there is λ^* ∈ R^m such that (5.19) and (5.20) hold. If f is pseudo-semilocal E-preinvex on K and $$ is quasi-semilocal E-preinvex on K with respect to η, then x^* is an optimal solution for (NP).

Proof. Due to $$, for all x ∈ K₀, it follows from Proposition 5.9(ii) that $$, for all x ∈ K₀, which together with $$ implies

Moreover, using λ^*Tg(x^*) = 0, g(x^*) ≤ 0, and λ^* ≥ 0, we get Therefore, from (5.19), we have f^′(x^*; η(E(x), x^*)) ≥ 0, for all x ∈ K₀.

Thus, from Proposition 5.9(iii), this implies f(x) ≥ f(x^*), for all x ∈ K₀.

That is, x^* is an optimal solution for (NP).

Theorem 5.11. Let x^* ∈ K₀ and E(x^*) = x^*. Suppose that f, g possess directional derivatives with respect to the direction η(E(x), x^*) at x^* for any x ∈ K, and there is a λ^* ∈ R^m such that (5.19) and (5.20) hold. If $$ is pseudo-semilocal E-preinvex on K with respect to η, then x^* is an optimal solution for (NP).

Proof. Considering λ^*Tg(x^*) = 0, g(x^*) ≤ 0, λ^* ≥ 0, and the given conditions, we have $$, for all x ∈ K₀. Hence, from Proposition 5.9(iii), we get

The inequality above together with $$ follows: On account of $$ and $$, we obtain Therefore, x^* is an optimal solution for (NP).

Corollary 5.12. Let x^* ∈ K₀ and E(x^*) = x^*. Suppose that f, g possess directional derivatives with respect to the direction η(E(x), x^*) at x^* for any x ∈ K, and assume that there is a λ^* ∈ R^m such that (5.19) and (5.20) hold. If f and $$ are semilocal E-preinvex functions on K with respect to η, then x^* is an optimal solution for (NP).

Theorem 5.13 (weak duality). Let x and (u, λ) be arbitrary feasible solutions of (NP) and (DP), respectively. If f and g are all semilocal E-preinvex functions on K with respect to η, and they possess directional derivatives with respect to the direction η(E(x), u) at u, where E(u) = u, x ∈ K, then f(x) ≥ f(u).

Proof. Considering f and g being semilocal E-preinvex on K with respect to η and E(u) = u, we get from Proposition 5.9(i)

Combining the first constraint condition of (DP) and the inequalities above, we have Hence, on account of λ ≥ 0, g(x) ≤ 0, and λ^Tg(u) ≥ 0, we obtain f(x) ≥ f(u).

Theorem 5.14 (strong duality). Assume that x^* is an optimal solution for (NP), E(x^*) = x^*, and E(u) = u for any feasible point (u, λ) of (DP). Suppose that f and g are semilocal E-preinvex on K with respect to η and g_j is continuous at x^* for any $$, and they possess directional derivatives with respect to the direction η(E(x), x^*) at x^* and the direction η(x^*, u) at u, respectively, where x ∈ K. Further, assume that there is $$ such that $$. If $$ is a convex-like function, then there is a λ^* ∈ R^m such that (x^*, λ^*) is an optimal solution for (DP).

Proof. From the assumptions and Theorem 5.5, we can conclude that there is λ^* ≥ 0 such that (x^*, λ^*) is a feasible point for (DP). Assume that (u, λ) is a feasible solution of (DP). On account of f and g being semilocal E-preinvex on K with respect to η and E(u) = u, we get from Proposition 5.9(i)

Combining the first constraint condition of (DP) and the relationships above, we have Noticing that λ ≥ 0, g(x^*) ≤ 0, and λ^Tg(u) ≥ 0, we know f(x^*) ≥ f(u).

Therefore, (x^*, λ^*) is an optimal solution for (DP).

Theorem 5.15 (converse duality). Suppose that x^* ∈ K₀ and $$ is a feasible point for (DP). Further, suppose that f and g are semilocal E-preinvex on K with respect to η, and f, g possess directional derivatives with respect to the direction $$ at $$ for any x ∈ K. If $$ and $$, then x^* is an optimal solution for (NP).

Proof. Since f and g are semilocal E-preinvex on K with respect to η and $$, we have from Proposition 5.9(i)

On account of $$ being a feasible point for (DP), we get from the first constraint inequality of (DP) and the two relationships above This together with $$, g(x) ≤ 0, $$, and $$ follows that Thus, x^* is an optimal solution for (NP).