Optimality and Duality with Respect to b-( E , m )-Convex Programming

Noticing that E -convexity, m-convexity and b-invexity have similar structures in their definitions, there are some possibilities to treat these three class of mappings uniformly. For this purpose, the definitions of the (E , m)-convex sets and the b-(E , m)-convex mappings are introduced. The properties concerning operations that preserve the (E , m)-convexity of the proposed mappings are derived. The unconstrained and inequality constrained b-(E , m)-convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the b-(E , m)-convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz–John necessary optimality criteria for nonlinear multi-objective b-(E , m)-convex programming are established. The Wolfe-type symmetric duality theorems under the b-(E , m)-convexity, including weak and strong symmetric duality theorems, are also presented. Finally, we construct two examples in detail to show how the obtained results can be used in b-(E , m)-convex programming.


Introduction
Convexity, as well as generalized convexity, has a vital position in optimality and has many consequences in different aspects of mathematical programming.Because of its significance, researchers made efforts towards generalized convexity.For instance, Youness [1] considered a class of sets as well as a family of mappings named E -convex sets along with E -convex mappings in 1999.Bector and Singh [2] introduced b-vex functions in 1991.Several years later, these kinds of generalized convex mappings aroused plenty of research enthusiasm.For example, Iqbal et al. [3] discussed a new family of sets as well as a new group of mappings named geodesic E -convex sets together with geodesic E -convex mappings, which are defined on a Riemannian manifold.Mishra et al. [4] studied a class of E -b-vex mappings, for which the elementary properties were observed and kinds of interrelations with other mappings were discussed.Syau et al. [5] defined a family of mappings, called E -b-vex mappings, via generalizing b-vex mappings and E -vex mappings.
These studies contribute to the evolution of the generalized convex mappings.However, there are also some unsatisfactory generalizations.For example, Yang [6] pointed out the drawback of the work in [1] by presenting some counterexamples.Therefore, it is urgent to consider wider families of generalized convex mappings and study the optimality conditions for nonlinear generalized convex programming.Recently, some significant results involving the properties of generalized convex mappings, optimality conditions for nonlinear generalized convex programming, and the duality theorems are developed.For example, in [7], the authors studied geodesic sub-b-s-convex mapping on the Riemann manifolds.In [8], the author considered roughly B-invex mappings as well as generalized roughly B-invex mapping.Sufficient optimality criteria for nonlinear programming involving these mappings are also investigated.The class of E -convex set, E -convex and E -quasiconvex mappings are extended to E -invex set, E -preinvex and E -prequasiconvex mappings in [9].Gulati and Verma introduced a pair of nondifferentiable higher-order symmetric dual models in [10].In [11], the authors investigated a class of mapping called geodesic semi-E -b-vex mappings as well as generalized geodesic semi-E -b-vex mappings.To solve E -convex multiobjective nonlinear programming, Megahed and his collaborators presented a combined interactive approach in [12].In addition, to estimate a possible impact on applied sciences, Pitea et al. studied generalized nonconvex multitime multiobjective variational problems [13] as well as its application such as minimizing a vector of functionals of curvilinear integral type [14].
Inspired by the above work and based on the work in [15][16][17], we introduce a new family of generalized convex sets as well as generalized convex mappings, named (E , m)-convex sets and b-(E , m)-convex mappings, and develop some interesting properties of this family of sets and mappings, respectively.Furthermore, we develop duality theorems and the optimality conditions for single objective programming as well as multi-objective programming, which are under the b-(E , m)-convexity.In addition, two detailed examples are provided to depict the results.
To end this section, let us evoke several concepts of generalized convexity.[18], where m ∈ [0, 1], if for all µ, ν ∈ [0, b] and τ ∈ [0, 1], the following holds: Definition 4. Let S be nonempty and convex in R n .The mapping g : S → R is called a b-vex mapping defined on S corresponding to mapping b :

(E, m)-Convex Sets and b-(E, m)-Convex Mappings
Before we present the notion of b-(E , m)-convex mappings, we give the concept of (E , m)-convex set in this section as follows.

Remark 1.
By definition, we can easily check that mE (x) ∈ M for all x ∈ M and some fixed m ∈ [0, 1].
In addition, each convex set M ⊂ R n is an (E , m)-convex set via choosing a mapping E : R n → R n to be the identify mapping and m = 1.Each E -convex set M ⊂ R n is an (E , m)-convex set through choosing m = 1.
To establish the optimal conditions and duality theory concerning b-(E , m)-convexity, we first understand the basic properties of (E , m)-convex set.We have the following proposition without proof.Proposition 1.The following statements are true: Assuming that E : R n → R n is a linear mapping and M 1 , M 2 ⊂ R n are two (E , m)-convex sets, we have that M 1 + M 2 is also an (E , m)-convex set.
Remark 2. If we assume that M 1 and M 2 are two (E , m)-convex sets, then M 1 M 2 is not necessarily an (E , m)-convex set.A counterexample is given below.

Example 1.
Let us consider the mapping E : R 2 → R 2 defined by and two sets The fact that the E -convex, m-convex, and b-vex functions have almost the same constructs invokes us to generalize these different classes of convexity.Now, let us introduce the b-(E , m)-convex function and (E , m)-quasiconvex function as follows.
Next, we explore if the b-(E , m)-convex mapping on the (E , m)-convex sets have some properties that are similar to those of the E -convex mapping.Let us present the first property in the following.Proposition 2. If g is a convex mapping that is defined on the convex set M, then g must be a b-(E , 1)-convex mapping, where E is the identity mapping and b(x, y, τ) ≡ 1.
Remark 4. Proposition 2 states a sufficient condition for g being a b-(E , 1)-convex mapping, but the converse may fail to hold.
To illustrate this fact, let us construct an example as follows.
Example 2. Suppose that g : R → R has the following expression and However, g is not a convex mapping.
Clearly, R is an (E , m)-convex set, and , then the following fact holds.That is, , then obviously we have that To explore the optimal conditions and duality theory on nonlinear b-(E , m)-convex programming as well as multi-objective b-(E , m)-convex programming, we consider the preserving property of b-(E , m)-convex mappings under positive linear combination, taking extremes, and composition.Since the proofs of these properties are straightforward, they are skipped.
Thus, g is an (E , m)-quasiconvex mapping on M.
Theorem 2. Let us assume that g : Noticing that α ≥ 0, the condition g is a b-(E , m)-convex mapping on M yields that ), the following result holds: Proof.According to the Taylor expansion of g, invoking the b-(E , m)-convexity of g yields that Notice that, when τ = 0, m f (E (µ)) = g(mE (µ)).Hence, by dividing the inequality above by τ and letting τ → 0 + , we deduce that which completes the proof of the desired result.

b-(E, m)-Convex Programming
To demonstrate the application of the results established in last section, the following nonlinear programming is considered in this section: where Proof.We prove this result by contradiction.Assume that µ 1 , µ 2 ∈ M, µ 1 = µ 2 are two global optimal solutions to (P ).Thus, g E (µ 1 ) = g E (µ 2 ) .Because g is a nonnegative strictly b-(E , m)-convex mapping, we have that which implies E (µ 1 ) and E (µ 2 ) are not global optimal solutions.This contradiction shows that the global optimal solution to the b-(E , m)-convex programming (P ) must be unique.
Theorem 5. Let us assume that g: ∈ M and the following inequality holds for every µ ∈ M and certain fixed m ∈ [0, 1], then mE ( μ) is the optimal solution to the b-(E , m)-convex programming (P ) corresponding to g on M.
Proof.Because g is a differentiable b-(E , m)-convex mapping, by Theorem 3, for every µ ∈ M, we obtain that At the same time, noticing that b(µ, μ, τ) ≥ 0 and we can conclude that g E (µ) − g mE ( μ) ≥ 0. Hence, mE ( μ) is the optimal solution to the b-(E , m)convex programming (P ).This completes the proof.Now, let us apply the results above to the nonlinear programming with the following inequality constraints: where g : R n → R and h i (i ∈ I) : R n → R are differentiable b-(E , m)-convex and b i -(E , m)-convex functions.For convenience, we denote the feasible set of (CP Proof.Since it is straightforward to prove Theorem 6, we skip it here. The following theorem presents the Karush-Kuhn-Tucker (KKT) sufficient conditions.Theorem 7. Let us assume that g is a differentiable b-(E , m)-convex mapping corresponding to b and h i are differentiable b i -(E , m)-convex corresponding to b i (i ∈ I).Suppose that mE (µ * ) ∈ M E is a KKT point of (CP), namely there are multipliers u i ≥ 0(i ∈ I) satisfying ∇g mE (µ * ) + ∑ i∈I u i ∇h i mE (µ * ) = 0, Then, for the problem (CP), we have a unique optimal solution mE (µ * ).
Proof.For every µ ∈ M E , Therefore, by the b-(E , m)-convexity of h i and Theorem 3, for i ∈ I(µ * ), we obtain that Thus, by using the KKT conditions and multipliers u i ≥ 0, (i ∈ I(µ * )), we deduce that Hence, by Theorem 5, for every µ ∈ M E with µ = µ * , we can conclude that g E (µ) ≥ g mE (µ * ) .This proves that, for the problem (CP), we have a unique optimal solution mE (µ * ) and ends the proof.

Multi-Objective b-(E, m)-Convex Programming
To consider an application of the results developed in Section 2 in multi-objective b-(E , m)-convex programming, let us assume that E : M → M(M ⊂ R n ) is a surjection in this section.Simultaneously, we define the mapping (g • E ) : M → R by (g • E )(µ) = g(E (µ)) for each µ ∈ M.
Consider the multi-objective nonlinear programming as follows: We also consider the b-(E , m)-convex programming corresponding to (MP E ) as follows: where g i • E, i ∈ P and h j • E, j ∈ J are differentiable functions on M.
for every i ∈ P along with inequality holding strictly for at least one i 0 ∈ P.
Theorem 8. Suppose that E : M → M is a surjective mapping.Then, μ being an effective solution to (MP E ) is equivalent to E ( μ) being an effective solution to (MP).
Proof.We omit the proof of Theorem 8 here because it is essentially the same as that of Theorem 3.1 in [19].
Based on Theorem 8, we present the following sufficient optimality conditions for multi-objective b-(E , m)-convex programming (MP) .Theorem 9. (Sufficient optimality condition) Let E : M → M be a surjective mapping and M be an (E , m)-convex set.Suppose that ( μ, τ, η) has the following properties: where τ ∈ R p , η ∈ R q , then mE ( μ) must be an effective solution to (MP).
Proof.On the contrary, if we assume that mE ( μ) is not an effective solution to (MP), then there is Since g i and h j are differentiable b-(E , m)-convex on M, combining with Theorem 3, for every µ ∈ M, we have that and where b i = b i (µ, μ, 0), i ∈ I and b j = b j (µ, μ, 0), j ∈ J .Due to τ > 0, η ≥ 0, from Equations ( 15) and ( 16), for every i ∈ I, j ∈ J , it follows that where b i max = max{b i |i ∈ P } and b j max = max{b j |j ∈ J }.By the conditions τ∇g mE ( μ) + η∇h mE ( μ) = 0, ηh mE ( μ) = 0 and h mE ( μ) ≤ 0, we get which is an contradiction to Equation ( 14).Thus, mE ( μ) has to be an effective solution to (MP).
Now, we are ready to give the following result which builds a bridge connecting the scalar with multi-objective nonlinear programming.
Theorem 10.The point μ ∈ E (M) being an effective solution to (MP E ) is equivalent to μ solving Proof.Assume contrarily that μ does not solve the problem (MP E ) k , then there exists a µ ∈ E (M) satisfying Thus, we have that μ does not solve the problem (MP E ) either.Conversely, if μ solves the problem (MP E ) k for each k ∈ P, then for any µ ∈ E (M) with the property that (g where the inequality should hold strictly for at least one i.This implies that μ is not a solution of (MP E ) k .The proof is complete.
Avoiding of loss of generality, let us assume P ∩ J = ∅.Setting and T = P k ∪ J , it is clear that (MP E ) k can be rewritten as: for each k ∈ P.
To establish the necessary optimal conditions, the following theorem is developed.We skip the proof due to Mangasarian in [20].
Theorem 11.Assume that μ ∈ E (M) is a local solution to (MP E ).Let g k • E, for each k ∈ I and G t • E and t ∈ T be differentiable functions.Furthermore, let Then, the system of inequalities does not have any solution z ∈ R n for every k ∈ P.
Using Theorem 10 and Theorem 11, we present Fritz-John necessary optimality criteria as follows.
Proof.Because μ is an effective solution to (MP E ), by Theorem 10, μ solves (MP E ) k for every k ∈ I.
Combining with Theorem 11, we get that the system of inequalities does not have any solution z ∈ R n for each k ∈ P. Thus, by Motzin's Theorem in [20], there are τk , τW , τV satisfying Therefore, for every k ∈ I it yields Noticing that μ ∈ E (M), (h • E )( μ) ≤ 0, the proof is finished.

Duality Theorems
As an another application of the results stated in Section 2, the Wolfe duality Theorems of (P E ) under the b-(E , m)-convexity are considered in this section.
Consider the following programming: Similarly, let A = diag(b i , i ∈ I), which yields that From Equation (21), we have that Since u ≥ 0, h(µ) ≤ 0 and b i ≥ 0, from Equations ( 23)-( 25), we get that According to b ≤ b 0 ≤ b and b 0 > 0, dividing both sides of the inequality above by b 0 yields that Hence, g E (µ) ≥ mg E (z) + mu T h E (z) and the proof is completed.(1) µ * is an optimal solution of (CP); and (2) for each (z, u) ∈ M E , t 0 > 0, t i > 0 and t ≤ t 0 ≤ t .
Then, (µ * , u * ) is an optimal solution to (D E ).Moreover, the optimal values to (CP) and (D E ) are the same.
Proof.Invoking Taylor expansion again, in term of the b-(E , m)-convexity of g, we get that Similarly, let H = diag(t i , i ∈ I), we have At the same time, since (µ * , u * ) is a KKT point to (CP), we have that ∇g mE (µ * ) + ∑ i∈I µ * i ∇h i mE (µ * ) = 0 and u * i h i mE (µ * ) = 0, which implies that (µ * , u * ) is a feasible solution to (D E ).Using the inequities in Equations ( 26) and ( 27) and noticing that u ≥ 0, h(E (µ * )) ≤ 0 and t ≤ t 0 ≤ t , we have Thus, in view of t 0 > 0, we obtain that That is to say, (µ * , u * ) is an optimal solution of (D E ).Furthermore, the optimal values of (CP) and those of (D E ) equal to each other.This completes the proof.Corollary 2. Let g and h i (i ∈ I) be differentiable b-(E , m)-convex function on R n corresponding to the same mapping b and (µ * , u * ) be a KKT point to (CP).Assume that (1) µ * is an optimal solution to (P E ); and (2) for each (z, u) ∈ M E , t 0 > 0.
We have that (µ * , u * ) is an optimal solution to (D E ).Moreover, the optimal values to (CP) and (D E ) are the same.(CP) min g(µ)

Proposition 3 .Remark 5 .Example 3 .
Every b-vex mapping g on the convex set M is a b-(E , 1)-convex mapping with E being an identical mapping.Proposition 3 states a sufficient condition for g being a b-(E , 1)-convex mapping, but the converse may fail to hold.A counterexample is given as follows.The mapping g considered in Example 2 is also a b-(E , m)-convex mapping but not a b-vex mapping for all mappings b : R ×

Proposition 5 .Theorem 1 .
are all b-(E , m)-convex mappings related to the mapping E and b on M. If g : M → R + is a b-(E , m)-convex mapping on the (E , m)-convex set M corresponding to the mapping b and ψ : R → R is an increasing homogeneous mapping, then the composite mapping ψ(g) is a b-(E , m)-convex mapping on M. Now, we are ready to give some theorems involving b-(E , m)-convex mappings.Each nonnegative b-(E , m)-convex mapping g on the

Theorem 4 .
Suppose that g : M → R is a nonnegative strictly b-(E , m)-convex mapping on an (E , m)-convex set M, then the global optimal solution to the b-(E , m)-convex programming (P ) is unique.

Theorem 6 .
Suppose that there exist mappings E : R n → R n and b, b i : R n × R n × [0, 1] → R + (i ∈ I) satisfying g and h i (i ∈ I) are b-(E , m)-convex and b i -(E , m)-convex on R n .If g is nonnegative strictly b-(E , m)-convex, then the global optimal solution to the b-(E , m)-convex programming (CP) must be unique.

Theorem 13 .
(Weak Duality Theorem) Suppose that g and h i (i ∈ I) are differentiable b-(E , m)-convex and b i -(E , m)-convex on R n corresponding to mappings b and b i(i ∈ I).If µ ∈ M E , (z, u) ∈ M E and b ≤ b 0 ≤ b , then g E (µ) ≥ mg E (z) + mu T h E (z) is correct for every feasible point x to (CP).Proof.Combining the Taylor expansion of g and the b-(E , m)-convexity of g obtains that

Corollary 1 .
Suppose that g and h i (i ∈ I) are differentiable b-(E , m)-convex mappings on R n corresponding to the same mapping b.If µ ∈ M E , (z, u) ∈ M E , then g E (µ) ≥ mg E (z) + mu T h E (z) holds for each feasible point µ of (P E ).Theorem 14. (Strong Duality Theorem) Let g be differentiable b-(E , m)-convex mapping corresponding to mappings b, h i (i ∈ I) be b i -(E , m)-convex functions on R n corresponding to mappings b i (i ∈ I), and (µ * , u * ) be a KKT point of (CP).Assume that To illustrate the optimality conditions proposed in this paper about b-(E , m)-convex programming and multi-objective b-(E , m)-convex programming, respectively, two examples are constructed in this section.