Generalized Differentiable 𝐸-Invex Functions and Their Applications in Optimization

Abstract

The concept of 𝐸-convex function and its generalizations is studied with differentiability assumption. Generalized differentiable 𝐸-convexity and generalized differentiable 𝐸-invexity are used to derive the existence of optimal solution of a general optimization problem.

1. Introduction

𝐸−convex function was introduced by Youness [1] and revised by Yang [2]. Chen [3] introduced Semi-𝐸-convex function and studied some of its properties. Syau and Lee [4] defined 𝐸-quasi-convex function, strictly 𝐸-quasi-convex function and studied some basic properties. Fulga and Preda [5] introduced the class of 𝐸-preinvex and 𝐸-prequasi-invex functions. All the above 𝐸-convex and generalized 𝐸-convex functions are defined without differentiability assumptions. Since last few decades, generalized convex functions like quasiconvex, pseudoconvex, invex, 𝐵-vex, (𝑝,𝑟)-invex, and so forth, have been used in nonlinear programming to derive the sufficient optimality condition for the existence of local optimal point. Motivated by earlier works on convexity and 𝐸-convexity, we have introduced the concept of differentiable 𝐸-convex function and its generalizations to derive sufficient optimality condition for the existence of local optimal solution of a nonlinear programming problem. Some preliminary definitions and results regarding 𝐸-convex function are discussed below, which will be needed in the sequel. Throughout this paper, we consider functions 𝐸∶𝑅^𝑛→𝑅^𝑛, 𝑓∶𝑀→𝑅, and 𝑀 are nonempty subset of 𝑅^𝑛.

Definition 1.1 (see [1]). 𝑀 is said to be 𝐸-convex set if (1−𝜆)𝐸(𝑥)+𝜆𝐸(𝑦)∈𝑀 for 𝑥,𝑦∈𝑀, 𝜆∈[0,1].

Definition 1.2 (see [1]). 𝑓∶𝑀→𝑅 is said to be 𝐸-convex on 𝑀 if 𝑀 is an 𝐸-convex set and for all 𝑥,𝑦∈𝑀 and 𝜆∈[0,1], 𝑓(1−𝜆)𝐸(𝑥)+𝜆𝐸𝑦≤(1−𝜆)𝑓(𝐸(𝑥))+𝜆𝑓𝐸𝑦. (1.1)

Definition 1.3 (see [3]). Let 𝑀 be an 𝐸-convex set. 𝑓 is said to be semi-𝐸-convex on 𝑀 if for 𝑥,𝑦∈𝑀 and 𝜆∈[0,1], 𝑓𝜆𝐸(𝑥)+(1−𝜆)𝐸𝑦≤𝜆𝑓(𝑥)+(1−𝜆)𝑓𝑦. (1.2)

Definition 1.4 (see [5]). 𝑀 is said to be 𝐸-invex with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛 if for 𝑥,𝑦∈𝑀 and 𝜆∈[0,1],  𝐸(𝑦)+𝜆𝜂(𝐸(𝑥),𝐸(𝑦))∈𝑀.

Definition 1.5 (see [6]). Let 𝑀 be an 𝐸-invex set with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛.  Also  𝑓∶𝑀→𝑅 is said to be 𝐸-preinvex with respect to 𝜂 on 𝑀 if for 𝑥,𝑦∈𝑀 and 𝜆∈[0,1], 𝑓𝐸𝑦+𝜆𝜂𝐸(𝑥),𝐸𝑦≤𝜆𝑓(𝐸(𝑥))+(1−𝜆)𝑓𝐸𝑦. (1.3)

Definition 1.6 (see [7]). Let 𝑀 be an 𝐸-invex set with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛. Also  𝑓∶𝑀→𝑅 is said to be semi-𝐸-invex with respect to 𝜂 at 𝑦∈𝑀 if 𝑓𝐸𝑦+𝜆𝜂𝐸(𝑥),𝐸𝑦≤𝜆𝑓(𝑥)+(1−𝜆)𝑓𝑦 (1.4) for all 𝑥∈𝑀 and 𝜆∈[0,1].

Definition 1.7 (see [7]). Let 𝑀 be a nonempty 𝐸-invex subset of 𝑅^𝑛 with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛, 𝐸∶𝑅^𝑛→𝑅^𝑛. Let 𝑓∶𝑀→𝑅 and 𝐸(𝑀) be an open set in 𝑅^𝑛. Also  𝑓 and 𝐸 are differentiable on 𝑀. Then, 𝑓 is said to be semi-𝐸-quasiinvex at 𝑦∈𝑀 if 𝑓(𝑥)≤𝑓𝑦∀𝑥∈𝑀⇒∇𝑓∘𝐸𝑦^𝑇𝜂𝐸(𝑥),𝐸𝑦≤0, (1.5) or ∇𝑓∘𝐸𝑦^𝑇𝜂𝐸(𝑥),𝐸𝑦>0∀𝑥∈𝑀⇒𝑓(𝑥)>𝑓𝑦. (1.6)

Lemma 1.8 (see [1]). If a set 𝑀⊆𝑅^𝑛 is 𝐸-convex, then 𝐸(𝑀)⊆𝑀.

Lemma 1.9 (see [5]). If 𝑀 is 𝐸-invex, then 𝐸(𝑀)⊆𝑀.

Lemma 1.10 (see [5]). If {𝑀_𝑖}_𝑖∈𝐼 is a collection of 𝐸-invex sets and 𝑀_𝑖⊆𝑅^𝑛, for all 𝑖∈𝐼, then ∩_𝑖∈𝐼𝑀_𝑖 is 𝐸-invex.

2. 𝐸-Convexity and Its Generalizations with Differentiability Assumption

𝐸-convexity and convexity are different from each other in several contests. From the previous results on 𝐸-convex functions, as discussed by our predecessors, one can observe the following relations between 𝐸-convexity and convexity. (1)All convex functions are 𝐸-convex but all 𝐸-convex functions are not necessarily convex. (In particular, 𝐸-convex function reduces to convex function in case 𝐸(𝑥)=𝑥 for all x in the domain of 𝐸.) (2)A real-valued function on 𝑅^𝑛 may not be convex on a subset of 𝑅^𝑛, but 𝐸-convex on that set. (3)An 𝐸-convex function may not be convex on a set 𝑀 but 𝐸-convex on 𝐸(𝑀). (4)It is not necessarily true that if 𝑀 is an 𝐸-convex set then 𝐸(𝑀) is a convex set.

In this section we study 𝐸-convex and generalized 𝐸-convex functions with differentiability assumption.

2.1. Some New Results on 𝐸-Convexity with Differentiability

𝐸-convexity at a point may be interpreted as follows.

Let 𝑀 be a nonempty subset of 𝑅^𝑛, 𝐸∶𝑅^𝑛→𝑅^𝑛. A function 𝑓∶𝑀→𝑅 is said to be 𝐸-convex at 𝑥∈𝑀 if 𝑀 is an 𝐸-convex set and 𝑓(𝜆𝐸(𝑥)+(1−𝜆)𝐸(𝑥))≤𝜆𝑓∘𝐸(𝑥)+(1−𝜆)𝑓∘𝐸(𝑥) (2.1) for all 𝑥∈𝑁_𝛿(𝑥) and 𝜆∈[0,1], where 𝑁_𝛿(𝑥) is 𝛿-neighborhood of 𝑥, for small 𝛿>0.

It may be observed that a function may not be convex at a point but 𝐸-convex at that point with a suitable mapping 𝐸.

Example 2.1. Consider 𝑀={(𝑥,𝑦)∈𝑅²∣𝑦≥0}. 𝐸∶𝑅²→𝑅² is 𝐸(𝑥,𝑦)=(0,𝑦) and 𝑓(𝑥,𝑦)=𝑥³+𝑦². Also 𝑓 is not convex at (−1,1). For all (𝑥,𝑦)∈𝑁_𝛿(−1,1),   𝛿>0, and 𝜆∈[0,1],𝑓(𝜆𝐸(𝑥,𝑦)+(1−𝜆)𝐸(−1,1))−𝜆(𝑓∘𝐸)(𝑥,𝑦)−(1−𝜆)(𝑓∘𝐸)(−1,1)=−𝜆(1−𝜆)(𝑦−1)²≤0. Hence, 𝑓 is 𝐸-convex at (−1,1).

Proposition 2.2. Let 𝑀⊆𝑅^𝑛, 𝐸∶𝑅^𝑛→𝑅^𝑛 be a homeomorphism. If 𝑓∶𝑀→𝑅 attains a local minimum point in the neighborhood of 𝐸(𝑥), then it is 𝐸-convex at 𝑥.

Proof. Suppose 𝑓 has a local minimum point in a neighborhood 𝑁_𝜖(𝐸(𝑥)) of 𝐸(𝑥) for some 𝑥∈𝑀,  𝜖>0. This implies 𝑓 is convex on 𝑁_𝜖(𝐸(𝑥)). That is, 𝑓(𝜆𝑧+(1−𝜆)𝐸(𝑥))≤𝜆𝑓(𝑧)+(1−𝜆)𝑓(𝐸(𝑥))∀𝑧∈𝑁_𝜖(𝐸(𝑥)). (2.2) Since 𝐸∶𝑅^𝑛→𝑅^𝑛 is a homeomorphism, so inverse of the neighborhood 𝑁_𝜖(𝐸(𝑥)) is a neighborhood of 𝑥 say 𝑁_𝛿(𝑥) for some 𝛿>0. Hence, there exists 𝑥∈𝑁_𝛿(𝑥) such that 𝐸(𝑥)=𝑧,  𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)). Replacing 𝑧 by 𝐸(𝑥) in the above inequality, we conclude that 𝑓 is 𝐸-convex at 𝑥.

In the above discussion, it is clear that if a local minimum exists in a neighborhood of 𝐸(𝑥), then 𝑓 is 𝐸-convex at 𝑥. But it is not necessarily true that if 𝑓 is 𝐸-convex at 𝑥 then 𝐸(𝑥) is local minimum point. Consider the above example where 𝑓 is 𝐸-convex at (−1,1) but 𝐸(−1,1) is not local minimum point of 𝑓.

Theorem 2.3. Let 𝑀 be an open 𝐸-convex subset of 𝑅^𝑛,  𝑓 and 𝐸 are differentiable functions, and let 𝐸 be a homeomorphism. Then, 𝑓 is 𝐸-convex at 𝑥∈𝑀 if and only if 𝑓∘𝐸(𝑥)≥𝑓∘𝐸(𝑥)+∇𝑓∘𝐸(𝑥)^𝑇(𝐸(𝑥)−𝐸(𝑥)) (2.3) for all 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)) where 𝑁_𝜖(𝐸(𝑥)) is 𝜖-neighborhood of 𝐸(𝑥),  𝜖>0.

Proof. Since 𝑀 is an 𝐸-convex set, by Lemma 1.8, 𝐸(𝑀)⊆𝑀. Also, 𝐸(𝑀) is an open set as 𝐸 is a homeomorphism. Hence, there exists 𝜖>0 such that 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)) for all 𝑥∈𝑁_𝛿(𝑥), 𝛿>0, very small. So, 𝑓 is differentiable on 𝐸(𝑀). Using expansion of 𝑓 at 𝐸(𝑥) in the neighborhood 𝑁_𝜖(𝐸(𝑥)), 𝑓(𝐸(𝑥)+𝜆(𝑧−𝐸(𝑥)))=𝑓(𝐸(𝑥))+𝜆∇𝑓(𝐸(𝑥))^𝑇(𝑧−𝐸(𝑥[𝐸())+𝛼𝑥),𝜆(𝑧−𝐸(𝑥))]𝜆‖𝑧−𝐸(𝑥)‖, (2.4) where 𝑧∈𝑁_𝜖(𝐸(𝑥)) and lim_𝜆→0𝛼[𝐸(𝑥),𝜆(𝑧−𝐸(𝑥))]=0. Since 𝑓 is 𝐸-convex at 𝑥∈𝑀, so for all 𝑥∈𝑁_𝛿(𝑥), 𝜆∈(0,1], 𝑥≠𝑥, 𝜆𝑓∘𝐸(𝑥)−𝑓∘𝐸(𝑥)≥𝑓(𝐸(𝑥)+𝜆(𝐸(𝑥)−𝐸(𝑥)))−𝑓∘𝐸(𝑥). (2.5) Since 𝐸 is a homeomorphism, there exists 𝑥∈𝑁_𝛿(𝑥) such that 𝐸(𝑥)=𝑧. Replacing 𝑧 by 𝐸(𝑥) in (2.4) and using above inequality, we get 𝑓∘𝐸(𝑥)−𝑓∘𝐸(𝑥)≥∇𝑓∘𝐸(𝑥)^𝑇(𝐸(𝑥)−𝐸(𝑥)+𝛼[𝐸(𝑥),𝜆(𝐸(𝑥)−𝐸(𝑥))]‖𝐸(𝑥)−𝐸(𝑥)‖), (2.6) where lim_𝜆→0𝛼[𝐸(𝑥),𝜆(𝐸(𝑥)−𝐸(𝑥))]=0. Hence, (2.3) follows.
The converse part follows directly from (2.4).

It is obvious that if 𝐸(𝑥) is a local minimum point of 𝑓, then ∇(𝑓∘𝐸)(𝑥)=0. The following result proves the sufficient part for the existence of local optimal solution, proof of which is easy and straightforward. We leave this to the reader.

Corollary 2.4. Let 𝑀⊆𝑅^𝑛 be an open 𝐸-convex set, and let 𝑓 be a differentiable 𝐸-convex function at 𝑥. If 𝐸∶𝑅^𝑛→𝑅^𝑛 is a homeomorphism and ∇(𝑓∘𝐸)(𝑥)=0, then 𝐸(𝑥) is the local minimum of 𝑓.

2.2. Some New Results on Generalized 𝐸-Convexity with Differentiability

Here, we introduce some generalizations of 𝐸-convex function like semi-𝐸-convex, 𝐸-invex, semi-𝐸-invex, 𝐸-pseudoinvex, 𝐸-quasi-invex and so forth, with differentiability assumption and discuss their properties.

2.2.1. Semi-𝐸-Convex Function

Chen [3] introduced a new class of semi-𝐸-convex functions without differentiability assumption. Semi-𝐸-convexity at a point may be understood as follows:

𝑓∶𝑀→𝑅 is said to be semi-𝐸-convex at 𝑥∈𝑀 if 𝑀 is an 𝐸-convex set and 𝑓(𝜆𝐸(𝑥)+(1−𝜆)𝐸(𝑥))≤𝜆𝑓(𝑥)+(1−𝜆)𝑓(𝑥) (2.7) for all 𝑥∈𝑁_𝛿(𝑥) and 𝜆∈[0,1], where 𝑁_𝛿(𝑥) is 𝛿-neighborhood of 𝑥.

The following result proves the necessary and sufficient condition for the existence of a semi-𝐸-convex function at a point.

Theorem 2.5. Suppose 𝑓∶𝑀→𝑅 and 𝐸∶𝑅^𝑛→𝑅^𝑛 are differentiable functions. Let 𝐸 be a homeomorphism and let 𝑥 be a fixed point of 𝐸. Then, 𝑓 is semi-𝐸-convex at 𝑥∈𝑀 if and only if 𝑓(𝑥)≥𝑓(𝑥)+∇𝑓∘𝐸(𝑥)^𝑇(𝐸(𝑥)−𝐸(𝑥)) (2.8) for all 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)), very small 𝜖>0.

Proof. Proceeding as in Theorem 2.3, we get the following relation from the expansion of 𝑓 at 𝐸(𝑥) in the neighborhood 𝑁_𝜖(𝐸(𝑥)), where 𝑥 is the fixed point of 𝐸. (Since 𝐸 is a homeomorphism, there exists 𝜖>0 such that 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)) for all 𝑥∈𝑁_𝛿(𝑥), very small 𝛿>0): 𝑓(𝐸(𝑥)+𝜆(𝐸(𝑥)−𝐸(𝑥)))=𝑓∘𝐸(𝑥)+𝜆∇𝑓∘𝐸(𝑥)^𝑇(𝐸(𝑥)−𝐸(𝑥[𝐸())+𝛼𝑥),𝜆(𝐸(𝑥)−𝐸(𝑥))]𝜆‖𝐸(𝑥)−𝐸(𝑥)‖, (2.9) where 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)),  lim_𝜆→0𝛼[𝐸(𝑥),𝜆(𝐸(𝑥)−𝐸(𝑥))]=0. Since 𝑓 is semi-𝐸-convex at 𝑥∈𝑀, and 𝑥 is a fixed point of 𝐸, so, for all 𝑥∈𝑁_𝛿(𝑥), 𝜆∈(0,1], 𝑥≠𝑥, 𝜆𝑓(𝑥)−𝑓(𝑥)≥𝑓(𝐸(𝑥)+𝜆(𝐸(𝑥)−𝐸(𝑥)))−𝑓∘𝐸(𝑥). (2.10) Using (2.9), the above inequality reduces to 𝑓(𝑥)−𝑓(𝑥)≥∇𝑓∘𝐸(𝑥)^𝑇(𝐸(𝑥)−𝐸(𝑥)+𝛼[𝐸(𝑥),𝜆(𝐸(𝑥)−𝐸(𝑥))]‖𝐸(𝑥)−𝐸(𝑥)‖), (2.11) where lim_𝜆→0𝛼[𝐸(𝑥),𝜆(𝐸(𝑥)−𝐸(𝑥))]=0. Hence Inequality (2.8) follows for all 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)).
Conversely, suppose Inequality (2.8) holds at the fixed point 𝑥 of 𝐸 for all 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)). Using (2.9) and 𝐸(𝑥)=𝑥 in (2.8), we can conclude that 𝑓 is semi-𝐸-convex at 𝑥∈𝑀.

2.2.2. Generalized 𝐸-Invex Function

The class of preinvex functions defined by Ben-Israel and Mond is not necessarily differentiable. Preinvexity, for the differential case, is a sufficient condition for invexity. Indeed, the converse is not generally true. Fulga and Preda [5] defined 𝐸-invex set, 𝐸-preinvex function, and 𝐸-prequasiinvex function where differentiability is not required (Section 1). Chen [3] introduced semi-𝐸-convex, semi-𝐸-quasiconvex, and semi-𝐸-pseudoconvex functions without differentiability assumption. Jaiswal and Panda [7] studied some generalized 𝐸-invex functions and applied these concepts to study primal dual relations. Here, we define some more generalized 𝐸-invex functions with and without differentiability assumption, which will be needed in next section. First, we see the following lemma.

Lemma 2.6. Let 𝑀 be a nonempty 𝐸-invex subset of 𝑅^𝑛 with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛.   Also 𝑓∶𝑀→𝑅 are differentiable on 𝑀. 𝐸(𝑀) is an open set in 𝑅^𝑛. If 𝑓 is 𝐸-preinvex on 𝑀 then (𝑓∘𝐸)(𝑥)≥(𝑓∘𝐸)(𝑦)+(∇(𝑓∘𝐸)(𝑦))^𝑇𝜂(𝐸(𝑥),𝐸(𝑦))forall𝑥,𝑦∈𝑀.

Proof. If 𝐸(𝑀) is an open set, 𝑓 and 𝐸 are differentiable on 𝑀, then 𝑓∘𝐸 is differentiable on 𝑀. From Taylor's expansion of 𝑓 at 𝐸(𝑦) for some 𝑦∈𝑀 and 𝜆>0, 𝑓𝐸𝑦+𝜆𝜂𝐸(𝑥),𝐸𝑦=𝑓∘𝐸𝑦+𝜆∇𝑓∘𝐸𝑦^𝑇𝜂𝐸(𝑥)𝑦𝜂𝐸(𝑥)𝑦𝛼𝐸𝑦𝐸(𝑥)𝑦,,𝐸+𝜆,𝐸,𝜆𝜂,𝐸 (2.12) where 𝐸(𝑥)≠𝐸(𝑦),  lim_𝜆→0𝛼(𝐸(𝑦),𝜆𝜂(𝐸(𝑥),𝐸(𝑦)))=0.
If 𝑓 is 𝐸-preinvex on 𝑀 with respect to 𝜂 (Definition 1.5), then as 𝜆→0⁺, the above inequality reduces to (𝑓∘𝐸)(𝑥)≥(𝑓∘𝐸)(𝑦)+(∇(𝑓∘𝐸)(𝑦))^𝑇𝜂(𝐸(𝑥),𝐸(𝑦)) for all 𝑥,𝑦∈𝑀.

As a consequence of the above lemma, we may define 𝐸-invexity with differentiability assumption as follows.

Definition 2.7. Let 𝑀 be a nonempty 𝐸-invex subset of 𝑅^𝑛 with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛. Also 𝑓∶𝑀→𝑅 are differentiable on 𝑀. 𝐸(𝑀) is an open set in 𝑅^𝑛. Then, 𝑓 is 𝐸-invex on 𝑀 if (𝑓∘𝐸)(𝑥)≥(𝑓∘𝐸)(𝑦)+(∇(𝑓∘𝐸)(𝑦))^𝑇𝜂(𝐸(𝑥),𝐸(𝑦)) for all 𝑥,𝑦∈𝑀.

From the above discussions on 𝐸-invexity and 𝐸-preinvexity, it is true that 𝐸-preinvexity with differentiability is a sufficient condition for 𝐸-invexity. Also a function which is not 𝐸-convex may be 𝐸-invex with respect to some 𝜂. This may be verified in the following example.

Example 2.8. 𝑀={(𝑥,𝑦)∈𝑅²∣𝑥,𝑦>0}, 𝐸∶𝑅²→𝑅² is 𝐸(𝑥,𝑦)=(0,𝑦) and 𝑓∶𝑀→𝑅 is defined by 𝑓(𝑥,𝑦)=−𝑥²−𝑦², and 𝜂𝑥₁,𝑦₁,𝑥₂,𝑦₂=𝑥²₁2𝑥₂,𝑦²₁2𝑦₂,𝑥₂≠0,𝑦₂𝑦≠0,0,²¹2𝑦₂,𝑥₂=0,𝑦₂𝑥≠0,₂₁2𝑥₂,0,𝑥₂≠0,𝑦₂(),=0,0,0otherwise. (2.13)

Definition 2.9. Let 𝑀 be a nonempty 𝐸-invex subset of 𝑅^𝑛 with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛, let 𝐸(𝑀) be an open set in 𝑅^𝑛. Suppose 𝑓 and 𝐸 are differentiable on 𝑀. Then, 𝑓 is said to be 𝐸-quasiinvex on 𝑀 if 𝑓∘𝐸(𝑥)≤𝑓∘𝐸𝑦∀𝑥,𝑦∈𝑀⇒∇𝑓∘𝐸𝑦^𝑇𝜂𝐸(𝑥),𝐸𝑦≤0 (2.14) or ∇𝑓∘𝐸𝑦^𝑇𝜂𝐸(𝑥),𝐸𝑦>0⇒𝑓∘𝐸(𝑥)>𝑓∘𝐸𝑦. (2.15)

A function may not be 𝐸-invex with respect to some 𝜂 but 𝐸-quasiinvex with respect to same 𝜂. This may be justified in the following example.

Example 2.10. Consider 𝑀={(𝑥,𝑦)∈𝑅²∣𝑥,𝑦<0}, 𝐸∶𝑅²→𝑅² is 𝐸(𝑥,𝑦)=(0,𝑦), and 𝑓∶𝑀→𝑅 is 𝑓(𝑥,𝑦)=𝑥³+𝑦³,  𝜂∶𝑅²×𝑅²→𝑅² is 𝜂((𝑥₁,𝑦₁),(𝑥₂,𝑦₂))=(𝑥₁−𝑥₂,𝑦₁−𝑦₂). Now for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑀, (𝑓∘𝐸)(𝑥₁,𝑦₁) − (𝑓∘𝐸)(𝑥₂,𝑦₂) − ∇(𝑓∘𝐸)(𝑥₂,𝑦₂)^𝑇𝜂(𝐸(𝑥₁,𝑦₁),𝐸(𝑥₂,𝑦₂)) = 𝑦³₁+𝑦³₂−3𝑦²₂(𝑦₁−𝑦₂), which is not always positive. Hence, 𝑓 is not 𝐸-invex with respect to 𝜂 on 𝑀.

If we assume that (𝑓∘𝐸)(𝑥₁,𝑦₁)≤(𝑓∘𝐸)(𝑥₂,𝑦₂) for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑀, then (∇(𝑓∘𝐸)(𝑥₂,𝑦₂))^𝑇𝜂(𝐸(𝑥₁,𝑦₁),𝐸(𝑥₂,𝑦₂))=3𝑦²₂(𝑦₁−𝑦₂)≤0. Hence, 𝑓 is 𝐸-quasiinvex with respect to same 𝜂 on 𝑀.

Definition 2.11. Let 𝑀 be a nonempty 𝐸-invex subset of 𝑅^𝑛 with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛, let 𝐸(𝑀) be an open set in 𝑅^𝑛. Suppose 𝑓 and 𝐸 are differentiable on 𝑀. Then, 𝑓 is said to be 𝐸-pseudoinvex on 𝑀 if ∇𝑓∘𝐸𝑦^𝑇𝜂𝐸(𝑥),𝐸𝑦≥0∀𝑥,𝑦∈𝑀⇒𝑓∘𝐸(𝑥)≥𝑓∘𝐸𝑦 (2.16) or 𝑓∘𝐸(𝑥)<𝑓∘𝐸𝑦∀𝑥,𝑦∈𝑀⇒∇𝑓∘𝐸𝑦^𝑇𝜂𝐸(𝑥),𝐸𝑦<0. (2.17)

A function may not be 𝐸-invex with respect to some 𝜂 but 𝐸-pseudoinvex with respect to same 𝜂. This can be verified in the following example.

Example 2.12. Consider 𝑀={(𝑥,𝑦)∈𝑅²∣𝑥,𝑦>0}. 𝐸∶𝑅²→𝑅² is 𝐸(𝑥,𝑦)=(0,𝑦) and 𝑓∶𝑀→𝑅 is 𝑓(𝑥,𝑦)=−𝑥²−𝑦². For 𝜂∶𝑅²×𝑅²→𝑅², defined by 𝜂((𝑥₁,𝑦₁),(𝑥₂,𝑦₂))=(𝑥₁−𝑥₂,𝑦₁−𝑦₂), and for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑀, 𝑦₁≠𝑦₂, (𝑓∘𝐸)(𝑥₁,𝑦₁)−(𝑓∘𝐸)(𝑥₂,𝑦₂)−∇(𝑓∘𝐸)(𝑥₂,𝑦₂)^𝑇𝜂(𝐸(𝑥₁,𝑦₁),𝐸(𝑥₂,𝑦₂))=−(𝑦₂−𝑦₁)²<0. Hence, 𝑓 is not 𝐸-invex with respect to 𝜂 on 𝑀. If ∇(𝑓∘𝐸)(𝑥₂,𝑦₂))^𝑇𝜂(𝐸(𝑥₁,𝑦₁),𝐸(𝑥₂,𝑦₂))≥0, then (𝑓∘𝐸)(𝑥₁,𝑦₁)−(𝑓∘𝐸)(𝑥₂,𝑦₂)−𝑓(0,𝑦₁)−𝑓(0,𝑦₂)=(𝑦₂+𝑦₁)(𝑦₂−𝑦₁)≥0. Hence, 𝑓 is 𝐸-pseudoinvex with respect to 𝜂 on 𝑀.

If a function 𝑓∶𝑀→𝑅 is semi-𝐸-invex with respect to 𝜂 at each point of an 𝐸-invex set 𝑀, then 𝑓 is said to be semi-𝐸-invex with respect to 𝜂 on 𝑀. Semi-𝐸-invex functions and some of its generalizations are studied in [7]. Here, we discuss some more results on generalized semi-𝐸-invex functions.

Proposition 2.13. If 𝑓∶𝑀→𝑅 is semi-𝐸-invex on an 𝐸-invex set 𝑀, then 𝑓(𝐸(𝑦))≤𝑓(𝑦) for each 𝑦∈𝑀.

Proof. Since 𝑓 is semi-𝐸-invex on 𝑀⊆𝑅^𝑛 and 𝑀 is an 𝐸-invex set so for 𝑥,𝑦∈𝑀 and 𝜆∈[0,1], we have 𝐸(𝑦)+𝜆𝜂(𝐸(𝑥),𝐸(𝑦))∈𝑀 and 𝑓(𝐸(𝑦)+𝜆𝜂(𝐸(𝑥),𝐸(𝑦)))≤𝜆𝑓(𝑥)+(1−𝜆)𝑓(𝑦). In particular, for 𝜆=0,  𝑓(𝐸(𝑦))≤𝑓(𝑦)foreach𝑦∈𝑀.

An 𝐸-invex function with respect to some 𝜂 may not be semi-𝐸-invex with respect to same 𝜂 may be verified in the following example.

Example 2.14. Consider the previous example where 𝑀={(𝑥,𝑦)∈𝑅²∣𝑥,𝑦<0}, 𝐸∶𝑅²→𝑅² is 𝐸(𝑥,𝑦)=(0,𝑦) and 𝑓∶𝑀→𝑅 is 𝑓(𝑥,𝑦)=𝑥³+𝑦³,  𝜂∶𝑅²×𝑅²→𝑅² is 𝜂((𝑥₁,𝑦₁),(𝑥₂,𝑦₂))=(𝑥₁−𝑥₂,𝑦₁−𝑦₂). It is verified that 𝑓 is 𝐸-invex with respect to 𝜂 on 𝑀. But 𝑓(𝐸(2,0))>𝑓(2,0). From Proposition 2.13 it can be concluded that 𝑓 is not semi-𝐸-invex with respect to same 𝜂. Also, using Definition 1.6, for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑀,  𝜆∈[0,1],𝑓(𝐸(𝑥₂,𝑦₂)+𝜆𝜂(𝐸(𝑥₁,𝑦₁),𝐸(𝑥₂,𝑦₂)))−𝜆𝑓(𝑥₁,𝑦₁)−(1−𝜆)𝑓(𝑥₂,𝑦₂)=−(𝑦₂+𝜆𝑦²₁/2𝑦₂)²+𝜆(𝑥²₁+𝑦²₁)+(1−𝜆)(𝑥²₂+𝑦²₂), which is not always negative. Hence, 𝑓 is not semi-𝐸-invex with respect to 𝜂 on 𝑀.

3. Application in Optimization Problem

In this section, the results of previous section are used to derive the sufficient optimality condition for the existence of solution of a general nonlinear programming problem. Consider a nonlinear programming problem (𝑃)min𝑓(𝑥)subjectto𝑔(𝑥)≤0, (3.1) where 𝑓∶𝑀→𝑅, 𝑔_𝑖∶𝑀→𝑅^𝑚, 𝑀⊆𝑅^𝑛,𝑔=(𝑔₁,𝑔₂,…,𝑔_𝑚)^𝑇.  𝑀^′={𝑥∈𝑀∶𝑔_𝑖(𝑥)≤0,𝑖=1,…,𝑚} is the set of feasible solutions.

Theorem 3.1 (sufficient optimality condition). Let 𝑀 be a nonempty open 𝐸-convex subset of 𝑅^𝑛,𝑓∶𝑀→𝑅,  𝑔∶𝑀→𝑅^𝑚, and 𝐸∶𝑅^𝑛→𝑅^𝑛 are differentiable functions. Let 𝐸 be a homeomorphism and let 𝑥 be a fixed point of 𝐸. If 𝑓 and 𝑔 are semi-𝐸-convex at 𝑥∈𝑀^′ and (𝑥,𝑦)∈𝑀^′×𝑅^𝑚 satisfies ∇(𝑥)+(𝑥𝑥𝑓∘𝐸𝑦,𝑔∘𝐸)=0,𝑦,𝑔∘𝐸()=0,𝑦≥0, (3.2) then 𝑥 is local optimal solution of (𝑃).

Proof. Since 𝑓 and 𝑔 are semi-𝐸-convex at 𝑥∈𝑀 by Theorem 2.5, 𝑓(𝑥)−𝑓(𝑥)≥∇𝑓∘𝐸(𝑥)^𝑇(𝐸(𝑥)−𝐸(𝑥))∀𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥,𝑔(𝑥)())−𝑔𝑥)≥∇𝑔∘𝐸(𝑥)^𝑇(𝐸(𝑥)−𝐸(𝑥))∀𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)). (3.3) Adding the above two inequalities, we have 𝑓(𝑥)−𝑓(𝑥)+𝑦^𝑇𝑔(𝑥)−𝑔(𝑥)≥∇𝑓∘𝐸(𝑥)+𝑦^𝑇𝑔∘𝐸(𝑥)^𝑇(𝐸(𝑥)−𝐸(𝑥)). (3.4)
If (3.2) hold, then 𝑓(𝑥)−𝑓(𝑥)+𝑦^𝑇𝑔(𝑥)≥0forall𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)). Since 𝑔(𝑥)≤0 for all 𝑥∈𝑀^′ and 𝑦≥0, so 𝑦^𝑇𝑔(𝑥)≤0. Hence, 𝑓(𝑥)−𝑓(𝑥)≥0 for all 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)). Since 𝐸 is a homeomorphism, there exists 𝛿>0 such that 𝑥∈𝑁_𝛿(𝑥) for all 𝐸(𝑥)∈𝑁_𝜖(𝐸(𝑥)), which means 𝑓(𝑥)≥𝑓(𝑥)forall𝑥∈𝑁_𝛿(𝑥). Hence, 𝑥 is a local optimal solution of (𝑃).

Also we see that a fixed point of 𝐸 is a local optimal solution of (𝑃) under generalized 𝐸-invexity assumptions.

Lemma 3.2. Let 𝑀 be a nonempty 𝐸-invex subset of 𝑅^𝑛 with respect to some 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛. Let 𝑔_𝑖∶𝑀→𝑅, 𝑖=1,…,𝑚 be semi-𝐸-quasiinvex functions with respect to 𝜂 on 𝑀. Then, 𝑀^′ is an 𝐸-invex set.

Proof. Let 𝑀_𝑖={𝑥∈𝑀∶𝑔_𝑖(𝑥)≤0},𝑖=1,…,𝑚.  𝑀^′=∩^𝑚_𝑖=1𝑀_𝑖 and 𝑀^′⊆𝑀. Since 𝑔_𝑖, 𝑖=1,…,𝑚 are semi-𝐸-quasiinvex function on 𝑀, so for all 𝑥,𝑦∈𝑀_𝑖 and 𝜆∈[0,1],𝑔_𝑖(𝐸(𝑦)+𝜆𝜂(𝐸(𝑥),𝐸(𝑦)))≤max{𝑔_𝑖(𝑥),𝑔_𝑖(𝑦)}≤0. Hence, 𝐸(𝑦)+𝜆𝜂(𝐸(𝑥),𝐸(𝑦))∈𝑀_𝑖 for all 𝑥,𝑦∈𝑀_𝑖. So 𝑀_𝑖 is 𝐸-invex with respect to same 𝜂. From Lemma 1.10, 𝑀^′=∩^𝑚_𝑖=1𝑀_𝑖 is 𝐸-invex with respect to same 𝜂.

Corollary 3.3. Let 𝑀 be a nonempty 𝐸-invex subset of 𝑅^𝑛 with respect to some 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛. Let 𝑔_𝑖∶𝑀→𝑅,𝑖=1,…,𝑚, be semi-𝐸-quasiinvex functions with respect to 𝜂 on 𝑀. If 𝑥 is a feasible solution of (𝑃), then 𝐸(𝑥) is also a feasible solution of (𝑃).

Proof. Since 𝑥 is a feasible solution of (𝑃), so 𝑥∈𝑀^′⇒𝐸(𝑥)∈𝐸(𝑀^′). Since each 𝑔_𝑖, 𝑖=1,…,𝑚 is semi-𝐸-quasiinvex function on 𝑀, from Lemma 3.2, 𝑀^′ is an 𝐸-invex set. Also 𝐸(𝑀^′)⊆𝑀^′. Hence, 𝐸(𝑥)∈𝑀^′. That is, 𝐸(𝑥) is a feasible solution of (𝑃).

Theorem 3.4 (sufficient optimality condition). Let 𝑀 be a nonempty 𝐸-invex subset of 𝑅^𝑛 with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛. Let 𝐸(𝑀) be an open set in 𝑅^𝑛. Suppose 𝑓∶𝑅^𝑛→𝑅, 𝑔∶𝑅^𝑛→𝑅^𝑚 and 𝐸 are differentiable functions on 𝑀. If 𝑓 is 𝐸-pseudoinvex function with respect to 𝜂 and for 𝑢≥0,𝑢^𝑇𝑔 is semi-𝐸-quasiinvex function with respect to the same 𝜂 at 𝑥∈𝑀^′, where 𝑥 is a fixed point of the map 𝐸 and (𝑥,𝑢)∈𝑀^′×𝑅^𝑚,𝑢≥0 satisfies the following system: ∇𝑓∘𝐸(𝑥)+𝑢,𝑔∘𝐸(𝑥)=0,𝑢,𝑔∘𝐸(𝑥)=0, (3.5)(3.6) then 𝑥 is a local optimal solution of (𝑃).

Proof. Suppose (𝑥,𝑢)∈𝑀^′×𝑅^𝑚 satisfies (3.5) and (3.6). For all 𝑦∈𝑀^′,  𝑔(𝑦)≤0. Also, 𝑢≥0. Hence, 𝑢^𝑇𝑔(𝑦)≤0 for all 𝑦∈𝑀^′. From (3.6), ⟨𝑢,(𝑔∘𝐸)(𝑥)⟩=0, that is, 𝑢^𝑇𝑔(𝐸(𝑥))=0.  𝑥 is a fixed point of 𝐸 that is 𝐸(𝑥)=𝑥. So 𝑢^𝑇𝑔(𝑥)=0. Hence, 𝑢^𝑇𝑔𝑦≤𝑢^𝑇𝑔(𝑥)∀𝑦∈𝑀^′. (3.7) Since 𝑢≥0 and 𝑢^𝑇𝑔 is semi-𝐸-quasiinvex function with respect to 𝜂 at 𝑥, so the above inequality implies ∇𝑢^𝑇𝑔(𝐸(𝑥))𝜂𝐸𝑦,𝐸(𝑥)≤0. (3.8) From (3.5), ∇𝑓(𝐸(𝑥))=−∇𝑢^𝑇𝑔(𝐸(𝑥)). Putting this value in the above inequality, we have ∇𝑓(𝐸(𝑥))𝜂(𝐸(𝑥),𝐸(𝑦))≥0.
𝑓 is 𝐸-pseudoinvex at 𝑥 with respect to 𝜂. Hence, ∇𝑓(𝐸(𝑥))𝜂(𝐸(𝑥),𝐸(𝑦))≥0 implies 𝑓𝐸𝑦≥𝑓(𝐸(𝑥))=𝑓(𝑥)∀𝑦∈𝑀^′. (3.9) Hence, 𝑥 is the optimal solution (𝑃) on 𝐸(𝑀^′).

The following example justifies the above theorem.

Example 3.5. Consider the optimization problem, (𝑃)min−𝑥²−𝑦²subjectto𝑥²+𝑦²−4≤0, (3.10) where 𝑀={(𝑥,𝑦)∈𝑅²∣𝑥,𝑦>0}. 𝐸∶𝑅²→𝑅² is 𝐸(𝑥,𝑦)=(0,𝑦). This is not a convex programming problem. Consider 𝜂∶𝑅²×𝑅²→𝑅² defined by 𝜂((𝑥₁,𝑦₁),(𝑥₂,𝑦₂))=(𝑥₁−𝑥₂,𝑦₁−𝑦₂). Here, 𝑀^′={(𝑥,𝑦)∈𝑀∶𝑥²+𝑦²−4≤0}, and 𝐸(𝑀^′)={(0,𝑦)∶𝑦≥0,𝑦²−4≤0}. The sufficient conditions (7-8) reduce to −2𝑦+𝑢2𝑦=0,𝑢𝑦²−4=0,𝑢≥0, (3.11) whose solution is 𝑦=2,𝑢=1 and 𝐸(0,2)=(0,2).
In Example 2.12, we have already proved that 𝑓(𝑥,𝑦)=−𝑥²−𝑦² is 𝐸-pseudoinvex function with respect to 𝜂. Using Definition 1.7, one can verify that 𝑢𝑔(𝑥,𝑦) is semi-𝐸-quasi-invex with respect to same 𝜂 at (0,2)∈𝑀^′, where 𝑢𝑔(𝑥,𝑦)=𝑥²+𝑦²−4. So (0,2) is the optimal solution of (𝑃) on 𝐸(𝑀^′).

4. Conclusion

𝐸-convexity and its generalizations are studied by many authors earlier without differentiability assumption. Here, we have studied the the properties of 𝐸-convexity, 𝐸-invexity, and their generalizations with differentiable assumption. From the developments of this paper, we conclude the following interesting properties. (1)A function may not be convex at a point but 𝐸-convex at that point with a suitable mapping 𝐸, and if a local minimum of 𝑓 exists in a neighborhood of 𝐸(𝑥), then 𝑓 is 𝐸-convex at 𝑥. But it is not necessarily true that if 𝑓 is 𝐸-convex at 𝑥 then 𝐸(𝑥) is local minimum point. (2)From the relation between 𝐸-invexity and its generalizations, one may observe that a function which is not 𝐸-convex may be 𝐸-invex with respect to some 𝜂 and 𝐸-preinvexity with differentiability is a sufficient condition for 𝐸-invexity. Moreover, a function may not be 𝐸-invex with respect to some 𝜂 but 𝐸-quasi-invex with respect to same 𝜂, a function may not 𝐸-invex with respect to some 𝜂 but 𝐸-pseudoinvex with respect to the same 𝜂 and an 𝐸-invex function with respect to some 𝜂 may not be semi-𝐸-invex with respect to same 𝜂.

Here, we have studied 𝐸-convexity for first-order differentiable functions. Higher-order differentiable 𝐸-convex functions may be studied in a similar manner to derive the necessary and sufficient optimality conditions for a general nonlinear programming problems.