Continuous Optimization
Criteria for generalized invex monotonicities☆

https://doi.org/10.1016/j.ejor.2003.11.017Get rights and content

Abstract

In this paper, under appropriate conditions, we establish that (i) if the gradient of a function is (strictly) pseudo-monotone, then the function is (strictly) pseudo-invex; (ii) if the gradient of a function is quasi-monotone, then the function is quasi-invex; and (iii) if the gradient of a function is strong pseudo-monotone, then the function is strong pseudo-invex.

Introduction

Invex functions and invex monotonicities are interesting topics in the study of generalized convexity. Generalized invexity and invex monotonicities have been investigated in [1], [2]. However, it is noted that some necessary conditions are not correct in [1]. The purpose of this note is to point out these errors and to suggest appropriate modifications. We also introduce the concept of strong pseudo-invex monotonicity, and give its necessary condition.

Section snippets

Pseudo-invex monotonicity

Let Γ be a nonempty subset of Rn, η a vector-valued function from Γ×Γ into Rn and F a vector-valued function from Γ into Rn. Throughout the paper, we let θ:Γ→R be a differentiable function.

Definition 2.1

A set Γ is said to be invex with respect to η if there exists an η:Rn×Rn→Rn, such that, for any x,y∈Γ, λ∈[0,1],y+λη(x,y)∈Γ.

Definition 2.2

Ref. [1]

Let Γ of Rn be an invex set with respect to η. Then, F:Γ→Rn is said to be (strictly) pseudo-invex monotone with respect to η on Γ of Rn if for every pair of distinct points x,y∈Γ,η(y,x)T

Quasi-invex monotonicity

In [1], necessary conditions of quasi-invex monotonicity are not discussed. Now we give a result on the aspect.

Definition 3.1

Ref. [1]

Let Γ of Rn be an invex set with respect to η. Then, F:Γ→Rn is said to be quasi-invex monotone with respect to η on Γ of Rn if for every pair of distinct points x,y∈Γ,η(y,x)TF(x)>0impliesη(y,x)TF(y)⩾0.

Definition 3.2

Ref. [1]

A differentiable function θ on an open invex subset Γ of Rn is a quasi-invex function with respect to η on Γ if for every pair of distinct points x,y∈Γ,θ(y)⩾θ(x)impliesη(x,y)T∇θ(y)⩽0.

Theorem 3.1

Strong pseudo-invex monotonicity

Finally, we introduce the new concepts of strong pseudo-invex monotonicity and strong pseudo-invexity which are the modifications of corresponding definitions in [1]. We will give a necessary condition for strong pseudo-invex monotonicity.

Definition 4.1

Let Γ of Rn be an invex set with respect to η. Then, F:Γ→Rn is said to be strong pseudo-invex monotone with respect to η on Γ of Rn if there exists a scalar β>0, such that for every pair of distinct points x,y∈Γ,η(y,x)TF(x)⩾0impliesη(y,x)TF(y)⩾β∥η(y,x)∥.

Definition 4.2

A

Conclusions

In this paper, we point out some errors appeared in [1] and give appropriate modifications. We have also defined the concepts of strong pseudo-invex monotone mapping and strong pseudo-invex function. Some new relations between generalized invex monotonicity and generalized invexity are established.

Acknowledgements

The authors are indebted to two anonymous referees whose comments and suggestions helped considerably to improve this paper.

References (3)

There are more references available in the full text version of this article.

Cited by (0)

☆

This research was partially supported by the National Natural Science Foundation of China, and Applied Basic Key Project Research Foundation of Chongqing.

View full text
Copyright © 2003 Elsevier B.V. All rights reserved.