European Journal of Operational Research 164 (2005) 115–119

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Continuous Optimization

Criteria for generalized invex monotonicities q

X.M. Yang a,*, X.Q. Yang b, K.L. Teo b

a Department of Mathematics, Chongqing Normal University, Chongqing 400047, Chinab Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PR China

Received 4 April 2003; accepted 24 November 2003

Available online 4 February 2004

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.

� 2003 Elsevier B.V. All rights reserved.

Keywords: Generalized invex monotonicity; Generalized invex functions; Mathematical programming

1. Introduction

Invex functions and invex monotonicities are

interesting topics in the study of generalized con-

vexity. Generalized invexity and invex monoto-nicities 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 mod-

ifications. We also introduce the concept of strong

pseudo-invex monotonicity, and give its necessary

condition.

qThis research was partially supported by the National

Natural Science Foundation of China, and Applied Basic Key

Project Research Foundation of Chongqing.* Corresponding author. Present address: Department of

Applied Mathematics, The Hong Kong Polytechnic University,

Hung Hom, Kowloon, Hong Kong, China.

E-mail address: maxmyang@polyu.edu.hk (X.M. Yang).

0377-2217/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.ejor.2003.11.017

2. Pseudo-invex monotonicity

Let C be a nonempty subset of Rn, g a vector-valued function from C � C into Rn and F a vec-

tor-valued function from C into Rn. Throughoutthe paper, we let h : C ! R be a differentiable

function.

Definition 2.1. A set C is said to be invex with

respect to g if there exists an g : Rn � Rn ! Rn,

such that, for any x; y 2 C, k 2 ½0; 1�,y þ kgðx; yÞ 2 C:

Definition 2.2 (Ref. [1]). Let C of Rn be an invex

set with respect to g. Then, F : C ! Rn is said to be

(strictly) pseudo-invex monotone with respect to gon C of Rn if for every pair of distinct points

x; y 2 C,

gðy; xÞTF ðxÞP 0 implies gðy; xÞT F ðyÞð>ÞP 0:

ed.

116 X.M. Yang et al. / European Journal of Operational Research 164 (2005) 115–119

Definition 2.3 (Ref. [1]). A differentiable function

h on an open invex subset C of Rn is a (strictly)

pseudo-invex function with respect to g on C if for

every pair of distinct points x; y 2 C,

gðy; xÞTrhðxÞP 0 implies hðyÞð>ÞP hðxÞ:In [1], Ruiz-Garz�on et al. gave two theorems as

follows:

Theorem A (see Theorem 4.6 in [1]). Let C be anopen convex subset of Rn. Suppose that:

1. rh : Rn ! Rn is strictly pseudo-invex monotonewith respect to gðy; xÞ > 0 8x; y 2 C,

2. g is a linear function in the first argument andskew,

then h is a strictly pseudo-invex function on C, withrespect to g.

Theorem B (see Theorems 4.7 in [1]). Let C be anopen convex subset of Rn, Suppose that:

1. rh : Rn ! Rn is pseudo-invex monotone with re-spect to gðy; xÞ > 0 8x; y 2 C,

2. g is a linear function in the first argument andskew,

then h is a pseudo-invex function on C, with respectto g.

Remark 2.1. There are two deficiencies of Theo-

rem A or B. That is, (1) assumptions of Theorem

A or B, (a) gðy; xÞ > 0 8x; y 2 C and (b) g is skew.But (a) and (b) are inconsistent; (2) In the process

of proof of Theorem 4.6 or 4.7 of [1],‘‘gðxðkÞ; xÞTrhðxðkÞÞ > 0 and vector gðxðkÞ; xÞ > 0

imply vector rhðxðkÞÞ > 0’’ is an error. For

example, a ¼ ð1; 2Þ, b ¼ ð�1; 1Þ, it is obvious thata > 0 and aTb ¼ 1 > 0, but b� 0.

In [1], Ruiz-Garz�on et al. obtained some suffi-cient conditions of generalized invex monotoni-

city. These results are very interesting. In thissection, we will correct the conditions of Theorem

4.6 or 4.7 in [1] and give necessary conditions of

generalized invex monotonicity under Condition C.

Condition C (Ref. [3]). Let g : X � X ! Rn. Then,

for any x; y 2 Rn and for any k 2 ½0; 1�,

gðy; y þ kgðx; yÞÞ ¼ �kgðx; yÞ;

gðx; y þ kgðx; yÞÞ ¼ ð1� kÞgðx; yÞ:

Remark 2.2. From Condition C, we have

gðy þ �kgðx; yÞ; yÞ ¼ �kgðx; yÞ: ð1Þ

In fact, we easily prove the result as follows:

gðy þ �kgðx; yÞ; yÞ ¼ gðy þ �kgðx; yÞ; y þ �kgðx; yÞþ gðy; y þ �kgðx; yÞÞÞ

¼ �gðy; y þ �kgðx; yÞÞ ¼ �kgðx; yÞ:

The following example shows that Condition C

is different from that g is a linear function in thefirst argument and skew.

Example 2.1. Let

gðx; yÞ ¼

x� y if xP 0; y P 0;

x� y if x6 0; y6 0;�2� y if x > 0; y6 0;2� y if x6 0; y > 0:

8>><>>:

Then, it is easy to verify that g satisfies ConditionC. However, g is not linear function in the first

argument and skew.

Theorem 2.1. Suppose that

1. C of Rn is an open invex set with respect to g;2. g satisfies Condition C;3. for each x 6¼ y, hðyÞ > hðxÞ implies gðx; yÞTr

hðy þ �kgðx; yÞÞ < 0 for some �k 2 ð0; 1Þ;4. rh is pseudo-invex monotone with respect to g

on C.

Then h is a pseudo-invex function with respect to gon C.

Proof. Let x; y 2 C, x 6¼ y be such that

gðx; yÞTrhðyÞP 0: ð2ÞWe need to show that

hðxÞP hðyÞ:Assume the contrary, i.e.,

hðxÞ < hðyÞ: ð3Þ

X.M. Yang et al. / European Journal of Operational Research 164 (2005) 115–119 117

By hypothesis 3,

gðx; yÞTrhðy þ �kgðx; yÞÞ < 0;

for some �k 2 ð0; 1Þ: ð4Þ

It follows from (4) and (1) that

gðy þ �kgðx; yÞ; yÞTrhðy þ �kgðx; yÞÞ < 0;

for some �k 2 ð0; 1Þ: ð5Þ

Since rh is a pseudo-invex monotone with respectto g, (5) implies

gðy þ �kgðx; yÞ; yÞTrhðyÞ < 0 for some �k 2 ð0; 1Þ:ð6Þ

From (5) and �k 2 ð0; 1Þ, (6) becomes

gðx; yÞTrhðyÞ < 0;

which contradicts (2). Hence, h is a pseudo-invexfunction with respect to g. h

Theorem 2.2. Suppose that

1. C of Rn is an open invex set with respect to g;2. g satisfies Condition C;3. for each x 6¼ y, hðyÞP hðxÞ implies gðx; yÞTrh

ðy þ �kgðx; yÞÞ6 0 for some �k 2 ð0; 1Þ;4. rh is strictly pseudo-invex monotone with re-

spect to g on C.

Then h is a strictly pseudo-invex function with re-spect to g on C.

Proof. Let x; y 2 C, x 6¼ y be such that

gðx; yÞTrhðyÞP 0: ð7ÞWe need to show that

hðxÞ > hðyÞ:Assume the contrary, i.e.,

hðxÞ6 hðyÞ: ð8ÞBy hypothesis 3,

gðx; yÞTrhðy þ �kgðx; yÞÞ6 0; for some �k 2 ð0; 1Þ:ð9Þ

From Condition C, we know

gðy; y þ �kgðx; yÞÞ ¼ ��kgðx; yÞ: ð10Þ

It follows from (9) and (10) that

gðy; y þ �kgðx; yÞÞTrhðy þ �kgðx; yÞÞP 0;

for some �k 2 ð0; 1Þ: ð11Þ

Since rh is a strictly pseudo-invex monotone withrespect to g, (11) implies

gðy; y þ �kgðx; yÞÞTrhðyÞ > 0: ð12ÞFrom Condition C and �k 2 ð0; 1Þ, (12) becomes

gðx; yÞTrhðyÞ < 0;

which contradicts (7). Hence, h is a strictly pseudo-invex function with respect to g. h

3. 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 C of Rn be an invex

set with respect to g. Then, F : C ! Rn is said to be

quasi-invex monotone with respect to g on C of Rn

if for every pair of distinct points x; y 2 C,

gðy; xÞTF ðxÞ > 0 implies gðy; xÞT F ðyÞP 0:

Definition 3.2 (Ref. [1]). A differentiable function

h on an open invex subset C of Rn is a quasi-invex

function with respect to g on C if for every pair of

distinct points x; y 2 C,

hðyÞP hðxÞ implies gðx; yÞTrhðyÞ6 0:

Theorem 3.1. Suppose that

1. C of Rn is an open invex set with respect to g;2. g satisfies Condition C;3. for each x 6¼ y, hðyÞP hðxÞ implies gðx; yÞTrh

ðy þ �kgðx; yÞÞ < 0 for some �k 2 ð0; 1Þ;4. rh is quasi-invexmonotone with respect to g onC.

Then h is a quasi-invex function with respect to gon C.

Proof. Assume that h is not quasi-invex with re-

spect to the same g. Then, there exist x; y 2 C such

that

118 X.M. Yang et al. / European Journal of Operational Research 164 (2005) 115–119

hðyÞP hðxÞ; ð13Þ

but

gðx; yÞTrhðyÞ > 0: ð14Þ

By hypothesis 3 and (13),

gðx; yÞTrhðy þ �kgðx; yÞÞ < 0; for some �k 2 ð0; 1Þ:ð15Þ

From Condition C, we know

gðy; y þ �kgðx; yÞÞ ¼ ��kgðx; yÞ: ð16Þ

It follows from (15) and (16) that

gðy; y þ �kgðx; yÞÞTrhðy þ �kgðx; yÞÞ > 0;

for some �k 2 ð0; 1Þ: ð17Þ

Sincerh is a quasi-invex monotone with respect tog, (17) implies

gðy; y þ �kgðx; yÞÞTrhðyÞP 0; for some �k 2 ð0; 1Þ:ð18Þ

From Condition C and �k 2 ð0; 1Þ, (18) becomes

gðx; yÞTrhðyÞ6 0;which contradicts (14). Hence, h is a quasi-invexfunction with respect to g. h

4. Strong pseudo-invex monotonicity

Finally, we introduce the new concepts of

strong pseudo-invex monotonicity and strongpseudo-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 C of Rn be an invex set with

respect to g. Then, F : C ! Rn is said to be strong

pseudo-invex monotone with respect to g on C ofRn if there exists a scalar b > 0, such that for every

pair of distinct points x; y 2 C,

gðy; xÞTF ðxÞP 0 implies

gðy; xÞT F ðyÞP bkgðy; xÞk:

Definition 4.2. A differentiable function h on an

open invex subset C of Rn is a strong pseudo-invex

function with respect to g on C if there exists a

scalar a > 0, such that for every pair of distinct

points x; y 2 C,

gðy; xÞTrhðxÞP 0 implies

hðyÞP hðxÞ þ akgðy; xÞk:

Theorem 4.1. Suppose that

1. C of Rn is an open invex set with respect to g;2. g satisfies Condition C;3. hðxþ gðy; xÞÞ6 hðyÞ, 8x; y 2 C;4. rh is a continuous strong pseudo-invex monotone

with respect to g on C.

Then, h is a strong pseudo-invex function with re-spect to g on C.

Proof. Let x; y 2 C be such that

gðy; xÞTrhðxÞP 0:

From hypothesis 1 and Condition C, for anyk 2 ð0; 1�,

gðxþ kgðy; xÞ; xÞTrhðxÞP 0:

By the strong pseudo-invex monotonicity of rhwith respect to g, there exists a scalar b > 0,

gðxþ kgðy; xÞ; xÞTrhðxþ kgðy; xÞÞP bkgðxþ kgðy; xÞ; xÞk:

Again from Condition C and k 2 ð0; 1�,

gðy; xÞTrhðxþ kgðy; xÞÞP bkgðy; xÞk: ð19Þ

Let gðkÞ ¼ hðxþ kgðy; xÞÞ. It follows from (19)

that

g0ðkÞP bkgðy; xÞk 8k 2 ð0; 1�:

Integrating the last expression between 0 and 1, we

have

gð1Þ � gð0ÞP bkgðy; xÞk:

That is,

hðxþ gðy; xÞÞ � hðxÞP bkgðy; xÞk:

X.M. Yang et al. / European Journal of Operational Research 164 (2005) 115–119 119

By hypothesis 3, we obtain

hðyÞ � hðxÞP bkgðy; xÞk;

Thus, h is a strong pseudo-invex function with

respect to g on C. h

5. Conclusions

In this paper, we point out some errors ap-

peared 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 be-

tween generalized invex monotonicity and gen-eralized invexity are established.

Acknowledgements

The authors are indebted to two anonymous

referees whose comments and suggestions helped

considerably to improve this paper.

References

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[2] X.M. Yang, X.Q. Yang, K.L. Teo, Generalized invexity and

generalized invariant monotonicity, Journal of Optimization

Theory and Applications 117 (2003) 607–625.

[3] S.R. Mohan, S.K. Neogy, On invex sets and preinvex

functions, Journal of Mathematical Analysis and Applica-

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