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J. Appl. Math. & Computing Vol. 16(2004), No. 1 - 2, pp. 371 - 381

FUZZY METRIC SPACES

ZUN-QUAN XIA AND FANG-FANG GUO∗

Abstract. In this paper, fuzzy metric spaces are redefined, different from

the previous ones in the way that fuzzy scalars instead of fuzzy numbers

or real numbers are used to define fuzzy metric. It is proved that every

ordinary metric space can induce a fuzzy metric space that is complete

whenever the original one does. We also prove that the fuzzy topology

induced by fuzzy metric spaces defined in this paper is consistent with the

given one. The results provide some foundations for the research on fuzzy

optimization and pattern recognition.

AMS Mathematics Subject Classification: 03E72, 90C70, 15A03.

Key words and phrases : Fuzzy metric space, completeness of fuzzy metric

space, fuzzy topology, fuzzy closed set.

1. Introduction

How to define fuzzy metric is one of the fundamental problems in fuzzy mathe-

matics which is wildly used in fuzzy optimization and pattern recognition. There

are two approaches in this field till now. One is using fuzzy numbers to define

metric in ordinary spaces, firstly proposed by Kaleva (1984)[12], following which

fuzzy normed spaces, fuzzy topology induced by fuzzy metric spaces, fixed point

theorem and other properties of fuzzy metric spaces are studied by a few re-

searchers, see for instance, Felbin (1992)[7], George (1994)[8], George (1997)[9],

Gregori (2000)[10], Hadzic (2002)[11] etc. The other one is using real numbers

to measure the distances between fuzzy sets. The references of this approach can

be referred to, for instance, Dia (1990) [5], Chaudhuri (1996)[4], Boxer (1997)[2],

Received July 27, 2003. Revised October 20, 2003.

This paper was supported by the National Foundations of Ph. D Units from the Ministry of

Education of China No. 20020141013, the Scientific Research Foundation of DUT No. 3004888.

c ? 2004 Korean Society For Computational & Applied Mathematics and Korean SIGCAM.

371

∗Corresponding author.

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372Zun-Quan Xia and Fang-Fang Guo

Fan (1998)[6], Voxmam (1998)[16], Przemyslaw, (1998)[14], Brass (2002)[3]. Re-

sults of these researches have been applied to many practical problems in fuzzy

environment. While, usually, different measures are used in different problems

in other words, there does not exist a uniform measure that can be used in all

kinds of fuzzy environments.

Therefore, it is still interesting to find some kind of new fuzzy measure such

that it may be useful for solving some problems in fuzzy environment. The

attempt of the present paper is using fuzzy scalars (fuzzy points defined on

the real-valued space R) to measure the distances between fuzzy points, which

is consistent with the theory of fuzzy linear spaces in the sense of Xia and

Guo (2003) [17] and hence more similar to the classical metric spaces. The

new definitions in this paper are different from the previous ones because fuzzy

scalars are used instead of fuzzy numbers or real numbers to measure the distance

between two fuzzy points. It is the first time that fuzzy scalars are introduced

in measuring the distances between fuzzy points. Some other properties of fuzzy

metric spaces, for instance, completeness and induced fuzzy topology are also

given in this paper.

For the convenience of reading, some basic concepts of fuzzy points and de-

notations are presented below.

Fuzzy points are the fuzzy sets being of the following form in the sense of Pu

(1980) [15],

?

where X is a nonempty set and λ ∈ [0,1].

In this paper, fuzzy points are usually denoted by (x,λ) and the set of all

the fuzzy points defined on X is denoted by PF(X). Particularly, when X = R,

fuzzy points are also called fuzzy scalars and the set of all the fuzzy scalars

is denoted by SF(R). A fuzzy set A can be regarded as a set of fuzzy points

belonging to it, i.e.,

A = {(x,λ)|A(x) ≥ λ}

or a set of fuzzy points on it,

xλ(y) =

λ,

0,

y = x,

y ?= x,

∀y ∈ X,

A = {(x,λ)|A(x) = λ}.

This paper is organized as follows: In Section 2, fuzzy metric spaces, strong

fuzzy metric spaces and fuzzy linear normed spaces are defined and some ex-

amples are given to show the existence of these kinds of spaces; In Section 3,

the convergence of sequences of fuzzy points and the completeness of induced

fuzzy metric spaces are considered; In the last section, it is proved that the fuzzy

topology induced by fuzzy metric spaces is consistent with the given one, see for

instance, Pu (1980), [15], which implies in another way the usefulness of the

fuzzy metric spaces defined in Section 2.

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Fuzzy metric spaces373

2. Fuzzy metric spaces

The purpose of this section mainly consists in defining fuzzy metric spaces,

strong fuzzy metric spaces and fuzzy normed linear spaces. To do so, we first

give some definitions related to fuzzy scalars.

Definition 1. Suppose (x,λ) and (y,γ) are two fuzzy scalars. A series of defi-

nitions contains the following ones:

(1) we say (a,λ) ? (b,γ) if a > b or (a,λ) = (b,γ);

(2) (a,λ) is said to be no less than (b,γ) if a ≥ b, denoted by (a,λ) ? (b,γ)

or (b,γ) ≺ (a,λ);

(3) (a,λ) is said to be nonnegative if a ≥ 0. The set of all the nonnegative

fuzzy scalars is denoted by S+

F(R).

Obviously, the orders defined in Definition 1(1) and Definition 1(2) are both

partial orders. Note that when R is considered as a subset of SF(R), (R,?) and

(R,?) are the same as (R,≥). Thus both ? and ? can be viewed as some kind

of generalization of the ordinary complete order ≥. It is obvious that the order

defined in Definition 1(1) is stronger than the one in Definition 1(2).

We now present the definition of fuzzy metric spaces. It will be seen that it is

very similar to the definition of ordinary metric spaces except that ≤ is replaced

by ≺ in the triangle inequality. This is because that there exist no reasonable

complete order in SF(R)+.

Definition 2. Suppose X is a nonempty set and

dF: PF(X) × PF(X) → S+

is a mapping. (PF(X),dF) is said to be a fuzzy metric space if for any {(x,λ),

(y,γ), (z,ρ)} ⊂ PF(X), dF satisfies the following three conditions,

(1) Nonnegative: dF((x,λ), (y,γ)) = 0 iff x = y and λ = γ = 1;

(2) Symmetric:

dF((x,λ), (y,γ)) = dF((y,γ), (x,λ));

(3) Triangle inequality:

dF((x,λ), (z,ρ)) ≺ dF((x,λ), (y,γ)) + dF((y,γ), (z,ρ)).

dF is called a fuzzy metric defined in PF(X) and dF((x,λ), (y,γ)) is called a

fuzzy distance between the two fuzzy points.

F(R)

Note that fuzzy metric spaces have fuzzy points as their elements, i.e., they

are sets of fuzzy points. There are many fuzzy metric spaces in the sense of

Definition 2. To show this, some examples are presented below.

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374Zun-Quan Xia and Fang-Fang Guo

Example 1. Suppose (X, d) is an ordinary metric space. The distance of any

two fuzzy points (x,λ), (y,γ) in PF(X) is defined by

dF((x,λ), (y,γ)) = (d(x,y), min{λ,γ}),

where d(x,y) is the distance between x and y defined in (X, d).Then (PF(X),dF)

is a fuzzy metric space.

Proof. It suffices to prove that dF satisfies the three conditions in Definition 2.

Nonnegative: Suppose (x,λ) and (y,γ) are two fuzzy points in PF(X). Since

d(x,y) is a distance between x and y, one has d(x,y) ≥ 0. It follows from

Definition 1 that dF((x,λ), (y,γ)) = (d(x,y), min{λ,γ}) is a nonnegative fuzzy

scalar. It is obvious that dF((x,λ), (y,γ)) = 0 iff d(x,y) = 0 and min{λ,γ} = 1

which is equal to that x = y and λ = γ = 1.

Symmetric: For any {(x,λ), (y,γ)} ⊂ PF(X), one has

dF((x,λ), (y,γ))=

=

=

Triangle inequality: For any {(x,λ), (y,γ), (z,ρ)} ⊂ PF(X), we have

dF((x,λ), (z,ρ))=(d(x,z), min{λ,ρ})

≺

=(d(x,y),min{λ,γ}) + (d(y,z),min{γ,ρ})

=

d((x,λ), (y,γ)) + d((y,γ), (z,ρ)).

(d(x,y), min{λ,γ})

(d(y,x), min{γ,λ})

dF((y,γ), (x,λ)).

(d(x,y) + d(y,z),min{λ,ρ,γ})

?

Example 2. We denote Rnthe usual n-dimensional Euclidean space. Suppose

L is a fuzzy linear space defined in Rn. The distance between arbitrary two fuzzy

points (x,λ), (y,γ) belonging to L, denoted by dFE((x,λ), (y,γ)), is defined by

dFE((x,λ), (y,γ)) = (dE(x,y), min{λ,γ}),

where dEis the usual Euclidean distance. Then (L, dEF) is also a fuzzy metric

space, where L is also viewed as the set of fuzzy points belonging to the fuzzy

set L.

Proof. Since Rnis a metric space in the ordinary sense and L can be regarded

as a subset of PF(Rn), dFEis a fuzzy metric from Example 1.

?

The two examples given above show that a fuzzy (linear) metric space can

be constructed by a (linear) metric space in the usual sense, called an induced

(linear) metric space of it and the metric of the space is called an induced metric

of the original one.

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Fuzzy metric spaces375

Since S+

nition 2, ≤ is replaced by ≺ which is much weaker than it. A natural question is

that whether there exist some kind of fuzzy metric spaces satisfying the triangle

inequality with some partial order stronger than ≺, for example, ?. The answer

is positive and they are called strong fuzzy metric spaces.

F(R) is not a complete ordered set, in the triangle inequality of Defi-

Definition 3. Suppose X is a nonempty set and dF : PF(X)×PF(X) → S+

is a mapping. (PF(X), dF) is said to be a strong fuzzy metric space if it satisfies

the first two conditions in Definition 2 and for any (x,λ), (y,γ), (z,ρ) in PF(X),

one has

(3?) dF((x,λ), (z,ρ)) ? dF((x,λ), (y,γ)) + dF((y,γ), (z,ρ)).

It is obvious from Definition 2 and Definition 3 that every strong fuzzy metric

space is a fuzzy metric space. The following example shows the existence of

strong fuzzy metric spaces and the difference between these two kinds of spaces.

F(R)

Example 3. L is a fuzzy linear space defined in Rn. The distance between

arbitrary two fuzzy points (x,λ) and (y,γ) on L is defined by

dFE((x,λ), (y,γ)) = (dE(x,y), min{λ,γ}),

where dE is the Euclidean distance. Then (L, dFE) is a strong fuzzy metric

space where L denote the set of fuzzy points on the fuzzy set L.

(1)

Proof. The first two conditions can be proved just as Example 1. Here we only

prove the third one.

Given arbitrarythree fuzzy points on L, (x,λ), (y,γ)and(z,ρ). Since (Rn, dE)

is a metric space, one has

dE(x,z) ≤ dE(y,z) + dE(x,y).

In the case of that inequality (2) holds strictly, it is obvious from Definition 1(1)

that condition (3?) is satisfied. In the other case, there must exists some λ ∈ F

such that y = (1−λ)x+λz. Let α = min{λ,ρ}. We have that {x,z} ⊂ Lα. Since

L is a fuzzy linear space, Lαis a linear subspace of Rn(see the Representation

Theorem of fuzzy linear spaces due to Lowen (1980), [13]). It follows that y ∈ Lα,

i.e., γ = L(y) ≥ α = min{λ,ρ}. This implies that min{λ,ρ,γ} = min{λ,ρ}.

Thus, one has

dFE((x,λ), (z,ρ)) = (dE(x,z), min{λ,ρ})

= (dE(x,y) + dE(y,z), min{λ,ρ,γ})

= dFE((x,λ), (y,γ)) + dFE((y,γ), (z,ρ)).

Consequently, condition (3?) is satisfied.

(2)

?

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376Zun-Quan Xia and Fang-Fang Guo

Note that the strong fuzzy metric space given above is a set of fuzzy points

on some fuzzy linear space. Different from it, the fuzzy metric space in Example

2 comprises fuzzy points belonging to a fuzzy linear space. The difference is

caused by that ≺ is replaced by the partial order ? which is much stronger than

it.

Definition 4. Suppose that L is a fuzzy linear space. (L,? · ?) is said to be a

fuzzy linear normed space if the mapping ? · ? : L → S+

(a) ?(x,λ)? = 0 if and only if x = 0 and λ = 1;

(b) For any k ∈ R and (x,λ) ∈ L, one has ?k(x,λ)? = |k| · ?(x,λ)?;

(c) For any {(x,λ), (y,γ)} ⊂ L, one has ?(x,λ)+(y,γ)? ? ?(x,λ)?+?(y,γ)?.

The mapping ?·? : x ?→ ?x? is called the fuzzy norm of (L,?·?). Note that a

fuzzy linear normed space L has fuzzy points belonging to the fuzzy set L as its

elements.

F(R) satisfies:

Example 4. Let (G, ||·||G) be a linear normed space defined on R. L is a fuzzy

linear space defined in G and ?·?FGis a mapping from L to SF(R)+defined by

?(x,λ)?FG:= (?x?G,λ),

Then (L,?·?FG) is a fuzzy linear normed space which can be verified similar to

Example 2.

∀(x,λ) ∈ L.

The following proposition given without proof shows the relationship between

fuzzy linear normed spaces and fuzzy metric spaces.

Proposition 1. Suppose (L,? · ?FG) is a fuzzy linear normed space.

(L,dFG) is a fuzzy metric space, where dFGis defined by

dFG((x,λ), (y,γ)) := ?(x,λ) − (y,γ)?FG.

Then

Proof. It is omitted.

?

Taking G = Rnin Example 4, we have the following proposition, which shows

the relationship between fuzzy norm and inner product of fuzzy points.

Proposition 2. Suppose (L,? · ?FE) is a fuzzy linear normed space defined in

Rn. For any (x,λ) ∈ L, one has

< (x,λ), (x,λ) >= ?(x,λ)?2

where the inner product is defined in the sense of Xia and Guo (2003) [17], i.e.,

< (x,λ), (y,γ) >= (< x,y >, min{λ,γ}).

FE,

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Fuzzy metric spaces377

Proof. From the definition of inner product of fuzzy points, one has

< (x,λ), (x,λ) >

=

=

=

=

(< x,x >, λ)

(?x?2

(?x?E, λ) · (?x?E, λ)

?(x,λ)?2

E, λ)

FE,

where ? · ?Eis the Euclidean norm.

?

3. The completeness of fuzzy metric spaces

In this section, we mainly consider the convergence of a sequence of fuzzy

points and the completeness of induced fuzzy metric spaces. Since fuzzy scalars

are used to measure the distances between fuzzy points, the convergence of a

sequence of fuzzy scalars is considered first.

Definition 5. Let {(an,λn)} be a sequence of fuzzy scalars. It is said to be

convergent to a fuzzy scalar (a,λ), λ ?= 0, denoted by limn→∞(an,λn) = (a,λ) if

limn→∞an= a, {λi|λi< λ, i ∈ N} is a finite set and there exists a subsequence

of {λi}, denoted by {λl}, such that limn→∞λl= λ.

The requirement that almost all the λi∈ N satisfy λi≥ λ is natural since we

hope that the degree of the convergence is not less than λ. A new definition of

the convergence of a sequence of fuzzy points is presented below based on the

fuzzy metric given in the last section.

Definition 6. Suppose (PF(X), dF) is the induced fuzzy metric space of (X, d)

and {(xn,λn)} is a sequence of fuzzy points in (PF(X), dF). {(xn,λn)} is said

to be convergent to a fuzzy point (x,λ), if limn→∞dF((xn,λn), (x,λ)) = 0λand

for any γ ∈ (0,1] such that limn→∞dF((xn,λn), (x,γ)) = 0γ, one has λ ≥ γ.

(x,λ) is called the limit of the sequence, denoted by limn→∞(xn,λn) = (x,λ).

Proposition 3. Suppose {(xn,λn)} is a sequence of fuzzy points in (PF(X), dF)

and (x,λ) ∈ (PF(X), dF), λ ?= 0. We have that limn→∞(xn,λn) = (x,λ) if and

only if limn→∞xn = x, {λi|λi < λ, i ∈ N} is a finite set and there exists a

subsequence of {λi}, denoted by {λl}, such that limn→∞λl= λ.

Proof. It is omitted.

?

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378Zun-Quan Xia and Fang-Fang Guo

Definition 7. A sequence of fuzzy points (xn,λn) ∈ (PF(X), dF) is said to be

a Cauchy sequence if there exists some λ ∈ (0,1] such that

lim

n→∞dF((xm+n,λm+n), (xn,λn)) = 0λ,

Note that every Cauchy sequence of fuzzy points defined above has a unique

fuzzy point as its limit, which is very similar to the classical one. We now begin

to consider the completeness of fuzzy metric spaces.

∀m ∈ N.

Definition 8. An induced fuzzy metric space is said to be complete if any

Cauchy sequence in it has a unique limit in the space.

Theorem 1. Suppose (PF(X), dF) is the induced fuzzy metric space of an or-

dinary metric space (X, d). Then it is complete iff (X, d) is complete.

Proof. Necessity (Only if) : It is obvious.

Sufficiency (If) : Suppose {(xn,λn)} is an arbitrary Cauchy sequence of

(PF(X), dF). Since (X,d) is complete and limn→∞d(xm+n, xn) = 0 for any

m ∈ N, there must exists some x ∈ X such that limn→∞xn = x. For any

m ∈ N, denote the index set {l|λl= min{λm+n,λn}, n = 1,2···} by Lm.

From the definition of Cauchy sequences of fuzzy points, there exists some

λ ∈ (0,1] such that for any m ∈ N, the set {λl|λl< λ, l ∈ Lm} is finite and there

exists a subsequence of {λl}l∈Lm, denoted by {λk}, which is also a subsequence

of {λn}, such that limk→∞λk= λ. It is obvious that

{l|λl< λ, l ∈ Lm} ⊃ {n|λn< λ, n = 1,2,···}.

Consequently, {λn|λn < λ, n = 1,2,···} is also a finite set. From the above

arguments, we have limn→∞(xn,λn) = (x,λ). It implies that there exists a

limit of {xn,λn} in PF(X).

contradiction, assume that there is another limit of the same Cauchy sequence

{(xn,λn)}. Since we know x is the unique limit of {xn}, we can denote by (x,γ)

the limit different from (x,λ), γ ?= λ, say γ > λ. Then we have {λn|λn< γ}

is a finite set. From the above arguments, we know that limk→∞λk = λ and

{λk|λk< λ} is a finite set. Thus, taking ρ =λ+γ

{λk} ∩ [λ,ρ] ⊂ {λn|λn< γ}

is an infinite set. This contradicts that {λn|λn< γ} is a finite set. Therefore,

there is only one limit of the Cauchy sequence.

In the following we prove the uniqueness. By

2, we have

?

Note that a strong fuzzy linear metric space is generally not complete. It can

be seen through the counter-example given below.

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Fuzzy metric spaces379

Example 5. Consider the strong fuzzy linear metric space (L, dFE), where

?

and dFEis induced by the ordinary Euclidean metric dE. The sequence {(1

in L is a Cauchy sequence in the sense of Definition 7. However, the limit of the

sequence, (0,1

L =(x,λ)|x ∈ R \ {0}, λ =1

2

??

{(0,1)}

n,1

2)}

2) is not on the space L.

4. The fuzzy topology spaces induced by fuzzy metric spaces

Fuzzy metric spaces given in this paper have many similar properties to the

ordinary metric spaces. Except the relationship between distances and inner

products of fuzzy points mentioned in the Section 2, a conclusion similar to that

every metric space can induce a topology will be proved in this section. Here

we introduce fuzzy topology in the sense of Pu (1980) [15] via fuzzy closed sets.

It provides a convenient method to construct a fuzzy topology of any ordinary

metric space. To do this, the definition of fuzzy closed sets with respect to

induced fuzzy metric spaces is given first. Suppose (X,d) is an ordinary metric

space. Since a fuzzy set A in X can be viewed as a set of fuzzy points belonging

to it, A can be regarded as a subset of PF(X), called a fuzzy set in the induced

fuzzy metric space (PF(X), dF) in the following definition.

Definition 9. A fuzzy set A in (PF(X), dF) is said to be closed if the limit of

any Cauchy sequence in A belongs to it. A fuzzy set A in (PF(X), dF) is said

to be open if A?is a fuzzy closed set, where A?is defined by A?(x) = 1 − A(x),

for any x ∈ X.

The following proposition shows that the new definition of fuzzy closed sets

is reasonable.

Proposition 4. A fuzzy set A in (PF(X), dF) is closed if and only if every

α-cut set of A, α ∈ [0,1], is a closed set in (X,d) in the ordinary sense.

Proof. It is omitted.

?

In the following we will show that every induced fuzzy metric space can induce

a fuzzy topology. To prove it, a lemma about Cauchy sequences of fuzzy points

is given.

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380Zun-Quan Xia and Fang-Fang Guo

Lemma 1. Any subsequence of a Cauchy sequence of fuzzy points is also a

Cauchy sequence and has the same limit as the original one.

Proof. It is obvious from Definition 7.

?

Theorem 2. Suppose (PF(X), dF) is the induced fuzzy metric space of a metric

space (X,d). Then (X, TF) is a fuzzy topology space in the sense of Pu (1980)

[15], called the fuzzy topology space induced by (PF(X), dF), where TFis defined

by

TF = {A ⊂ PF(X)|A is a fuzzy closed set in (PF(X), dF)}.

Proof. It suffices to prove that TFsatisfies the three conditions in the definition

of fuzzy topology due to Pu (1980) [15].

(1) It is obvious that X and ∅ are fuzzy closed sets.

(2) For any {A,B} ⊂ TF, we prove in the following that A ∪ B ∈ TF. For

any Cauchy sequence of fuzzy points {(yn,γn)} included in A ∪ B, A or B,

say A, must contain a subsequence {(ym,γm)} of {(yn,γn)}. From Lemma 1,

{(ym,γm)} is also a Cauchy sequence and hence has a limit. Since A is a closed

fuzzy set, the limit of {(ym,γm)} which is also the limit of {(yn,γn)} is included

in A. In consequence, the limit of {(yn,γn)} is included in A∪B, which implies

that A ∪ B ∈ TF.

(3) For any {Ai}i∈I⊂ TF, where I is an arbitrary index set, it only need to

be proved that?

fuzzy set, the limit of {(xn,λn)} is in Aifor any i ∈ I. It follows that?

The proof is completed.

i∈IAi∈ TF. For any Cauchy sequence in?

a closed fuzzy set in the sense of Definition 9. Therefore one has?

From the theorem given above we know that every fuzzy metric space can

induce a fuzzy topology space, which implies in another way that the fuzzy

measure defined in this paper is not only reasonable but also significant.

i∈IAi, denoted by

{(xn,λn)}, we have that {(xn,λn)} ⊂ Aifor any i ∈ I. Since every Aiis a closed

i∈IAiis

i∈IAi∈ TF.

?

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100 (1998), 353-365.

17. Z. Q. Xia and F. F. Guo, Fuzzy linear spaces, Int. J. Pure and Applied Mathematics, to

appear.

Fang-Fang Guo is a student for Ph.D. under the supervision of Prof. Xia. She received her

master’s degree from Liaoning Normal University in 2001. Her research interesting focus

on fuzzy convex analysis, fuzzy optimization and numerical method and fuzzy reasoning.

Laboratory 2, CORA, Department of Applied Mathematics, Dalian University of Technol-

ogy, Dalian 116024, P. R. China

e-mail:gracewuo@163.com

Zun-Quan Xia is a professor in Department of Applied Mathematics, Dalian University of

Technology, Dalian, China. He graduated from Institute of Mathematics, Fudan University,

Shanghai, as a graduate student in 1968, His research areas are (smooth, nonsmooth,

discrete and numerical) optimization and applications (in science and technology) OR

methods and applications.

Laboratory 2, CORA, Department of Applied Mathematics, Dalian University of Technol-

ogy, Dalian 116024, P. R. China

e-mail: zqxiazhh@dlut.edu.cn