AbstractIn this study, using the properties of limiting subdifferentials in nonsmooth analysis and regarding a separation theorem, some weak Pareto-optimality (necessary and sufficient) conditions for nonsmooth multiobjective optimization problems are proved

Jahanshahloo, G.R.

Soleimani-damaneh, M.

Elsevier - Publisher Connector 

J. Math. Anal. Appl. 328 (2007) 281–286
www.elsevier.com/locate/jmaa
Nonsmooth multiobjective optimization using
limiting subdifferentials
M. Soleimani-damaneh ∗, G.R. Jahanshahloo
Department of Mathematics, Teacher Training University, Tehran, Iran
Received 4 January 2006
Available online 13 June 2006
Submitted by William F. Ames
Abstract
In this study, using the properties of limiting subdifferentials in nonsmooth analysis and regarding
a separation theorem, some weak Pareto-optimality (necessary and sufficient) conditions for nonsmooth
multiobjective optimization problems are proved.
© 2006 Elsevier Inc. All rights reserved.
Keywords: Limiting subdifferential; Lipschitz function; Invex function; Multiobjective optimization; Weak
Pareto-optimality
1. Introduction
Recently, much attention has been paid to characterize the Pareto-optimality conditions for
multiple-objective optimization problems, under various types of conditions. Under smooth
conditions (say, convexity and generalized convexity as well as differentiability) optimality con-
ditions for these problems have been studied by some scholars, see, e.g., Hanson and Mond [8],
Giorgi and Guerraggio [5], Kaul et al. [10], Rueda and Hanson [18], Aghezzaf and Hachimi [1,2],
Mishra et al. [11–13,15], Soleimani-damaneh [19] and references therein. The nondifferentiable
case has been studied by some scholars, too. See, e.g., Hachimi and Aghezzaf [6], Mishra et al.
[14,16], and Yang et al. [20], among others.
* Corresponding author.
E-mail address: m_soleimani@tmu.ac.ir (M. Soleimani-damaneh).0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2006.05.026
282 M. Soleimani-damaneh, G.R. Jahanshahloo / J. Math. Anal. Appl. 328 (2007) 281–286In this paper we deal with the limiting subdifferentials for Lipschitz functions following
[4,17] as well as the concept of invexity along the lines of [7,9] and characterize the weak
Pareto-optimal solutions under locally Lipschitz and invexity conditions. Our idea is based on
an algebraic construction for the set of improving directions and the set of feasible directions
using limiting subdifferential properties in nonsmooth analysis and finally using a separation
theorem in convex analysis to derive the optimality conditions. Section 2 contains some basic
definitions and primary results. In Section 3 the main results of our study have been sketched and
established through four theorems.
2. Preliminaries
For x, y ∈ Rn, x < y means xi < yi for all i = 1, . . . , n. x  y means xi  yi for all
i = 1, . . . , n, but x = y. x  y allows equality. In this paper we consider the multiobjective
optimization problem
min
{
f (x) = (f1(x), . . . , fm(x)): x ∈ X, gj (x) 0; j = 1,2, . . . , p}, (1)
where X is a nonempty open set in Rn, f :Rn →Rm, and g = (g1, . . . , gp) :Rn →Rp .
This section contains some basic definitions and results which are useful for the rest of the
paper.
Consider the feasible set of (1) as follows:
S = {x ∈ X: g(x) = (g1(x), . . . , gp(x)) 0}.
Definition 2.1. Let φ = S ⊆Rn and let x¯ ∈ clS. The set of feasible directions of S at x¯, denoted
by Dx¯ , is given by
Dx¯ = {d ∈Rn: d = 0 and ∃δ > 0 such that x¯ + λd ∈ S ∀λ ∈ (0, δ)}.
Note that if x¯ ∈ intS, then Dx¯ =Rn.
Definition 2.2. Let φ = S ⊆Rn and let x¯ ∈ S. The set of descent directions at x¯, denoted by F x¯ ,
is given by
F x¯ = {d ∈Rn: ∃δ > 0 such that f (x¯ + λd) < f (x¯) ∀λ ∈ (0, δ)}.
Definition 2.3. We say that x¯ ∈ S is a weak Pareto-optimal solution (WPOS) of problem (1) if
there exists no x ∈ S such that f (x) < f (x¯).
Lemma 2.1. If x¯ is a WPOS of (1) then F x¯ ∩ Dx¯ = φ.
Proof. Straightforward. 
Let A be a nonempty subset of Rn and y ∈Rn. The metric projection of y on A is defined as
ProjA(y) =
{
x¯ ∈A: ‖y − x¯‖ ‖y − x‖ ∀x ∈A}.
The proximal cone to A at x¯, denoted by NPA(x¯), is given by
NPA(x¯) =
{
ζ ∈Rn: ∃(λ 0, y ∈Rn) such that x¯ ∈ ProjA(y) and ζ = λ(y − x¯)}.
M. Soleimani-damaneh, G.R. Jahanshahloo / J. Math. Anal. Appl. 328 (2007) 281–286 283Considering h :A⊆Rn →R, the set of proximal subdifferential vectors of h at x¯, denoted by
∂P h(x¯), is defined as
∂P h(x¯) =
{
ζ ∈Rn: (ζ,−1) ∈ NPepih
(
x¯, h(x¯)
)}
.
Definition 2.4. A vector η ∈ Rn is a limiting subdifferential vector of h at x¯ if there exist two
sequences {ζi} and {xi} in Rn such that ζi ∈ ∂P h(xi), ζi → η, xi → x¯, and h(xi) → h(x¯). The
set of all limiting subdifferential vectors of h at x¯ is denoted by ∂Lh(x¯).
Definition 2.5. A function h :A ⊆ Rn → R is (V ,α)-invex at x¯ ∈ A if there exist a positive
real-valued function α and an n-dimensional vector-valued function V :A×A→Rn such that
h(x) − h(x¯) α(x, x¯)ηT V (x, x¯),
for every x ∈A and every η ∈ ∂Lh(x¯). h is said to be invex near x¯ if it is invex at each point of a
neighborhood of x¯.
Recall that a function h :A⊆Rn →R is said to be Lipschitz onA if there exists a k ∈R such
that ∣∣h(x) − h(y)∣∣ k‖x − y‖ ∀x, y ∈A.
h is said to be Lipschitz near x if it is Lipschitz on a neighborhood of x. Also, h is locally
Lipschitz on A if it is Lipschitz near x for every x ∈A.
The following results are known in nonsmooth analysis (see [4,17]).
Lemma 2.2. Let h be Lipschitz near x, then ∂Lh(x¯) = φ.
Proof. See [17, Remark 2.1]. 
Lemma 2.3. Let h be locally Lipschitz on x¯ ∈ S, then ∂Lh(x¯) is a convex and closed set. In fact,
if xi → x¯, ηi ∈ ∂Lh(xi), ηi → η, then η ∈ ∂Lh(x¯).
Theorem 2.1. Let function h be locally Lipschitz on a neighborhood of line segment [x, y]. Then
for every 	 > 0 there exists a point z in the 	-neighborhood of [x, y] and ζ ∈ ∂P h(z) such that
h(x) − h(y) ζ T (y − x) + 	.
The following separation theorem is known in convex analysis (see [3]).
Theorem 2.2. Let A be a nonempty closed convex set in Rn and y /∈ A. Then, there exists a
nonzero p ∈Rn and a scalar α such that pT y > α and pT x  α for each x ∈A.
3. Main results
The first two theorems of this section give an algebraic representation of one of the subsets of
F x¯ and Dx¯ , respectively.
Theorem 3.1. Consider f = (f1, . . . , fm) :Rn → Rm such that fi for each i, 1  i  m, is
Lipschitz near x¯, and there exists a d ∈Rn such that ηT d < 0 for all η ∈⋃1im ∂Lfi(x¯). Then
d ∈ F x¯ .
284 M. Soleimani-damaneh, G.R. Jahanshahloo / J. Math. Anal. Appl. 328 (2007) 281–286Proof. By the assumption of the theorem for each 1 i m there exists a γi > 0 such that fi is
Lipschitz on Bγi (x¯) = {x ∈Rn: ‖x − x¯‖ < γi}. Considering λ ∈ (0, γi2‖d‖ ), fi is locally Lipschitz
on a neighborhood of the line segment [x¯, x¯ +λd]. Using Theorem 2.1 for each 	 > 0, there exist
z	 in the 	-neighborhood of [x¯, x¯ + λd] and ζ	 ∈ ∂P fi(z	) such that
fi(x¯ + λd) − fi(x¯)
λ
 dT ζ	 + 	
λ
.
If 	 → 0, then the sequence {ζ	} has a subsequence, say {ζ¯	}, and there exists a correspond-
ing subsequence of {z	}, say {z¯	}, such that ζ¯	 ∈ ∂P fi(z¯	), z¯	 → xˆ, and ζ¯	 → ζˆ , where
xˆ ∈ [x¯, x¯ + λd] and hence ζˆ ∈ ∂Lfi(xˆ) by Lemma 2.3. Now λ → 0 implies xˆ → x¯ and ζˆ →
η ∈ ∂Lfi(x¯) by Lemma 2.3. Therefore
lim
λ→0+
lim
	→0+
dT ζ	 + 	
λ
= ηT d
and thus
lim
λ→0+
lim
	→0+
fi(x¯ + λd) − fi(x¯)
λ
< 0.
This implies that there exists a δi > 0 such that fi(x¯+λd) < fi(x¯) for each λ ∈ (0, δi). Therefore
f (x¯ + λd) < f (x¯) for each λ ∈ (0, δ), where δ = min1im{δi}. Thus d ∈ F x¯ and the proof is
complete. 
Theorem 3.2. Let x¯ ∈ S be a feasible solution for (1) and I (x¯) = {j : gj (x¯) = 0}. Suppose that
gj (x) for j ∈ I (x¯) is Lipschitz near x¯ and gj (x) for j /∈ I (x¯) is continuous at x¯. Also suppose
that there exists a d ∈Rn such that ηT d < 0 for all η ∈⋃j∈I (x¯) ∂Lgj (x¯). Then d ∈ Dx¯ .
Proof. By the assumption of the theorem and similar to the proof of Theorem 3.1, it can be
shown that for each j ∈ I (x¯) there exists a δj > 0 such that gj (x¯ + λd) < gj (x¯) = 0 for each
λ ∈ (0, δj ).
For each j /∈ I (x¯), since gj (x) is continuous at x¯ and gj (x¯) < 0, there exists a δ′j > 0 such
that gj (x¯ + λd) < 0 for each λ ∈ (0, δ′j ).
Since X is open, there exists a δ′′ > 0 such that x¯ + λd ∈ X for each λ ∈ (0, δ′′).
Now by setting
δ = min
{
δ′′, min
j∈I (x¯){δj }, minj /∈I (x¯)
{
δ′j
}}
we get x¯ + λd ∈ S for each λ ∈ (0, δ). Hence d ∈ Dx¯ and the proof is complete. 
The rest of this section contains two theorems which characterize the WPOSs of problem (1).
Theorem 3.3 (Necessary condition). Let x¯ ∈ S be a feasible solution for (1) and I (x¯) =
{j : gj (x¯) = 0}. Suppose that fi(x) for i = 1,2, . . . ,m and gj (x) for j ∈ I (x¯) are Lipschitz
near x¯ and gj (x) for j /∈ I (x¯) is continuous at x¯. If x¯ is a WPOS of (1), then there exists a
u = (v1, . . . , vm,u1, . . . , up) 0 such that
0 ∈
m∑
vi∂Lfi(x¯) +
p∑
uj∂Lgj (x¯)i=1 j=1
M. Soleimani-damaneh, G.R. Jahanshahloo / J. Math. Anal. Appl. 328 (2007) 281–286 285and
ujgj (x¯) = 0, j = 1,2, . . . , p.
Proof. To start the proof we define
F x¯0 =
{
d ∈Rn: ηT d < 0 ∀η ∈
⋃
1im
∂Lfi(x¯)
}
, (2)
Dx¯0 =
{
d ∈Rn: ηT d < 0 ∀η ∈
⋃
j∈I (x¯)
∂Lgj (x¯)
}
. (3)
Since x¯ is a WPOS, then F x¯ ∩ Dx¯ = φ by Lemma 2.1 and hence F x¯0 ∩ Dx¯0 = φ regarding Theo-
rems 3.1 and 3.2. Furthermore we form the set
B =
{
b ∈Rn: ∃u = (v1, . . . , vm,uj ; j ∈ I (x¯)) 0 and ∃(d1, . . . , dm, d ′j ; j ∈ I (x¯))
such that di ∈ ∂Lfi(x¯), d ′j ∈ ∂Lgj (x¯) and b =
m∑
i=1
vidi +
∑
j∈I (x¯)
uj d
′
j
}
.
This set is convex and closed. If 0 ∈ B, then the claim of the theorem is proved by setting uj = 0
for j /∈ I (x¯). By contradiction suppose that 0 /∈ B, then using Theorem 2.2 there exist a nonzero
p ∈ Rn and a scalar α such that pT (0) > α and pT b  α for each b ∈ B. Hence pT b < 0 for
each b ∈ B.
For each η ∈ ⋃1im ∂Lfi(x¯) as well as for each η ∈ ⋃j∈I (x¯) ∂Lgj (x¯), we have η ∈ B.
Therefore pT η < 0 for each η ∈ ⋃1im ∂Lfi(x¯) as well as for each η ∈ ⋃j∈I (x¯) ∂Lgj (x¯).
Hence p ∈ F x¯0 ∩ Dx¯0 which contradicts the result obtained in the first part of the proof and com-
pletes the proof. 
Theorem 3.4 (Sufficient condition). Let x¯ ∈ S be a feasible solution for (1) and I (x¯) =
{j : gj (x¯) = 0}. Suppose that fi(x) for i = 1,2, . . . ,m and gj (x) for j ∈ I (x¯) are (αi,V )-
invex and (βj ,V )-invex near x¯, respectively, for some real-valued functions αi,βj , defined
on X × X and the same vector-valued function V :X × X → Rn. F x¯0 is as defined in (2). If
F x¯0 ∩ {d ∈Rn: ηT d  0 ∀η ∈
⋃
j∈I (x¯) ∂Lgj (x¯)} = φ, then x¯ is a local WPOS of (1).
Proof. By the assumption of the theorem, for each 1 i m there exists an 	i > 0 such that fi
is Lipschitz on B	i (x¯) and
fi(x) − fi(x¯) αi(x, x¯)ηTi V (x, x¯), (4)
for each x ∈ B	i (x¯) and for every ηi ∈ ∂Lfi(x¯). Also for each j ∈ I (x¯) there exists a δj > 0 such
that gj is Lipschitz on Bδj (x¯) and
βj (x, x¯)η
T
j V (x, x¯) gj (x) − gj (x¯) = gj (x) 0, j ∈ I (x¯), (5)
for every x ∈ Bδj (x¯) and for every ηj ∈ ∂Lgj (x¯).
Setting 	 = min{min1im{	i},minj∈I (x¯){δj }}, relations (4) and (5) are valid for each
x ∈ B	(x¯).
286 M. Soleimani-damaneh, G.R. Jahanshahloo / J. Math. Anal. Appl. 328 (2007) 281–286If there exists an xˆ ∈ B	(x¯) such that f (xˆ) < f (x¯), then by (4) and (5) and regarding the fact
that αj s and βj s are positive, we get
ηTi V (xˆ, x¯) < 0, ∀ηi ∈ ∂Lfi(x¯), i = 1,2, . . . ,m,
ηTj V (xˆ, x¯) 0, ∀ηj ∈ ∂Lgj (x¯), j ∈ I (x¯).
These imply that
V (xˆ, x¯) ∈ F x¯0 ∩
{
d ∈Rn: ηT d  0 ∀η ∈
⋃
j∈I (x¯)
∂Lgj (x¯)
}
which contradicts the assumption. Therefore there exists no x ∈ B	(x¯) such that f (x) < f (x¯),
which completes the proof. 
References
[1] B. Aghezzaf, On generalized convexity and duality in multiobjective programming problems, Adv. Modeling Op-
tim. 3 (2001) 7–15.
[2] B. Aghezzaf, M. Hachimi, Generalized invexity and duality in multiobjective programming problems, J. Global.
Optim. 18 (2000) 91–101.
[3] M.S. Bazaraa, H.D. Sherali, C.M. Shetty, Nonlinear Programming Theory and Algorithms, second ed., Wiley, New
York, 1993.
[4] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
[5] G. Giorgi, A. Guerraggio, The notion of invexity in vector optimization: Smooth and nonsmooth case, in:
J.-P. Crouzeix, J.-E. Martinez Legaz, M. Volle (Eds.), Fifth Symposium on Generalized Convexity, Luminy–
Marseille, France, June 17–21, 1996, in: Nonconvex Optim. Appl., vol. 27, Kluwer Academic, Dordrecht, 1998,
pp. 389–405.
[6] M. Hachimi, B. Aghezzaf, Sufficiency and duality in nondifferentiable multiobjective programming involving gen-
eralized type I functions, J. Math. Anal. Appl. 319 (2006) 110–123.
[7] M.A. Hanson, On sufficiency of Kuhn–Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545–550.
[8] M.A. Hanson, B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Program. 37 (1987)
51–58.
[9] V. Jeyakumar, B. Mond, On generalized convex mathematical programming, J. Aust. Math. Soc. Ser. B 34 (1992)
43–53.
[10] R.N. Kaul, S.K. Suneja, N.K. Srivastava, Optimality criteria and duality in multiple-objective optimization involving
generalized invexity, J. Optim. Theory Appl. 80 (1994) 465–482.
[11] S.K. Mishra, V -invex functions and applications to multiobjective programming problems, PhD thesis, Banaras
Hindu University, Varanasi, India, 1995.
[12] S.K. Mishra, On multiple-objective optimization with generalized univexity, J. Math. Anal. Appl. 224 (1998) 131–
148.
[13] S.K. Mishra, Second order symmetric duality with F -convexity, European J. Oper. Res. 127 (2000) 507–518.
[14] S.K. Mishra, M.A. Noor, Some nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 316
(2006) 472–482.
[15] S.K. Mishra, Shou-Yang Wang, K.K. Lai, Optimality and duality for multiple-objective optimization under gener-
alized type I univexity, J. Math. Anal. Appl. 303 (2005) 315–326.
[16] S.K. Mishra, Shou-Yang Wang, K.K. Lai, Nondifferentiable multiobjective programming under generalized
d-univexity, European J. Oper. Res. 160 (2005) 218–226.
[17] S. Nobakhtian, Proximal characterization of Pareto optima, in: 13th Seminar on Analysis and Its Applications,
March 16–17, 2003, University of Isfahan, Iran.
[18] N.G. Rueda, M.A. Hanson, Optimality criteria in mathematical programming involving generalized invexity,
J. Math. Anal. Appl. 130 (1988) 375–385.
[19] M. Soleimani-damaneh, On the optimality and duality for multiple-objective optimization under generalized type I
univexity, submitted for publication.
[20] X.M. Yang, K.L. Teo, X.Q. Yang, Higher-order generalized convexity and duality in nondifferentiable multiobjec-
tive mathematical programming, J. Math. Anal. Appl. 197 (2004) 48–55.


Nonsmooth multiobjective optimization using limiting subdifferentials 

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Nonsmooth multiobjective optimization using limiting subdifferentials

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