Nonlinear Analysis 64 (2006) 1939–1945www.elsevier.com/locate/na

Vector variational-like inequalities and non-smoothvector optimization problems�

S.K. Mishraa,∗, S.Y. Wangb

aCollege of Basic Sciences and Humanities, Department of Mathematics, Statistics and Computer Science, G.B.pant University of Agriculture and Technology, Pantnagar-263 145, India

bInstitute of Systems Sciences, Academy of Mathematics and Systems Sciences,Chinese Academy of Sciences,Beijing 1000800, China

Received 10 January 2005; accepted 21 July 2005

Abstract

In this paper, we establish relationships between vector variational-like inequality problems andnon-smooth vector optimization problems under non-smooth invexity. We identify the vector criticalpoints, the weakly efficient points and the solutions of the non-smooth weak vector variational-likeinequality problems, under non-smooth pseudo-invexity assumptions. These conditions are moregeneral than those existing in the literature.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Non-smooth vector optimization; Pseudo-invex functions; Vector variational-like inequality problems

1. Introduction

Convexity is one of the most frequently used hypotheses in optimization theory. It isusually introduced to give global validity to propositions otherwise only locally true, forinstance, a local minimum is also a global minimum for a convex function. Moreover,

� This research is supported by the Department of Science and Technology, Ministry of Science and Technology,Government of India under the SERC Fast Track Scheme for Young Scientists 2001–2002, through Grant no.SR/FTP/MS-22/2001 and The National Natural Science Foundation of China.

∗ Corresponding author. Present address: Department of Management Sciences, City University of Hong Kong,Tat Chee Avenue, Kowloon, Hong Kong. Tel.: +91 59 44 23 33 99; fax: +91 59 44 23 34 73.

E-mail addresses: shashikmishra@rediffmail.com (S.K. Mishra), sywang@amss.ac.cn (S.Y. Wang).

0362-546X/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2005.07.030

1940 S.K. Mishra, S.Y. Wang / Nonlinear Analysis 64 (2006) 1939–1945

convexity is also used to obtain sufficiency for conditions that are only necessary, as withthe classical Fermat theorem or with Kuhn–Tucker conditions in non-linear programming.In microeconomics, convexity plays a fundamental role in general equilibrium theory and induality results. For more applications and historical reference, see [3,8,25]. In recent years,several extensions have been considered for classical convexity. A significant generalizationof convex functions is that of invex functions introduced by Hanson [9]. Hanson’s initialresult inspired a great deal of subsequent work which has greatly expanded the role andapplications of invexity in non-linear optimization and other branches of pure and appliedsciences.

Since it has been introduced by Giannessi [5], the theory of vector variational inequali-ties has shown many applications in vector optimization problems and traffic equilibriumproblems; see [6,11,19]. In fact, some recent works in vector optimization have shown thatoptimality conditions of some multi-objective optimization problems can be characterizedby vector variational inequalities; see [1,12,18,21,23]. A useful and important generaliza-tion of the variational inequalities is called the variational-like inequalities, which has beenstudied and investigated extensively, see [4,10,16–18,21,24] and the references therein.The variational-like inequalities are closely related to the concept of the invex and pre-invex functions [22], which generalize the notion of convexity of functions, for some moreextensions of generalized convexity see Mishra et al. [14,15].Yang and Chen [24] and Noor[16,17] have shown that the minimum of invex functions on the invex sets can be character-ized by variational-like inequalities. Noor [16,17] has pointed out that the variational-likeinequalities are defined in the setting of invexity. Furthermore, Noor [16,17] emphasizedthe fact that the kernel function in the definition of the invex function plays a significantrole in defining the variational-like inequalities. As a matter of fact, the concept of invexityplays exactly the same role in variational-like inequalities as the classical convexity playsin variational inequalities.

In this paper, we shall establish a relationship between vector variational-like inequalityand non-smooth vector optimization problems. We shall also extend an earlier work ofRuiz–Garzon et al. [21] to the non-smooth case as well as make some corrections to someresults obtained by Ruiz–Garzon et al. [21].

2. Preliminaries

In this section, we recall some notions of non-smooth analysis. For more details, see[2,7]. Let Rn be the n-dimensional Euclidean space and Rn+ be its non-negative orthant. Inthe sequel, let X be a non-empty subset of Rn.

The following convention for equalities and inequalities will be used throughout thispaper. If x, y ∈ Rn, then

x�y ⇔ xi �yi, i = 1, 2, . . . , n with strict inequality holding for at least one i;

x�y ⇔ xi �yi, i = 1, . . . , n;

x = y ⇔ xi = yi, i = 1, . . . , n;

x < y ⇔ xi < yi, i = 1, . . . , n.

S.K. Mishra, S.Y. Wang / Nonlinear Analysis 64 (2006) 1939–1945 1941

Definition 2.1. A function f : X → R is said to be Lipschitz near x ∈ X if for someK > 0,

|f (y) − f (z)|�K‖y − z‖, ∀y, z within a neighbourhood of x.

We say that f : X → R is locally Lipschitz on X if it is Lipschitz near any point of X.

Definition 2.2. If f : X → R is Lipschitz at x ∈ X, the generalized derivative (in the senseof Clarke) of f at x ∈ X in the direction v ∈ Rn, denoted by f 0(x; v), is given by

f 0(x; v) = lim supy→x�↓0

f (y + �v) − f (y)

�.

Definition 2.3. The Clarke’s generalized gradient of f at x ∈ X, denoted by �f (x), isdefined as follows:

�f (x) = {� ∈ Rn : f 0(x; v)��T v for all v ∈ Rn}.It follows that, for any v ∈ Rn

f 0(x; v) = max{�T v : � ∈ �f (x)}.

These definitions and properties can be extended to a locally Lipschitz vector-valuedfunction f : X → Rp. Denote by fi, i = 1, 2, . . . , p the components of f . The Clarkegeneralized gradient of f at x ∈ X is the set �f (x) = �f1(x) × �f2(x) × · · · × �fp(x).

Definition 2.4. Let u ∈ X, the set X is said to be invex at u with respect to � : X×X → Rn

if, for all x, u ∈ X, t ∈ [0, 1], u + t�(x, u) ∈ X.

From now onward we assume that the set X is a non-empty, closed and invex set, unlessotherwise specified.

Definition 2.5. The non-differentiable function f : X → Rp is invex with respect to� : X × X → Rn if

fi(x) − fi(u)��Ti �(x, u), ∀�i ∈ �fi(u), ∀x, u ∈ X.

Definition 2.6. The non-differentiable function f : X → Rp is strictly-invex with respectto � : X × X → Rn if

fi(x) − fi(u) > �Ti �(x, u), ∀�i ∈ �fi(u), ∀x �= u ∈ X.

Definition 2.7. The non-differentiable function f : X → Rp is pseudo-invex with respectto � : X × X → Rn if

fi(x) − fi(u) < 0 ⇒ �Ti �(x, u) < 0, ∀�i ∈ �fi(u), ∀x, u ∈ X.

1942 S.K. Mishra, S.Y. Wang / Nonlinear Analysis 64 (2006) 1939–1945

We consider the following non-smooth vector optimization problem: (VOP) Min f (x)=(f1(x), . . . , fp(x))

s.t x ∈ X,

where fi : X → R, i = 1, 2, . . . , p are non-differentiable locally Lipschitz functions.In vector optimization problems, multiple objectives are usually non-commensurable

and cannot be combined into a single objective. Moreover, often the objectives conflictwith each other. Consequently, the concept of optimality for single-objective optimizationproblems cannot be applied directly to vector optimization. In this regard the concept ofefficient solutions is more useful for vector optimization problems.

Definition 2.8. Given an open subset X ⊆ Rn and a function f : X → Rp a point x̄ ∈ X

is said to be efficient (Pareto), if there exists no y ∈ X such that f (y)�f (x̄).

Definition 2.9. Given an open subset X ⊆ Rn and a function f : X → Rp a point x̄ ∈ X

is said to be weakly efficient (Pareto), if there exists no y ∈ X such that f (y) < f (x̄).

(VVLIP) A vector variational-like inequality problem for non-smooth case, is to find apoint y ∈ X, and for any � ∈ �f (y), there exists no x ∈ X, such that �T �(x, y)�0.

(WVVLIP) A weak vector variational-like inequality problem, is to find a point y ∈ X,and for any � ∈ �f (y), there exists no x ∈ X, such that �T �(x, y) < 0.

3. Relationship between vector variational-like inequality and non-smooth vectoroptimization problems

In this section, using the tools of non-smooth analysis and the concept of non-smoothvector pseudo-invexity, we shall extend the results given by Ruiz-Garzon et al. [21] to thenon-smooth case.

Theorem 3.1. Let f : X → Rp be locally Lipschitz and invex function with respect to �.If y ∈ X solves the VVLIP with respect to the same �, then y is an efficient solution to thenon-smooth VOP.

Proof. Suppose that y is not an efficient solution to VOP. Thus, there exists a x ∈ X suchthat f (x) − f (y)�0. Since f is invex with respect to �, we know that ∃x ∈ X such that

�T �(x, y)�0, ∀� ∈ �f (y),

therefore, y cannot be a solution to the VVLIP. This contradiction leads to the result. �

In order to see the converse of the above theorem, we must impose stronger conditions,as can be observed in the following two theorems.

Theorem 3.2. Let f : X → Rp be locally Lipschitz and −f is strictly-invex with respectto �. If y ∈ X is a weak efficient solution for VOP, then y solves the VVLIP.

S.K. Mishra, S.Y. Wang / Nonlinear Analysis 64 (2006) 1939–1945 1943

Proof. Suppose that y does not solve VVLIP. Thus, there exists a point x ∈ X such that�T �(x, y)�0, ∀� ∈ �f (y). By the strict invexity of the non-smooth function −f withrespect to �, we have

f (x) − f (y) < �T �(x, y)�0, ∀� ∈ �f (y),

therefore, there exists a x ∈ X such that f (x) − f (y) < 0, which contradicts y ∈ X beinga weakly efficient solution of VOP. This contradiction leads to the result. �

As every efficient solution is also a weakly efficient solution to VOP, the following resultis trivial to prove.

Corollary 3.1. Let f : X → Rp be locally Lipschitz and −f is strictly-invex with respectto �. If y ∈ X is an efficient solution for VOP, then, y also solves the VVLIP.

Theorem 3.3. If y ∈ X is a weakly efficient solution for VOP, then y solves the weakWVVLIP.

Proof. Let y ∈ X be a weakly efficient solution for VOP. Since X is an invex set, we havethat �x ∈ X, such that f (y+ t�(x, y))−f (y) < 0, 0 < t < 1. Dividing the above inequalityby t and taking the limit as t tends to zero, we get to �x ∈ X such that �T �(x, y) < 0, ∀� ∈�f (y). �

Theorem 3.4. If f is a locally Lipschitz and pseudo-invex with respect to � and y solvesthe WVVLIP with respect to the same �, then y is a weakly efficient solution to VOP.

Proof. Suppose that y is not a weakly efficient solution to VOP. Thus, there exists a x ∈ X,such that f (x) < f (y). By the pseudo-invexity of f with respect to �, we have that, thereexists x ∈ X such that �T �(x, y) < 0, ∀� ∈ �f (y). This contradicts the fact that y is asolution to WVVLIP. �

Theorem 3.5. If f is a locally Lipschitz and strictly-invex with respect to � and y is a weakefficient solution of the VOP, then y is an efficient solution to the VOP.

Proof. Suppose that y is a weakly efficient solution of VOP, but not an efficient solutionto VOP. Hence, there exists x ∈ X such that f (x)�f (y) By the strict-invexity of thenon-smooth function f with respect to � we have

0�f (x) − f (y) > �T �(x, y), ∀� ∈ �f (y).

That is to say, there exists x ∈ X such that �T �(x, y) < 0, ∀� ∈ �f (y). Therefore, y doesnot solve WVVLIP. Then, by Theorem 3.4 we get a contradiction. Hence, y is an efficientsolution to the VOP. �

The following definition is a simple extension of the concept of vectorial critical pointgiven for the differentiable case by Osuna et al. [20] to the non-smooth case.

1944 S.K. Mishra, S.Y. Wang / Nonlinear Analysis 64 (2006) 1939–1945

Definition 3.1. A feasible solution y ∈ X is said to be a vectorial critical point for VOP ifthere exists a vector � ∈ Rp with ��0 such that �T � = 0, ∀� ∈ �f (y).

Lemma 3.1. Let y ∈ X be a vector critical point for VOP, and let f is pseudo-invex on Xwith respect to �. Then, y ∈ X is a weakly efficient solution to VOP.

Proof. The proof is obvious using the pseudo-invexity of the non-smooth function f andthe definition of vector critical point for VOP. �

Theorem 3.6. All vector critical points are weakly efficient solutions to VOP if and only iff is pseudo-invex on X

Proof. The proof follows the lines of the proof of Theorem 2.2 [20] in light of the earlierdiscussion in this paper. �

We can relate the vector critical points to the solutions of the WVVLIP using Theorems3.4 and 3.6.

Corollary 3.2. If the objective function f is locally Lipschitz and pseudo-invex with respectto �, then the vector critical points, the weakly efficient points and the solutions of theWVVLIP are equivalent.

4. Conclusions

We have extended an earlier result of Ruiz-Gorzon et al. [21] to the non-smooth case.Under the assumptions of pseudo-invexity, we have proven the relationship between thevector variational-like inequality problems and the vector optimization problems. Furtheran earlier work of Mishra and Noor [13] can be extended to non-smooth case.

References

[1] G.Y. Chen, B.D. Craven, Existence and continuity of solutions for vector optimization, J. Optim. TheoryAppl. 81 (1994) 459–468.

[2] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York, 1983.[3] S. Dafermos, Exchange price equilibrium and variational inequalities, Math. Programming 46 (1990)

391–402.[4] Y.P. Fang, N.J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces,

J. Optim. Theory Appl. 118 (2003) 327–338.[5] F. Giannessi, Theorem of Alternative, in: R.W. Cottle, F. Giannessi, J.L. Lion (Eds.), Variational Inequalities

and Complementarity Problems, Wiley, New York, 1980.[6] F. Giannessi, A. Maugeri, P.M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational

inequality Models, Kluwer Academic, Dordrecht, Holland, 2001.[7] G. Giorgi, A. Guerraggio, Various types of non-smooth invex functions, J. Inform. Optim. Sci 17 (1996)

137–150.[8] A. Guerraggio, E. Molho, The origin of quasi-concavity: a development between mathematics and economics,

Hist. Math. 31 (2004) 62–75.

S.K. Mishra, S.Y. Wang / Nonlinear Analysis 64 (2006) 1939–1945 1945

[9] M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545–550.[10] M.F. Khan, Salahuddin, On generalized vector variational-like inequalities, Nonlinear Anal. 59 (2004)

879–889.[11] D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications,

Academic Press, London, 1980.[12] O.G. Mancino, G. Stampacchia, Convex programming and variational inequalities, J. Optim. Theory Appl.

9 (1972) 3–23.[13] S.K. Mishra, M.A. Noor, On vector variational-like inequality problems, Journal of Mathematical Analysis

and Applications, in press, corrected proof available online 31 May 2005.[14] S.K. Mishra, S.Y. Wang, K.K. Lai, Multiple objective fractional programming involving semilocally type

I-preinvex and related functions. Journal of MathematicalAnalysis andApplications, in press, corrected proofavailable online 30 March 2005.

[15] S.K. Mishra, S.Y. Wang, K.K. Lai, Optimality and duality for multiple-objective optimization undergeneralized type I univexity, J. Math. Anal. Appl. 303 (2005) 315–326.

[16] M.A. Noor, Preinvex functions and variational inequalities, J. Nat. Geometry 9 (1996) 63–76.[17] M.A. Noor, Variational-like inequalities, Optimization 30 (1994) 323–330.[18] J. Parida, M. Sahoo, A. Kumar, A variational-like inequality problem, Bull. Aust. Math. Soc. 39 (1989)

225–231.[19] M. Patricksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach, Kluwer

Academic, Dordrecht, 1998.[20] R. Osuna, A. Rufian, G. Ruiz, Invex functions and generalized convexity in multiobjective programming,

J. Optim. Theory Appl. 98 (1998) 651–661.[21] G. Ruiz-Garzon, R. Osuna-Gomez,A. Rufian-Lizana, Relationships between vector variational-like inequality

and optimization problems, Eur. J. Oper. Res. 157 (2004) 113–119.[22] T. Weir, B. Mond, Preinvex functions in multiobjective optimization, J. Math. Anal. Appl. 136 (1988)

29–38.[23] X.Q. Yang, Vector variational inequality and vector pseudolinear optimization, J. Optim. Theory Appl. 95

(1997) 729–734.[24] X.Q. Yang, G.Y. Chen, A class of nonconvex functions and prevariational inequalities, J. Math. Anal. Appl.

169 (1992) 359–373.[25] X.Q. Yang, C.J. Goh, On vector variational inequalities: application to vector equilibria, J. Optim. Theory

Appl. 95 (1997) 431–443.