Necessary Optimality Conditions for a Class of Nonsmooth
Vector Optimization Problems
Xuan-wei ZHOU
*School of Basic Courses, Zhejiang Shuren University, Hangzhou 310015, China *Corresponding author
Keywords: Vectoroptimization,Necessaryoptimality conditions, Nonsmooth optimization, Clarke's generalized directional derivative.
Abstract. A class of nonsmooth vector optimization problems are considered, where the feasible set defined by cone constraint and the objective and constraint functions are locally Lipschitz. The concept of the Clarke's generalized directional derivative for a locally Lipschitz function is introduced. By using this concept, necessary optimality condition for the unconstrained optimization problem is established. Furthermore, a Slater-type constraint qualification is given in such a way that they generalize the classical one, when the constraint functions are differentiable. Then, Kuhn–Tucker necessary optimality condition in terms of the Clarke subdifferentials is obtained.
Introduction
Vector optimization is an extension of mathematical programming where a scaler valued objective function is replaced by a vector function. Necessary optimality conditions for this kind of programming are studied by several authors in the smooth and nonsmooth cases [1–11].
Vector optimization problems with differentiable functions is considered in [12] and then the Kuhn–Tucker condition for a Pareto efficient solution of a vector function over a feasible set defined by inequality constraints is obtained. Considering quasiconvex and directionally differentiable functions, the result obtained by [12] is extended in [13].
A vector optimization problem with only inequality constraints and suppose that the involved functions are locally Lipschitz is studied in [14]. The KT optimality conditions is given under a generalized Abadie type constraint qualification assuming that the Clarke subdifferentials of the objective and constraint functions are polytopes.
The result of [14] is also extended in [15] by considering locally Lipschitz functions, Fréchet differentiable equality constraints, and an abstract set constraint. These authors establish the KT optimality conditions under an extended generalized Guignard constraint qualification is established, but in this case the objective function is assumed to be Fréchet differentiable.
In this paper, the cone efficient solution of a class of nonsmooth vector optimization problems is considered, where the feasible set defined by cone constraint and the objective and constraint functions are locally Lipschitz. By using Clarke's generalized directional derivative, necessary optimality condition for the unconstrained optimization problem is established. Furthermore, some constraint qualifications are given in such a way that they generalize the classical one, when the constraint functions are differentiable. Then, Kuhn–Tucker necessary optimality condition for the cone efficient solution in terms of the Clarke subdifferentials is obtained.
Notations and Preliminaries
Consider the following vector minimization problem:
(VMP) 1
1
min ( ) ( ( ), , ( )), . . ( ) ,
p
V f x f x f x
s t g x K
where ( , ,1 ) : n p
p
p
K R and K1Rm are closed convex cones with nonempty interior. Let
1
{ p| ( ) }
D x R g x K be the feasible set, and denote by intK the interior ofK.
Definition 1 The weakly local efficient solution is with respect to the cone K, thus x0D is a weakly local efficient solution of (VMP) if, for some 0,
0
( ) ( ) int
f x f x Kwhenever xD and||xx0|| .
The functions f and g are not assumed differentiable; the Lipschitz hypothesis, with Rademacher's theorem, ensures that they are differentiable except on a set of zero measure.
For a set p
SR and point x0S , define
0 { ( 0) | 0, }
x
S xx xS . The dual
cone of S is defined by * {S Rp|Ts 0, s S}. Note that *S is a closed convex cone, even though S need not be. It is well known that, since intK , the dual cone K* has a compact convex base B,
Lemma 1 If K is a closed convex cone with nonempty interior, B is a compact convex base of the dual cone K*, and y0Rp satisfies
0 0( )
Ty B
, theny0 intK. Proof If T 0 0( )
y B
, then y0 K by the convex sets separation theorem; if
0 int
y K
, then y0 is a generator of the cone K, thus 0T(y0) 0 for some nonzero0 for which 0y 0( y K), therefore 0K*; consequently y0 intK .
For a locally Lipschitz function hat the point x0, Clarke's generalized directional derivative in the direction h is denoted by
0 0
0
0
( ) ( )
( , ) lim sup
x x t
h x tv h x
h x v
t
,
and Clarke's generalized gradient is denoted by
0
0 0
( ) { n| ( , ) T , n}
h x R h x v v v R
.
For a Lipschitz vector function : n p
f R R , f x( )0 is the convex hull of all limits of
sequences f x( )j , where xjx0 and the gradient f x( )j exists at xj. From [16 ,Theorem
2.6.6], ( )( )0 ( )( )0
T T
f x f x
for each linear functional .
The following lemma will be required.
Lemma 2 Let m
AR be nonempty; letBRn be compact, with 0B; let PconeB; let
: n
f A R R be continuous, with ( v Rn) ( , )f v convex, and ( u A f u) ( , ) concave and positively homogeneous. Then exactly one of the following systems is consistent:
(1)( u A)( v P\{0}) ( , ) 0f u v ;
(2)( v P\{0})( u A f u v) ( , ) 0 .
Proof It is an immediate special case of [17], Theorem 2.1.
Now the contrapositive of (1) is(3) ( u A)( v P\{0}) ( , ) 0f u v . It follows that (3) holds if and only if (2) holds.
Lemma 3 Let p
KR be a closed convex cone with nonempty interior, the dual cone *
K , B a compact set with 0B and K*coneB. If, for some direction vRn, and all
B
, the generalized gradient 0
0
(f) ( , ) 0x v , then there is ( )B such that
0 0
( K* \{0})( ' t (0, ( ))) B T( (f x tv) f x( )) 0, where the symbol ( ' t) denotes "for all t except a set of zero measure".
Proof Suppose that 0
0
( v Rn\{0})( K*)(T f) ( , ) 0x v , Using Fubini's theorem, there is some 1( ) such that ( t [0, ( )]) 1 f x( 0tv) exists except a set of zero measure. Since
0 0
(Tf)( )x T f x( )
Since f x( )0 consists of all convex combinations of limits of f at points xx0 ,
2 0
( ' [0, ( )]) T T( ) 0
t f x tv v
,where 0 2( ) 1( ). Hence
2
( B)( ' t [0, ( )]) T( (f x0tv) f x( ))0
TfT(x0tv vdt) 0.Since B is compact, 2( ) may be replaced by ( )B , independent of B. Then
0 0
( * \{0})( ' (0, ( ))) T( ( ) ( )) 0,
K t B f x tv f x
Optimality Conditions
In the following necessary optimality condition for the unconstrained optimization problem is established, and Fritz John and Kuhn–Tucker necessary optimality conditions in terms of the Clarke subdifferentials are obtained.
Theorem 1 (Unconstrained necessary optimality condition) If : n p
f R R satisfies local Lipschitz condition, and x0 is a local weakly efficient solution with respect to the cone K, then, there exists B such that 0 T f x( )0 .
Proof Suppose that, for some direction vRn\{0} , and all B , there holds
0 0
(f) ( , ) 0x v . From Lemma3, there is a sequence { } 0tj , and all B ,
0 0
( ( ) ( )) 0
T
j
f x t v f x
. Hence, from Lemma 1,
0 0
( (f x t vj ) f x( )) int ,K for j1, 2,,
contradicting x0 the local weakly efficient solution with respect to the cone K. Consequently,
0 0
( v Rn\{0})( K* \{0})(T f) ( , ) 0x v .
Now the hypotheses of the Lemma2 hold here with 0
0
( , ) ( ) ( , )
f v f x v , since this function is convex in v, and also concave (and positively homogeneous ) in . Therefore
0 0
( * \{0})( n\{0})( T ) ( , ) 0
K v R f x v
.
Consequently Clarke's generalized gradient satisfies 0 ( T )( )0 T ( )0
f x f x
for some
* \{0} K
, and hence for someB from K*coneB.
Theorem2 (Fritz John optimality condition) If : n p
f R R , :g Rn Rm satisfy local Lipschitz
conditions, p
K R and K1Rmare closed convex cones with nonempty interior, and x0 is a local weakly efficient solution with respect to the cone K and the constrained condition is
1
( )
g x K , then, there exist Lagrange mu1 tipliers K*and K1*, not all zero, such that
0 0
0 T ( ) T ( )
f x g x
, T ( ) 00
g x
.
Proof Let F( , )f g T; let
0 1 ( )
( ) g x
Q K K . Suppose that
0 0
( n\{0})( * \{0})( T ) ( , ) 0
v R Q F x v
.
From Lemma 3, we have
0 0
( * \{0})( ' [0, ( )]) ( (T ) ( )) 0
Q t F x tv F x
.
In particular, fixing B, and setting ( , ) with 0, then
1 0 0
( ' [0, ( )]) T( ( ) ( )) 0
t g x tv g x
,
where 1( ) may be taken independent of in the compact set A. Then, from Lemma 1,g x( 0tv)g x( )0 intK1. So that
0 1 0 1 1 1
( ) int ( ) int
By a similar argument, setting ( , ) with 0, then
2 0 0
( ' [0, ( )]) T( ( ) ( )) 0
t f x tv f x
,
where 2( ) may be taken independent of in the compact set B. Then, from Lemma
1, f x( 0tv) f x( )0 intK . Thus, for a sequence of tj 0 , g x( 0t vj ) K1 and
0 0
( j ) ( ) int
f x t v f x K, contradicting that x0 is a local weakly efficient solution. Therefore
0 0
( n\{0})( * \{0})( T ) ( , ) 0
v R Q F x v
.
An application of Lamme2 shows, as in the proof of Theorem 1, that there exists
1
( , ) K* K *
such that 0 0
0 0
( T ) ( , ) ( T T ) ( , ) 0
F x v f g x v
. Therefore, for this
1
( , ) K* K *
, 0 T ( )0 T ( )0
f x g x
. There remains that the case T ( ) 00
g x
is
obvious.
Now we obtain a necessary condition of Kuhn-Tucker type. Firstly, we give a constraint qua1ification that ensures that 0. The following Slater-type constraint qualification does this.
Slater-type constraint qualification : ( ( ))(0 n) ( ) int 1
G g x x R G x K
.
Theorem 3 (Kuhn-Tucker optimality condition) If f R: nRp, :g RnRm satisfy local
Lipschitz conditions, p
K R and K1Rmare closed convex cones with nonempty interior, x0 is a local weakly efficient solution with respect to the coneKand the constrained condition is
1
( )
g x K , and Slater-type constraint qualification : ( G g x( ))(0 x R G xn) ( ) intK1 holds, then, there exist Lagrange mu1tipliers K* \{0}and K1*, such that
0 0
0 T f x( )Tg x( ),
0
( ) 0
Tg x
Proof Assume that ( , ) 0 , and( ( ))(0 n) ( ) int 1
G g x x R G x K
.
Suppose, if possible that 0; then 0 from the hypothesis, and T 0 G
for some
0
( )
Gg x ), from Theorem 2. Then TG x( ) 0 by Slater-type constraint qualification, contradicting the hypothesis. Therefore 0, this is K* \{0}.
Conclusion
This paper presents a new type of nonsmooth vector optimization problems. This new vector optimization problems extend Pareto efficient solution to cone efficient solution and only inequality constraints to cone constraint. The objective and constraint functions are locally Lipschitz that generalizes the classical one, when the objective and constraint functions are differentiable. The new concept of the Clarke's generalized directional derivative for a locally Lipschitz vector function is introduced. By using this concept, Kuhn–Tucker necessary optimality condition for the cone efficient solution in terms of the Clarke subdifferentials is obtained.
As pointed out by an anonymous referee, we will study this type of nonsmooth vector optimization problems by considering locally Lipschitz functions, Fréchet differentiable equality constraints, locally Lipschitz inequality constraints and an abstract set constraint. We will study new constraint qualification that has a significant role in optimization problems. By using this constraint qualification, Kuhn–Tucker necessary optimality condition for the cone efficient solution will be given.
References
[1] T.Q. Bao, P. Gupta, B.S. Mordukhovich, Necessary conditions in multiobjective optimization with equilibrium constraints, J. Optim. Theory Appl. 135(2) (2007)179–203.
multiobjective optimization, J. Glob. Optim. 43(4) (2009) 533–552.
[4] T.Q. Bao, B.S. Mordukhovich, Relative Pareto minimizers for multiobjective problems: existence and optimality conditions, Math. Program. 122(2, Ser. A) (2010) 301–347.
[5] Z.K. Xu. Constraint qualifications in a class of nondifferentiable mathematical programming problems, J. Math. Anal. Appl., 302( 2005)282–290.
[6] H.Z. Luo, H.X. Wu, K-T necessary conditions for a class of nonsmooth multiobjective programming, OR Transactions, 7( 2003)62–68.
[7] H.X. Wu, H.Z. Luo, Necessary optimality conditions for a class of nonsmooth vector optimization problems, Acta Math Appl Sinica (English Series), 25(1) (2009) 87-94.
[8] X.M. Yang, X.Q.Yang, K.L. Teo, Higher-order generalized convexity and duality in nondifferentiable multiobjective mathematical programming, J Math Anal Appl, 2004,297: 48-55. [9] D.B. Bae, D. S.Kim, Optimality and duality theorems in nonsmooth multiobjective optimization, Fixed Point Theory and Appl, 2011, 2011: 42.
[10] Nobakhtian, S., Pouryayevali, M.R.: Optimality criteria for nonsmooth continuous-time problems of multiobjective optimization. J. Optim. Theory Appl. 136(1) (2008)69–76.
[11] C. Singh, Optimality conditions in multiobjective differentiable programming, J. Optim. Theory Appl. 53(1) (1987) 115–123.
[12] T. Maeda, Constraint qualifications in multiobjective optimization problems: differentiable case, J. Optim. Theory Appl. 80(3) (1994)483–500.
[13] V. Preda, Chi¸ I. tescu, On constraint qualification in multiobjective optimization problems: semi differentiable case, J. Optim. Theory Appl. 100(2) (1999)417–433.
[14] X.F.Li, Constraint qualifications in nonsmooth multiobjective optimization, J. Optim. Theory Appl.106(2) (2000)373–398.
[15] G. Giorgi, B. Jiménez, V.Novo, Strong Kuhn–Tucker conditions and constraint qualifications in locally Lipschitz multiobjective optimization problems, TOP 17(2) (2009) 288–304.
