Higher-Order Cone-pseudoconvex, Quasiconvex and Other Related Functions in Vector Optimization

Optim Lett (2013) 7:647664DOI 10.1007/s11590-012-0447-y

ORIGINAL PAPER

Higher-order cone-pseudoconvex, quasiconvexand other related functions in vector optimization

S. K. Suneja Pooja Louhan Meetu Bhatia Grover

Received: 26 January 2011 / Accepted: 11 January 2012 / Published online: 2 February 2012 Springer-Verlag 2012

Abstract Recently Bhatia (Optim. Lett. doi:10.1007/s11590-010-0248-0, 2010)introduced higher-order cone-convex functions and used them to obtain higher-ordersufficient optimality conditions and duality results for a vector optimization prob-lem over cones. The concepts of higher-order (strongly) cone-pseudoconvex andcone-quasiconvex functions were also defined by Bhatia (Optim. Lett. doi:10.1007/s11590-010-0248-0, 2010). In this paper we introduce the notions of higher-order nat-urally cone-pseudoconvex, strictly cone-pseudoconvex and weakly cone-quasiconvexfunctions and study various interrelations between the above mentioned functions.Higher-order sufficient optimality conditions have been established by using thesefunctions. Generalized MondWeir type higher-order dual is formulated and variousduality results have been established under the conditions of higher-order stronglycone-pseudoconvexity and higher-order cone quasiconvexity.

Keywords Vector optimization Cones Higher-order (naturally, strictly)cone-pseudoconvexity Higher-order weakly cone-quasiconvexity Optimality Duality

Mathematics Subject Classification (2000) 90C25 90C46 90C30

S. K. Suneja M. B. GroverDepartment of Mathematics, Miranda House, University of Delhi, New Delhi 110 007, India

S. K. Sunejae-mail: [email protected]. B. Grovere-mail: [email protected]

P. Louhan (B)Department of Mathematics, University of Delhi, New Delhi 110 007, Indiae-mail: [email protected]

123

648 S. K. Suneja et al.

1 Introduction

The field of multiobjective programming also known as vector optimization has grownremarkably in different directions in the setting of pareto optimality, game theory, equi-libria and variational inequalities as can be seen in [7] and [17]. It has been enrichedby the applications of convexity and its various generalizations as convex functionsplay a very important role in vector optimization. Convex functions and generalizedconvex functions have a wide range of applications in the field of economics, engi-neering, management and optimization. Various generalizations of convex functionshave been considered in literature. Weir et al. [22] gave the definition of cone-convexfunction. Cambini [5] introduced several classes of concave vector-valued functionswhich are possible extensions of scalar generalized concavity using the order relationsgenerated by a cone or the interior of a cone or a cone without origin. Suneja et al.[21] defined second-order cone-convex, pseudoconvex, strongly pseudoconvex andquasiconvex functions.

Mangasarian [14] introduced second and higher-order duality for a nonlinear pro-gramming problem. After that second and higher-order duality has been discussedby many authors. Mond [15] proved the second-order duality results for a nonlin-ear program established by Mangasarian [14] under different and less complicatedassumptions and also gave duality results for a pair of second-order symmetric dualproblems. Bector et al. [1] introduced four models of second-order duality for a mini-max problem and established duality theorems for each of them under generalizedbinvexity assumptions. Egudo and Hanson [8] used second-order invexity conditionsto formulate second-order duality for multiobjective programming problem. Hanson[11] introduced second-order type 1 functions and applied them to study second-orderduality in mathematical programming. Srivastava and Govil [18] defined second-order(F, , )-type I functions and their generalizations and formulated second-order dualfor a multiobjective nonlinear programming problem and Suneja et al. [21] estab-lished the second-order duality in vector optimization problem over cones. Chen [6]formulated MondWeir type and Gulati and Gupta [10] Wolfe type higher-order sym-metric dual pairs for nondifferentiable multiobjective programming problems andestablished duality results. Yang et al. [23] studied the higher-order duality in nondif-ferentiable multiobjective mathematical programming problem and Suneja et al. [20]introduced a new class of higher-order (F, , )-type I functions and used them todiscuss higher-order duality in multiobjective fractional programming with supportfunctions. The study of second and higher-order duality is significant due to the com-putational advantage over the first-order duality as it provides tighter bounds for thevalue of the objective function when approximations are used.

Recently Bhatia [3] obtained higher-order sufficient optimality conditions and dual-ity results for a vector optimization problem over cones by defining higher-order cone-convex functions. In this paper we define higher-order naturally cone-pseudoconvex,strictly cone-pseudoconvex and weakly cone-quasiconvex functions and study theirrelations with higher-order (strongly) cone-pseudoconvex and cone-quasiconvex func-tions introduced by Bhatia [3]. Kuhn and Tucker [13] and Geoffrion [9] introduced theconcept of proper efficiency which was later generalized to cones by Borwein [4] andBenson [2]. We have discussed the relation between the strong minimum and Benson

123

Higher-order cone-pseudoconvex, quasiconvex and other related functions 649

proper minimum of a vector optimization problem over cones with example. Higher-order sufficient optimality conditions for weak minimum, minimum, Benson properminimum and strong minimum are established using the above mentioned functions.

Generalized MondWeir type higher-order dual is formulated and various dualityresults have been established using higher-order strongly cone-pseudoconvex functionand higher-order cone-quasiconvex function. We have also proved duality results forthe higher-order dual considered by Bhatia [3] under the conditions of higher-order(strictly) cone-pseudoconvexity and higher-order cone quasiconvexity. In this paperwe give a constraint qualification which is weaker than Slater type constraint qualifi-cation considered by Bhatia [3] and Suneja et al. [21] and used it to prove the StrongDuality result.

2 Notations and definitions

Let K Rm be a closed convex pointed cone with vertex at origin and intK = ,where intK denotes the interior of K . The positive dual cone K + and the strict positivedual cone K s+ of K are defined as follows:

K + = {y Rm : xT y 0, for all x K },

and K s+ = {y Rm : xT y > 0, for all x K\0}.Let S be a non-empty open subset of Rn and f : S Rm, H : S Rn Rm

be vector-valued differentiable functions, where f = ( f1, f2, . . . , fm)T and H =(H1, H2, . . . , Hm)T .

On the lines of Suneja et al. [21] and Bhatia [3] we give the following definitions.

Definition 2.1 The function f is said to be higher-order K -pseudoconvex at x Swith respect to H , if for every (x, p) S Rn ,

(x x)T [ f (x) + p H(x, p)] / intK [ f (x) f (x) H(x, p) + pT p H(x, p)] / intK .

where f (x) = [ f1(x), f2(x), . . . , fm(x)].

Remark 2.1 If we take p = 0, H(x, 0) = 0 = p H(x, 0), S to be a convex set, K aclosed cone with vertex at origin such that intK = and we replace f by f and x byany y S, then Definition 2.1 reduces to the definition of (intK , intK )-pseudoconcavefunction given by Cambini [5].

Definition 2.2 The function f is said to be higher-order strongly K-pseudoconvex atx S with respect to H , if for every (x, p) S Rn ,

(x x)T [ f (x) + p H(x, p)] / intK f (x) f (x) H(x, p) + pT p H(x, p) K .

Remark 2.2 Every higher-order strongly K -pseudoconvex function is higher-orderK-pseudoconvex function but the converse is not true.

123


Definition 2.3 The function f is said to be higher-order K -quasiconvex at x S,with respect to H , if for every (x, p) S Rn ,

f (x) f (x) H(x, p) + pT p H(x, p) / intK (x x)T [ f (x) + p H(x, p)] K .

We now introduce higher-order naturally K -pseudoconvex function based on thedefinition of pseudoconvex function considered by Jahn [12].Definition 2.4 The function f is said to be higher-order naturally K -pseudoconvexat x S with respect to H , if for every (x, p) S Rn ,

(x x)T [ f (x) + p H(x, p)] K f (x) f (x) H(x, p) + pT p H(x, p) K .Remark 2.3 If we take p = 0, H(x, 0) = 0 = p H(x, 0) and if instead of Rn wetake any real normed space (X, .X ), instead of Rm any partially ordered normedspace (Y, .Y ), S to be any subset of X with nonempty interior, K to be any orderingcone in Y and f : S Y to be Frechet differentiable at x , then Definition 2.4reduces to that of pseudoconvex function at x defined by Jahn [12].

Below we give two examples to show that there is no relation between higher-ordernaturally K -pseudoconvex function and higher-order K -pseudoconvex function.Example 2.1 Let S = {(x1, x2) R2 : x1 < 2, x2 R}, K = {(x1, x2) R2 : x1 0, x1 x2},

f (x) = f (x1, x2) = (x31 x22 ,x22 ) and H(x, p) = H((x1, x2), (p1, p2)) =(p1(x1 + 1) p22,p22).Let x = (0, 0), then

(x x)T [ f (x) + p H(x, p)] K , x1 0, x1 2p2x2 0 and p1 R, f (x) f (x) H(x, p) + pT p H(x, p) K .Hence f is higher-order naturally K -pseudoconvex at x with respect to H but f is nothigher-order K -pseudoconvex at x , because for x =

(32, 1

) S and p = (1, 1) R2

(x x)T [ f (x) + p H(x, p)] =(1

2, 2

)/ intK

where as [ f (x) f (x) H(x, p) + pT p H(x, p)] =(

118

, 2)

intK .Example 2.2 Let S = {(x1, x2) R2 : x1 < 1, x2 R}, K = {(x1, x2) R2 : x2 0, x2 x1},

f (x) = (x31 x22 ,x22 ) and H(x, p) = (p1(x1 + 1), p1(x2 + 1)).Let x = (0, 0), then

(x x)T [ f (x) + p H(x, p)] / intK , x1 0, x2 R and (p1, p2) R2, [ f (x) f (x) H(x, p) + pT p H(x, p)] / intK .Hence f is higher-order K -pseudoconvex at x with respect to H but f fails to behigher-order naturally K -pseudoconvex at x , because for x = (1, 1) S and forany p R2

(x x)T [ f (x) + p H(x, p)] = (1,1) Kwhere as f (x) f (x) H(x, p) + pT p H(x, p) = (2, 1) / K .

123


Remark 2.4 Every higher-order strongly K -pseudoconvex function is higher-ordernaturally K -pseudoconvex function but the converse is not true which can be seen inthe next example.

Example 2.3 In Example 2.1 we have seen that the function f is higher-order naturallyK -pseudoconvex at x with respect to H and it is not higher-order K -pseudoconvex atx , therefore f is not higher-order strongly K -pseudoconvex at x .

A higher-order naturally K -pseudoconvex function need not be a higher-orderK -quasiconvex function, as shown in the following example.

Example 2.4 In Example 2.1 the function f is higher-order naturally K -pseudocon-vex at x with respect to H but it is not higher-order K -quasiconvex at x because forx = (0, 1) S and p = (1, 1) R2

f (x) f (x) H(x, p) + pT p H(x, p) = (2,2) / intKwhere as (x x)T [ f (x) + p H(x, p)] = (2, 2) / K .Remark 2.5 Every higher-order K -quasiconvex function is higher-order naturallyK\{0}-pseudoconvex function.

Now following the lines of Cambini [5] we give the definition of higher-orderstrictly K -pseudoconvex and higher-order weakly K -quasiconvex function.

Definition 2.5 The function f is said to be higher-order strictly K -pseudoconvex atx S with respect to H , if for every (x, p) S Rn ,

(x x)T [ f (x) + p H(x, p)] / intK [ f (x) f (x) H(x, p) + pT p H(x, p)] / K\{0}.Remark 2.6 If all the conditions of Remark 2.1 are satisfied then Definition 2.5 reducesto the definition of (K\{0}, intK )-pseudoconcave function given by Cambini [5].Remark 2.7 Every higher-order strictly K -pseudoconvex function is higher-orderK -pseudoconvex function but the converse is not true as is evident from followingexample.

Example 2.5 In Example 2.2 the function f is higher-order K -pseudoconvex at xwith respect to H but f is not higher-order strictly K -pseudoconvex at x because forx = (0, 1) S and for any p R2

(x x)T [ f (x) + p H(x, p)] = (0, 0) / intKwhere as [ f (x) f (x) H(x, p) + pT p H(x, p)] = (1,1) K\{0}.

In the next two examples we will show that neither a higher-order strictly K -pseudoconvex function is higher-order naturally K -pseudoconvex function nor is theconverse true.

Example 2.6 Let S = {(x1, x2) R2 : x1 < 1, x2 R}, K = {(x1, x2) R2 : x1 0, x1 x2},

f (x) = (x31 x22 ,2x22 ) and H(x, p) = (p1(x1 + 1),p1(x2 + 1)). Letx = (0, 0), then

123


(x x)T [ f (x) + p H(x, p)] / intK , x1 0, x2 R and (p1, p2) R2, [ f (x) f (x) H(x, p) + pT p H(x, p)] / K\{0}.Hence f is higher-order strictly K -pseudoconvex at x with respect to H but it isnot higher-order naturally K -pseudoconvex because for x = (0, 1) S and for anyp R2

(x x)T [ f (x) + p H(x, p)] = (0, 0) Kwhere as f (x) f (x) H(x, p) + pT p H(x, p) = (1,2) / K .Example 2.7 We have seen that the function f in Example 2.1 is higher-order naturallyK -pseudoconvex at x with respect to H and it fails to be higher-order K -pseudoconvexat x , therefore it is not higher-order strictly K -pseudoconvex at x .

Remark 2.8 Every higher-order strongly K -pseudoconvex function is higher-orderstrictly K -pseudoconvex function but the converse is not true as is evident from fol-lowing example.

Example 2.8 The function in Example 2.6 is higher-order strictly K -pseudoconvexat x with respect to H but it fails to be higher-order naturally K -pseudoconvex andhence it is not higher-order strongly K -pseudoconvex at x .

Now in the next two examples we show that higher-order strictly K -pseudoconvexfunctions are not related to higher-order K -quasiconvex functions.Example 2.9 Consider the function f in Example 2.6, it is higher-order strictlyK -pseudoconvex at x with respect to H but it is not higher-order K -quasiconvexat x because for x = (1,1) S and for any p R2

f (x) f (x) H(x, p) + pT p H(x, p) = (2,2) / intKwhere as (x x)T [ f (x) + p H(x, p)] = (1,1) / K .Example 2.10 Let S = {(x1, x2) R2 : x1 < 1, x2 R}, K = {(x1, x2) R2 :x1 0, x1 x2},

f (x) = (x31 x22 ,x22 ) and H(x, p) = (p1(x1 + 1), p1(x2 + 1)).Let x = (0, 0), then

f (x) f (x) H(x, p) + pT p H(x, p) / intK , 0 x1 < 1, x2 R and (p1, p2) R2, (x x)T [ f (x) + p H(x, p)] K .Hence f is higher-order K -quasiconvex at x with respect to H but it is not higher-order strictly K -pseudoconvex at x because for x =

(12, 0

) S and for any p R2

(x x)T [ f (x) + p H(x, p)] =(

12,1

2

)/ intK

where as [ f (x) f (x) H(x, p) + pT p H(x, p)] =(

18, 0

) K\{0}.

Definition 2.6 The function f is said to be higher-order weakly K -quasiconvex atx S with respect to H , if for every (x, p) S Rn ,

[ f (x) f (x) H(x, p) + pT p H(x, p)] K (x x)T [ f (x) + p H(x, p)] K .

123


Remark 2.9 If all the conditions of Remark 2.1 are satisfied then Definition 2.6 reducesto the definition of weakly (K , K )-quasiconcave function given by Cambini [5].

The following example shows that a higher-order weakly K -quasiconvex func-tions is not higher-order K -pseudoconvex functions and hence is neither higher-orderstrictly K -pseudoconvex nor higher-order strongly K -pseudoconvex.

Example 2.11 Let S = {(x1, x2) R2 : x1 < 2, x2 R}, K = {(x1, x2) R2 :x1 x2, x1 0},

f (x) = (x31 x22 ,x22 ) and H(x, p) = (p1(x1 + 1),p1(x2 + 1)).Let x = (0, 0), then

[ f (x) f (x) H(x, p) + pT p H(x, p)] K . x1 0, x2 R and (p1, p2) R2, (x x)T [ f (x) + p H(x, p)] K ,Hence f is higher-order weakly K -quasiconvex at x with respect to H but f fails tobe higher-order K -pseudoconvex at x with respect to H because for x = (1, 0) Sand for any p R2

(x x)T [ f (x) + p H(x, p)] = (1,1) / intKwhere as [ f (x) f (x) H(x, p) + pT p H(x, p)] = (1, 0) intK .Remark 2.10 Every higher-order strictly K -pseudoconvex function and hence everyhigher-order strongly K -pseudoconvex function is higher-order weakly K\{0}-quasi-convex.

In the next two examples we will show that higher-order weakly K -quasiconvexfunction and higher-order naturally K -pseudoconvex function are not related.

Example 2.12 In Example 2.11 f is higher-order weakly K -quasiconvex at x withrespect to H but f fails to be higher-order naturally K -pseudoconvex because forx = (1, 2) S and for any p R2

(x x)T [ f (x) + p H(x, p)] = (1,1) Kwhere as f (x) f (x) H(x, p) + pT p H(x, p) = (3, 4) / K .Example 2.13 Let S = {(x1, x2) R2 : x1 < 2, x2 R}, K = {(x1, x2) R2 :x1 x2 0}.

f (x) = (x31 x2,x2) and H(x, p) = (p1(x1 + 1) + p2(x2 + 1), x1).Let x = (0, 0), then

(x x)T [ f (x) + p H(x, p)] K , x1 0, x2 0 and (p1, p2) R2, f (x) f (x) H(x, p) + pT p H(x, p) K .Hence f is higher-order naturally K -pseudoconvex at x with respect to H but f isnot higher-order weakly K -quasiconvex function, because for x = (1,2) S andfor any p R2

[ f (x) f (x) H(x, p) + pT p H(x, p)] = (3,2) Kwhere as (x x)T [ f (x) + p H(x, p)] = (1,2) / KRemark 2.11 Every higher-order K -quasiconvex function is higher-order weaklyK -quasiconvex but converse is not true, as can be seen from the following example.

123


Example 2.14 The function f considered in Example 2.11 is higher-order weaklyK -quasiconvex but it fails to be higher-order K -quasiconvex, because for x = (1, 1) S and for any p R2

f (x) f (x) H(x, p) + pT p H(x, p) = (0, 1) / intKwhere as (x x)T [ f (x) + p H(x, p)] = (1, 1) / K

3 Optimality conditions

Consider the vector optimization problem(VP) K -Minimize f (x)

subject to g(x) Q,where f : S Rm and g : S Rp are differentiable vector-valued functionsand S is non-empty open subset of Rn . K and Q are closed convex pointed cones inR

m and Rp respectively with non-empty interiors. The feasible set of (VP) is givenby S0 = {x S : g(x) Q}.Definition 3.1 A point x S0 is called1. a weak minimum of (VP), if for all x S0, f (x) f (x) / intK .2. a minimum of (VP), if for all x S0, f (x) f (x) / K\{0}.3. a strong minimum of (VP), if for all x S0, f (x) f (x) K .4. a Benson proper minimum of (VP), if

(K ) cone( f (S0) + K f (x)) = {0}

where for any subset D of Rm , cone(D) denotes the closure of cone generated by D.

Remark 3.1 Every strong minimum of (VP) is a Benson proper minimum of (VP) butthe converse is not true which is evident from the following example.

Example 3.1 Let S = {(x1, x2) R2 : x1 < 1, x2 R}.f (x) = (x31 x22 ,x22 ), K = {(x1, x2) R2 : x2 x1 0}.g(x) = (x31 , x31 x2), Q = {(x1, x2) R2 : 0 x1 x2}.

Then, the feasible region is,S0 = {(x1, x2) S : x1 0, x2 0}.

Let x = (0, 0), then f (x) = (0, 0) and(K ) cone( f (S0) + K f (x)) = {0}

Therefore x is a Benson proper minimum of (VP), but it is not a strong minimumbecause for x = (1, 0) S0

f (x) f (x) = (1, 0) / K .Bhatia [3] established higher-order sufficient optimality conditions for (VP) by

using the concept of higher-order cone-convex functions. We now obtain higher-ordersufficient optimality conditions by using higher-order (strongly, strictly, naturally)cone-pseudoconvex and (weakly) cone-quasiconvex functions. Let H : S Rn R

m and G : S Rn Rp be differentiable vector-valued functions.

123


Theorem 3.1 Let f be higher-order strongly K -pseudoconvex at x S0 with respectto H and g be higher-order Q-quasiconvex at x with respect to G. Suppose that thereexist K +\{0} and Q+, such that for all (x, p) S0 Rn,

(xx)T [(T f )(x) + p(T H)(x, p) + (T g)(x) + p(T G)(x, p)] 0,(3.1)

(T g)(x) + (T G)(x, p) pT p(T G)(x, p) 0, (3.2)[(T H)(x, p) pT p(T H)(x, p)] 0. (3.3)

Then x is a weak minimum of (VP).

Proof For all x S0,g(x) Q and therefore

(T g)(x) 0 x S0.

From (3.2) and above inequality, we get

[(T g)(x)(T g)(x)(T G)(x, p)+ pT p(T G)(x, p)]0 (x, p) S0Rn .

If = 0, then we have

g(x) g(x) G(x, p) + pT pG(x, p) / intQ (x, p) S0 Rn .

Since g is higher-order Q-quasiconvex at x with respect to G, therefore

(x x)T [g(x) + pG(x, p)] Q (x, p) S0 Rn,

which implies that

(xx)T [(T g)(x)+p(T G)(x, p)]0 (x, p) S0Rn . (3.4)

If = 0, then also (3.4) holds. Now using (3.4) in (3.1), we get

(x x)T [(T f )(x) + p(T H)(x, p)] 0 (x, p) S0 Rn, (3.5)

which implies that

(x x)T [ f (x) + p H(x, p)] / intK (x, p) S0 Rn . (3.6)

Since f is higher-order strongly K -pseudoconvex at x with respect to H , therefore

f (x) f (x) H(x, p) + pT p H(x, p) K (x, p) S0 Rn,

123


which implies that

[(T f )(x) (T f )(x) (T H)(x, p)+pT p(T H)(x, p)]0 (x, p) S0 Rn .

Adding (3.3) and above inequality, we get

[(T f )(x) (T f )(x)] 0 x S0. (3.7)

which implies that

f (x) f (x) / intK x S0.

Therefore, x is a weak minimum of (VP). unionsqTheorem 3.2 Let f be higher-order strictly K -pseudoconvex at x S0 with respectto H and g be higher-order Q-quasiconvex at x with respect to G. Suppose that thereexist K s+ and Q+, such that for all (x, p) S0 Rn, (3.1) and (3.2)are satisfied and (3.3) holds (with replaced by ) for every K s+. Then x is aminimum of (VP).Proof Proceeding on the lines of proof of Theorem 3.1, we get

(x x)T [ f (x) + p H(x, p)] / intK (x, p) S0 Rn .

Since f is higher-order strictly K -pseudoconvex at x with respect to H , therefore

[ f (x) f (x) H(x, p) + pT p H(x, p) / K\{0} x S0.

Then for each x S and p Rn we have at least one K s+ such that

(T f )(x) (T f )(x) (T H)(x, p) + pT p(T H)(x, p) 0.


(T f )(x) (T f )(x) 0.

which implies that

f (x) f (x) / K\{0} x S0.

Hence x is a minimum of (VP). unionsqTheorem 3.3 Let f be higher-order strongly K -pseudoconvex at x S0 with respectto H and g be higher-order Q-quasiconvex at x with respect to G. Suppose that thereexist K s+ and Q+, such that for all (x, p) S0 Rn, (3.1), (3.2) and (3.3)hold. Then x is a Benson proper minimum of (VP).

123


Proof Let, if possible, x be not a Benson proper minimum of (VP), then there exists

0 = k (K ) cone( f (S0) + K f (x))

Therefore,

T k < 0. (3.8)

and there exists a sequence {xn}nN in S0, {kn}nN in K and {n}nN in the interval[0,) such that

n[ f (xn) + kn f (x)] k as n

Therefore,

n[(T f )(xn) + T kn (T f )(x)] T k as n .

Since f and g satisfy all the conditions of Theorem 3.1, so by using (3.7) we get

[(T f )(xn) (T f )(x)] 0 n N.

Further, kn K and K s+ implies that T kn > 0 n N.Thus, we get that n[(T f )(xn) + T kn (T f )(x)] 0 n N and hence

T k 0, which is a contradiction to (3.8). Hence x is a Benson proper minimum of(VP). unionsqTheorem 3.4 Let f be higher-order naturally K -pseudoconvex at x S0 with respectto H and g be higher-order weakly Q-quasiconvex at x with respect to G. Supposethat for all (x, p) S0 Rn, (3.1), (3.2) and (3.3) hold (with replaced by and replaced by ) for every K +, Q+. Then x is a strong minimum of (VP).Proof For all x S0,g(x) Q and therefore

(T g)(x) 0 x S0 & Q+.

From (3.2) and above inequality, we get

[(T g)(x) (T g)(x) (T G)(x, p) + pT p(T G)(x, p)] 0(x, p) S0 Rn & Q+.

which implies that

[g(x)g(x)G(x, p)+ pT pG(x, p)] (Q+)+ = Q = Q (x, p) S0Rn .

123


Since g is higher-order weakly Q-quasiconvex at x with respect to G, therefore

(x x)T [g(x) + pG(x, p)] Q (x, p) S0 Rn,

which implies that

(x x)T [(T g)(x) + p(T G)(x, p)] 0 (x, p) S0 Rn & Q+.

Now using (3.1) in above inequality, we get

(x x)T [(T f )(x) + p(T H)(x, p)] 0 (x, p) S0 Rn & K +,

which implies that

(x x)T [ f (x) + p H(x, p)] K (x, p) S0 Rn .

Since f is higher-order naturally K -pseudoconvex at x with respect to H , therefore

f (x) f (x) H(x, p) + pT p H(x, p) K (x, p) S0 Rn,

which implies that

[(T f )(x) (T f )(x) (T H)(x, p) + pT p(T H)(x, p)] 0(x, p) S0 Rn & K +.


[(T f )(x) (T f )(x)] 0 x S0 & K +,

which implies that

f (x) f (x) K . x S0.

Therefore, x is a strong minimum of (VP). unionsqThe above theorem is illustrated in the following example.

Example 3.2 Let S = {(x1, x2) R2 : x1 < 2, x2 R}.f (x) = (x31 x22 ,x22 ), H(x, p)) = (2p1(x1 + 1),p1(x2 + 1)),K = {(x1, x2) R2 : x1 0, x1 x2},g(x) = (x31 , x31 + x22 ), G(x, p) = (x2, p2(x1 + 1)),Q = {(x1, x2) R2 : 0 x1 x2}.

Let x = (0, 0), then f is higher-order naturally K -pseudoconvex at x with respect toH because,

(x x)T [ f (x) + p H(x, p)] K , x1 0, x2 R and (p1, p2) R2,

123


f (x) f (x) H(x, p) + pT p H(x, p) K .and g is higher-order weakly Q-quasiconvex at x with respect to G because,

[g(x) g(x) G(x, p) + pT pG(x, p)] Q, x1 0, x2 = 0 and (p1, p2) R2, (x x)T [g(x) + pG(x, p)] Q.The feasible region for (VP) is S0 = {(x1, x2) S : x1 0, x2 = 0}.

K + = {(x1, x2) R2 : 0 x2 x1} and Q+ = {(x1, x2) R2 : x2 x1, x2 0}.Then for all (x, p) S0 R2, K + and Q+

(xx)T [(T f )(x)+p(T H)(x, p)+(T g)(x)+p(T G)(x, p)] 0,[(T g)(x) + (T G)(x, p) pT p(T G)(x, p)] = 0,[(T H)(x, p) pT p(T H)(x, p)] = 0.

So (3.1), (3.2) and (3.3) are satisfied for all K +, Q+ and (x, p) S0 R2therefore x is a strong minimum of (VP).

4 Higher-order duality

We associate the following generalized MondWeir type higher-order dual with (VP)(HD1) K -Maximize f (w)

subject to(xw)T [(T f )(w)+p(T H)(w, p)+(T g)(w)+p(T G)(w, p)]

0 x S0,[(T g)(w) + (T G)(w, p) pT p(T G)(w, p)] 0 and(T H)(w, p) pT p(T H)(w, p) 0.where K +\{0}, Q+, w S, p Rn .

Now we will establish Weak Duality relation for (VP) and its higher-order dual(HD1).Theorem 4.1 (Weak Duality) Let x be feasible for (VP) and (w, , , p) for (HD1).Suppose that f is higher-order strongly K -pseudoconvex at w with respect to H andg is higher-order Q-quasiconvex at w with respect to G, then

f (w) f (x) / intK .

Proof Since x is feasible for (VP) and (w, , , p) is feasible for (HD1), therefore

[(T g)(x) (T g)(w) (T G)(w, p) + pT p(T G)(w, p)] 0.If = 0, then we have

g(x) g(w) G(w, p) + pT pG(w, p) / intQ.Since g is higher-order Q-quasiconvex at w with respect to G, therefore

(x w)T [g(w) + pG(w, p)] Q,

123


which implies that

(x w)T [(T g)(w) + p(T G)(w, p)] 0.

If = 0, then also the above inequality holds.Since (w, , , p) is feasible for (HD1), so from above inequality we get

(x w)T [(T f )(w) + p(T H)(w, p)] 0,

which implies that

(x w)T [ f (w) + p H(w, p)] / intK .

Further, f is higher-order strongly K -pseudoconvex at w with respect to H , therefore

f (x) f (w) H(w, p) + pT p H(w, p) K .

and hence

(T f )(x) (T ) f (w) (T H)(w, p) + pT p(T )H(w, p) 0.

Again using the fact that (w, , , p) is feasible for (HD1), we get

(T ) f (w) (T f )(x) 0

which implies that

f (w) f (x) / intK .

unionsqSuneja et al. [19] gave the following Fritz John type necessary optimality conditions

for a point to be a weak minimum of (VP).

Lemma 4.1 If x is a weak minimum of (VP), then there exist K + and Q+with (, ) = 0 such that

(x x)T [(T f )(x) + (T g)(x)] 0, x S,(T g)(x) = 0.

In order to establish Strong Duality result we will prove the following Kuhn-Tuckertype necessary optimality conditions for a point to be a weak minimum of (VP).

Lemma 4.2 If x is a weak minimum of (VP), then there exist K + and Q+with (, ) = 0 such that

123


(x x)T [(T f )(x) + (T g)(x)] 0, x S, (4.1)(T g)(x) = 0. (4.2)

Moreover, if g is Q-convex at x and there exists x S such that (T g)(x) < 0,then = 0.Proof Let x be a weak minimum of (VP), then we invoke Lemma 4.1 to deduce thatthere exist K + and Q+ with (, ) = 0 such that (4.1) and (4.2) hold.

Suppose now that g is Q-convex at x and there exists x S such that (T g)(x)


Let x = (1, 0) then f (x) = (1, 0),Now for all x S0, f (x) f (x) = (x31 1, 0) / intK .Therefore x is a weak minimum of (VP). So, there exist = (2, 1) K + and = (2, 0) Q+, such that(x x)T [(T f )(x) + (T g)(x)] = 0, x S and(T g)(x) = 0.Hence the necessary conditions are satisfied. Also, since

g(x)g(x)(xx)T g(x) = (x31 3x12, x31 3x12x22 ) Q x S,so g is Q-convex at x .and there exists x = (1, 0) S such that (T g)(x) = 4 < 0.Hence g satisfies the generalized Slater cone constraint qualification at x . But if forsome x = (x1, x2) S, g(x) intQ, then x22 < 0 which is not possible. Thereforeg does not satisfy the Slater type constraint qualification.

Theorem 4.2 (Strong Duality) Let x be a weak minimum for (VP), then there exist K + and Q+ with (, ) = 0 such that (4.1) and (4.2) hold. Moreover, ifH(x, 0) = 0 = G(x, 0),p H(x, 0) = 0 = pG(x, 0), g is Q-convex at x and thereexist x S such that (T g)(x) < 0, then (x, , , p = 0) is feasible for (HD1).Further, if the conditions of Weak Duality Theorem 4.1 are satisfied for each feasiblesolution (w, , , p) of (HD1), then (x, , , p = 0) is weak maximum for (HD1).Proof Since x is a weak minimum for (VP), therefore by Lemmas 4.1 and 4.2 thereexist K +\{0} and Q+ such that

(x x)T [(T f )(x) + (T g)(x)] 0, x S,(T g)(x) = 0.

Thus, (x, , , p = 0) is feasible for (HD1).Now if possible, let (x, , , p = 0) be not a weak maximum of (HD1), then thereexists a feasible solution (w, , , p) of (HD1) such that

f (w) f (x) intK .

which is a contradiction to the Weak Duality Theorem 4.1 for the feasible solution xfor (VP) and (w, , , p) of (HD1). Therefore (x, , , p = 0) is weak maximumof (HD1). unionsq

Now we associate the following higher-order dual with (VP) which was also con-sidered by Bhatia [3].(HD2) K -Maximize f (w) + H(w, p) pT p H(w, p)

subject to(xw)T [(T f )(w)+p(T H)(w, p)+(T g)(w)+p(T G)(w, p)]

0 x S0 and[(T g)(w) + (T G)(w, p) pT p(T G)(w, p)] 0.where K +\{0}, Q+, w S, p Rn .

123


Bhatia [3] proved Weak Duality by assuming the concerned functions to be higher-order cone-convex. Now we will establish Weak Duality relations between the primalproblem (VP) and its higher-order dual (HD2) by assuming the objective function to behigher-order (strictly) cone-pseudoconvex and constraint function to be higher-ordercone-quasiconvex.

Theorem 4.3 (Weak Duality) Let x be feasible for (VP) and (w, , , p) for (HD2).Suppose that f is higher-order K -pseudoconvex at w with respect to H and g ishigher-order Q-quasiconvex at w with respect to G then

f (w) + H(w, p) pT p H(w, p) f (x) / intK .

Proof Proceeding on the lines of proof of Theorem 4.1, we get

(x w)T [ f (w) + p H(w, p)] / intK .

Further, f is higher-order K -pseudoconvex at w with respect to H , therefore

f (w) + H(w, p) pT p H(w, p) f (x) / intK .

unionsqRemark 4.2 If, for each i = 1, 2, . . . , m, fi is twice continuously differentiable andwe take Hi (x, p) = 12 p

T 2 fi (x)p and for each i = 1, 2, . . . , p, gi is twice contin-uously differentiable and we take Gi (w, p) = 12 p

T 2gi (w)p then the above WeakDuality result reduces to that of Suneja et al. [21].Theorem 4.4 (Weak Duality) Let x be feasible for (VP) and (w, , , p) for (HD2).Suppose that f is higher-order strictly K -pseudoconvex at w with respect to H andg is higher-order Q-quasiconvex at w with respect to G then

f (w) + H(w, p) pT p H(w, p) f (x) / K\{0}.

Proof Proceeding on the lines of proof of Theorem 4.3 and using the higher-orderstrict K -pseudoconvexity of f , we get the desired result. unionsq

We now give a Strong Duality result which can be proved exactly on the lines ofthe Theorem 4.2.Theorem 4.5 (Strong Duality) Let x be a weak minimum for (VP), then there exist K + and Q+ with (, ) = 0 such that (4.1) and (4.2) hold. Moreover, ifH(x, 0) = 0 = G(x, 0),p H(x, 0) = 0 = pG(x, 0), g is Q-convex at x and thereexists x S such that (T g)(x) < 0, then (x, , , p = 0) is feasible for (HD2).Further, if the conditions of Weak Duality Theorem 4.3 are satisfied for each feasiblesolution (w, , , p) of (HD2), then (x, , , p = 0) is weak maximum for (HD2) and

123


if the conditions of Weak Duality Theorem 4.4 are satisfied for each feasible solution(w, , , p) of (HD2), then (x, , , p = 0) is maximum for (HD2).Acknowledgments The authors are thankful to the anonymous referees whose constructive suggestionshave improved the presentation of the paper. The second author is also thankful to Council of Scientificand Industrial Research (CSIR), India for the financial support provided during the work. The other twoauthors are grateful to University Grants Commission (UGC), India for offering financial support.

References

1. Bector, C.R., Chandra, S., Husain, I.: Second-order duality for minimax programming problem.Opsearch 28, 249263 (1991)

2. Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect tocones. J. Math. Anal. Appl. 71, 232241 (1979)

3. Bhatia, M.: Higher order duality in vector optimization over cones. Optim. Lett. (2010). doi:10.1007/s11590-010-0248-0

4. Borwein, J.: Proper efficient points for maximization with respect to cones. SIAM J. Control.Optim. 15, 5763 (1977)

5. Cambini, R.: Some new classes of generalized concave vector-valued functions. Optimization 36, 1124 (1996)

6. Chen, X.: Higher-order symmetric duality in non-differentiable multiobjective programming prob-lems. J. Math. Anal. Appl. 290, 423435 (2004)

7. Chinchuluun, A., Migdalas, A., Pardalos, P.M., Pitsoulis, L.: Pareto Optimality, Game Theory andEquilibria. Springer, New York (2008)

8. Egudo, R.R., Hanson, M.A.: Second order duality in multiobjective programming. Opsearch 30, 223230 (1993)

9. Geoffrion, A.N.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 613630 (1968)

10. Gulati, T.R., Gupta, S.K.: Higher-order non-differentiable symmetric duality with generalized F-con-vexity. J. Math. Anal. Appl. 329, 229237 (2007)

11. Hanson, M.: Second-order invexity and duality in mathematical programming. Opsearch 30, 313320 (1993)

12. Jahn, J.: Vector Optimization. Springer, Heidelberg (2004)13. Kuhn, H.W., Tucker, A.W.: Nonlinear Programming. In: Neyman, J. Second Barkeley symposium on

mathematical statistics and probability., University of California Press, Berkeley (1951)14. Mangasarian, O.L.: Second- and higher-order duality in nonlinear programming. J. Math. Anal.

Appl. 51, 607620 (1975)15. Mond, B.: Second-order duality for nonlinear programs. Opsearch 11, 9099 (1974)16. Nguyen, M.H., Luu, D.V.: On constraint qualification with generalized convexity and optimality con-

ditions. Cah. de la MSE (2005)17. Oveisiha, M., Zafarani, J.: Vector optimization problem and generalized convexity. J. Glob. Optim.

52, 2943 (2012)18. Srivastava, M.K., Govil, M.G.: Second-order duality for multiobjective programming involving

(F, , )-type-I functions. Opsearch 37, 316326 (2000)19. Suneja, S.K., Agarwal, S., Davar, S.: Multiobjective symmetric duality involving cones. Eur. J. Oper.

Res. 141, 471479 (2002)20. Suneja, S.K., Srivastava, M., Bhatia, M.: Higher-order duality in multiobjective fractional program-

ming with support functions. J. Math. Anal. Appl. 347, 817 (2008)21. Suneja, S.K., Sharma, S., Vani: Second-order duality in vector optimization over cones. J. Appl. Math.

Inform. 26, 251261 (2008)22. Weir, T., Mond, B., Craven, B.D.: Weak minimization and duality. Numer. Funct. Anal. Optim. 9, 181

192 (1987)23. Yang, X.M., Teo, K.L., Yang, X.Q.: Higher-order generalized convexity and duality in non-differen-

tiable multiobjective mathematical programming. J. Math. Anal. Appl. 297, 4855 (2004)

123

Higher-order cone-pseudoconvex, quasiconvex and other related functions in vector optimizationAbstract1 Introduction2 Notations and definitions3 Optimality conditions4 Higher-order dualityAcknowledgmentsReferences

Documents

Higher-Order Cone-pseudoconvex, Quasiconvex and Other Related Functions in Vector Optimization