Optimality Conditions in Nondifferentiable G-Invex Multiobjective Programming

Abstract

We consider a class of nondifferentiable multiobjective programs with inequality and equality constraints in which each component of the objective function contains a term involving the support function of a compact convex set. We introduce G-Karush-Kuhn-Tucker conditions and G-Fritz John conditions for our nondifferentiable multiobjective programs. By using suitable G-invex functions, we establish G-Karush-Kuhn-Tucker necessary and sufficient optimality conditions, and G-Fritz John necessary and sufficient optimality conditions of our nondifferentiable multiobjective programs. Our optimality conditions generalize and improve the results in Antczak (2009) to the nondifferentiable case.

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Recommended by Jong Kyu Kim

Division of Mathematical Sciences, Pukyong National University, Busan 608-737, South Korea

Received 29 October 2009; Revised 10 March 2010; Accepted 14 March 2010

1. Introduction and Preliminaries

A number of different forms of invexity have appeared. In [1], Martin defined Kuhn-Tucker invexity and weak duality invexity. In [2], Ben-Israel and Mond presented some new results for invex functions. Hanson [3] introduced the concepts of invex functions, and Type I, Type II functions were introduced by Hanson and Mond [4]. Craven and Glover [5] established Kuhn-Tucker type optimality conditions for cone invex programs, and Jeyakumar and Mond [6] introduced the class of the so-called V-invex functions to proved some optimality for a class of differentiable vector optimization problems than under invexity assumption.Egudo [7] established some duality results for differentiable multiobjective programming problems with invex functions. Kaul et al. [8] considered Wolfe-type and Mond-Weir-type duals and generalized the duality results of Weir [9] under weaker invexity assumptions.

Based on the paper by Mond and Schechter [10], Yang et al. [11] studied a class of nondifferentiable multiobjective programs. They replaced the objective function by the support function of a compact convex set, constructed a more general dual model for a class of nondifferentiable multiobjective programs, and established only weak duality theorems for efficient solutions under suitable weak convexity conditions. Subsequently, Kim et al. [12] established necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems.

Recently, Antczak [13, 14] studied the optimality and duality for G-multi-objective programming problems. They defined a new class of differentiable nonconvex vector valued functions, namely, the vector G-invex (G-incave) functions with respect to η . They used vector G-invexity to develop optimality conditions for differentiable multiobjective programming problems with both inequality and equality constraints. Considering the concept of a (weak) Pareto solution, they established the so-called G-Karush-Kuhn-Tucker necessary optimality conditions for differentiable vector optimization problems under the Kuhn-Tucker constraint qualification.

In this paper, we obtain an extension of the results in [13],which were established in the differentiable to the nondifferentiable case. We proposed a class of nondifferentiable multiobjective programming problems in which each component of the objective function contains a term involving the support function of a compact convex set. We obtain G-Karush-Tucker necessary and sufficient conditions and G-Fritz John necessary and sufficient conditions for weak Pareto solution. Necessary optimal theorems are presented by using alternative theorem [15] and Mangasarian-Fromovitz constraint qualification [16]. In addition, we give sufficient optimal theorems under suitable G-invexity conditions.

We provide some definitions and some results that we shall use in the sequel. Throughout the paper, the following convention will be used.

For any x=^{(_x1 ,_x2 ,...,_xn )T} , y=^{(_y1 ,_y2 ,...,_yn )T} , we write

[figure omitted; refer to PDF]

Throughout the paper, we will use the same notation for row and column vectors when the interpretation is obvious. We say that a vector z∈^...n is negative if z[<, double =]0 and strictly negative if z<0 .

Definition 1.1.

A function f:...[arrow right]... is said to be strictly increasing if and only if [figure omitted; refer to PDF]

Let f=(_f1 ,...,_fk ):X[arrow right]^...k be a vector-valued differentiable function defined on a nonempty open set X⊂^...n , and _Ifi (X),i=1,...,k , the range of _fi , that is, the image of X under _fi .

Definition 1.2 (see [11]).

Let C be a compact convex set in ^...n . The support function s(x|"C) is defined by [figure omitted; refer to PDF] The support function s(x|"C) , being convex and everywhere finite, has a subdifferential, that is, there exists z such that [figure omitted; refer to PDF] Equivalently, [figure omitted; refer to PDF] The subdifferential of s(x|"C) at x is given by [figure omitted; refer to PDF]

Now, in the natural way, we generalize the definition of a real-valued G-invex function. Let f=(_f1 ,...,_fk ):X[arrow right]^...k be a vector-valued differentiable function defined on a nonempty open set X⊂^...n , and _Ifi (X),i=1,...,k, the range of _fi , that is, the image of X under _fi .

Definition 1.3.

Let f:X[arrow right]^...n be a vector-valued differentiable function defined on a nonempty set X⊂^...n and u∈X . If there exist a differentiable vector-valued function _Gf =(_Gf1 ,...,_Gfk ):...[arrow right]^...k such that any of its component _Gfi :_Ifi (X)[arrow right]... is a strictly increasing function on its domain and a vector-valued function η:X×X[arrow right]^...n such that, for all x∈X (x≠u) and for any i=1,...,k, [figure omitted; refer to PDF] then f is said to be a (strictly) vector _Gf -invex function at u on X (with respect to η ) (or shortly, G -invex function at u on X ). If (1.7) is satisfied for each u∈X , then f is vector _Gf -invex on X with respect to η .

Lemma 1.4 (see [13]).

In order to define an analogous class of (strictly) vector _Gf -incave functions with respect to η , the direction of the inequality in the definition of _Gf -invex function should be changed to the opposite one.

We consider the following multiobjective programming problem.

[figure omitted; refer to PDF] where _fi :X[arrow right]..., i∈I={1,...,k}, _gj :X[arrow right]..., j∈J={1,...,m}, _ht :X[arrow right]..., t∈T={1,...,p} , are differentiable functions on a nonempty open set X⊂^...n . Moreover, _GFi ,i∈I, are differentiable real-valued strictly increasing functions, _Ggj ,j∈J, are differentiable real-valued strictly increasing functions, and _Ght ,t∈T , are differentiable real-valued strictly increasing functions. Let D={x∈X:_Ggj (_gj (x))[<, double =]0, j∈J,_Ght (_ht (x))=0,t∈T} be the set of all feasible solutions for problem (NMP), and _Fi =_fi (·)+(·^)T_wi . Further, we denote by J(z):={j∈J:_Ggj (_gj (z))=0} the set of inequality constraint functions active at z∈D and by I(z):={i∈I:_λi >0} the objective functions indices set, for which the corresponding Lagrange multiplier is not equal 0 . For such optimization problems, minimization means in general obtaining weak Pareto optimal solutions in the following sense.

Definition 1.5.

A feasible point x¯ is said to be a weak Pareto solution (a weakly efficient solution, a weak minimum) of (NMP) if there exists no other x∈D such that [figure omitted; refer to PDF]

Definition 1.6 (see [17]).

Let W be a given set in ^...n ordered by [<, double =] or by < . Specifically, we call the minimal element of W defined by ≤ a minimal vector, and that defined by < a weak minimal vector. Formally speaking, a vector z¯∈w is called a minimal vector in W if there exists no vector z in W such that z≤z¯ ; it is called a weak minimal vector if there exists no vector z in W such that z<z¯ .

By using the result of Antczak [13] and the definition of a weak minimal vector, we obtain the following proposition.

Proposition 1.7.

Let x¯ be feasible solution in a multiobjective programming problem and let _{Gfi (·)+(·^)Twi} ,i=1,...,k , be a continuous real-valued strictly increasing function defined on _{Ifi +(·^)Twi} (X) . Further, we denote W={_{Gf1 (·)+(·^)Tw1} (_f1 (x)+s(x|"_C1 )),...,_{Gfk (·)+(·^)Twk} (_fk (x)+s(x|"_Ck )):x∈X}⊂^...k and z¯=(_{Gf1 (·)+(·^)Tw1} (_f1 (x¯)+s(x¯|"_C1 )),...,_{Gfk (·)+(·^)Twk} (_fk (x¯)+s(x¯|"_Ck ))∈W . Then, x¯ is a weak Pareto solution in the set of all feasible solutions X for a multiobjective programming problem if and only if the corresponding vector z¯ is a weak minimal vector in the set W .

Proof.

Let x¯ be a weak Pareto solution. Then there does not exist^x* such that [figure omitted; refer to PDF] By the strict increase of _{Gfi (·)+(·^)Twi} involving the support function, we have [figure omitted; refer to PDF] Therefore, z¯=(_{Gf1 (·)+(·^)Tw1} (_f1 (x¯)+s(x¯|"_C1 )),...,_{Gfk (·)+(·^)Twk} (_fk (x¯)+s(x¯|"_Ck ))) is a weak minimal vector in the set W. The converse part is proved similarly.

Lemma 1.8 (see [13]).

In the case when _GFi (a)≡a, i=1,...,k , for any a∈_IFi (X) , we obtain a definition of a vector-valued invex function.

2. Optimality Conditions

In this section, we establish G-Fritz John and G-Karush-Kuhn-Tucker necessary and sufficient conditions for a weak Pareto optimal point of (NMP).

Theorem 2.1 (G-Fritz John Necessary Optimality Conditions).

Suppose that _GFi , i∈I, are differentiable real-valued strictly increasing functions defined on _IFi (D),_Ggj ,j∈J, are differentiable real-valued strictly increasing functions defined on _Igj (D),and _Ght , t∈T , are differentiable real-valued strictly increasing functions defined on _Iht (D) , and let _Fi =_fi (·)+(·^)T_wi . Let x¯∈D be a weak Pareto optimal point in problem (NMP). Then there exist λ∈^...+k ,ξ∈^...+m , μ∈^...p , and _wi ∈_Ci such that [figure omitted; refer to PDF]

Proof.

Let _bi (x¯)=s(x¯|"_Ci ), i=1,...,k . Since _Ci is convex and compact, [figure omitted; refer to PDF] is finite. Also, for all d∈^...n , [figure omitted; refer to PDF]

Since x¯ is a weak Pareto optimal point in (NMP) [figure omitted; refer to PDF] has no solution d∈^...n .By [15, Corollary 4.2.2 ], there exist _λi [>, double =]0, i=1,...,k, _ξj [>, double =]0, j∈J(x¯) , and _μt ,t=1,...,p , not all zero, such that for any d∈^...n , [figure omitted; refer to PDF] Let A = {^∑i=1k_λi [_{^{G[variant prime]}Fi} (_fi (x¯)+_bi (x¯))(∇_fi (x¯)+_wi )]+_{∑j∈J(x¯)ξj^{G[variant prime]}gj} (_gj (x¯))∇_gj (x¯)+^∑t=1p_μt^{G_ht [variant prime]} (_ht (x¯))∇_ht (x¯)|"_wi ∈∂_bi (x¯),i=1,...,k} . Then 0∈A . Assume to the contrary that 0∉A . By separation theorem, there exists ^d* ∈^...n , ^d* ≠(0,...,0) such that for all a∈A,...a,^d* ...<0 , that is, for all _wi ∈_bi (x¯) [figure omitted; refer to PDF] This contradicts (2.5).

Letting _ξj =0, for all j∉J(x¯) , we get [figure omitted; refer to PDF] Since ∂_bi (x¯)={_wi ∈_Ci |"..._wi ,x¯...=s(x¯|"_Ci )} , we obtain the desired result.

Theorem 2.2 (G-Karush-Kuhn-Tucker Necessary Optimality Conditions).

Suppose that _GFi , i∈I, are differentiable real-valued strictly increasing functions defined on _IFi (D), _Ggj , j∈J, are differentiable real-valued strictly increasing functions defined on _Igj (D),and _Ght , t∈T , are differentiable real-valued strictly increasing functions defined on _Iht (D) , and _Ght , t∈T , are linearly independent, and let _Fi =_fi (·)+(·^)T_wi . Moreover, we assume that there exists ^z* ∈^...n such that ...^{G_gj [variant prime]} (_gj (x¯))∇_gj (x¯),^z* ...<0, j∈J(x¯) , and ...^{G_ht [variant prime]} (_ht (x¯))∇_ht (x¯),^z* ...=0, t=1,...,p . If x¯∈D is a weak Pareto optimal point in problem (NMP), then there exist λ∈^...+k ,ξ∈^...+m , μ∈^...p , and _wi ∈_Ci , i=1,...,k such that [figure omitted; refer to PDF]

Proof.

Since x¯ is a weak Pareto optimal point of (NMP), by Theorem 2.1, there exist λ...∈^...+k ,ξ...∈^...+m , μ...∈^...p , and _wi ∈_Ci ,i=1,...,k such that [figure omitted; refer to PDF] Assume that there exists ^z* ∈^...n such that ...^{G_gj [variant prime]} (_gj (x¯))∇_gj (x¯),^z* ...<0, j∈J(x¯) , and ...^{G_ht [variant prime]} (_ht (x¯))∇_ht (x¯),^z* ...=0, t=1,...,p . Then (_λ...1 ,...,_λ...k ) ≠(0,...,0) . Assume to the contrary that (_λ...1 ,...,_λ...k )=(0,...,0) . Then (_ξ...1 ,...,_ξ...m ,_μ...1 ,...,_μ...p )≠(0,...,0) . If ξ...=0 , then μ...≠0 . Since _Ght , t∈T , are linearly independent, _μ...1Gh1 (_h1 (x¯))+...+_μ...pGhp (_hp (x¯))=0 has a trivial solution μ...=0 , this contradicts to the fact that μ...≠0 . So ξ...≥0 . Define _{ξ...j∈J(x¯)} >0, _{ξ...j∉J(x¯)} =0 . Since ...^{G_gj [variant prime]} (_gj (x¯))∇_gj (x¯),^z* ...<0, j∈J(x¯) , we have ^∑j=1m ...^{G_gj [variant prime]} (_gj (x¯))∇_gj (x¯),^z* ...<0 and so ^∑j=1m ...^{G_gj [variant prime]} (_gj (x¯))∇_gj (x¯),^z* ......+^∑t=1p ...^{G_ht [variant prime]} (_ht (x¯))∇_ht (x¯),^z* ...<0 . This is a contradiction. Hence (_λ...1 ,...,_λ...k )≠(0,...,0) . Indeed, it is sufficient only to show that there exist λ∈^...+k , ξ∈^...+m , and μ∈^...p such that ^∑i=1k_λi =1 . we set [figure omitted; refer to PDF] It is not difficult to see that the G-Karush-Kuhn-Tucker necessary optimality conditions are satisfied with Lagrange multipliers, there exist λ∈^...+k , ξ∈^...+m ; and μ∈^...p given by (2.10).

We denote by ^T+ (x¯) and ^T- (x¯) the sets of equality constraints indices for which a corresponding Lagrange multiplier is positive and negative, respectively, that is, ^T+ (x¯)={t∈T:_μt >0} and ^T- (x¯)={t∈T:_μt <0} .

Theorem 2.3 (G-Fritz John Sufficient Optimality Conditions).

Let (x¯,λ,ξ,μ,w) satisfy the G-Fritz John optimality conditions as follow: [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Further, assume that F(=f(·)+(·^)T w) is vector _GF -invex with respect to η at x¯ on D , g is strictly _Gg -invex with respect to η at x¯ on D , _ht ,t∈^T+ (x¯),is _Ght -invex with respect to η at x¯ on D , and _ht ,t∈^T- (x¯),is _Ght -incave with respect to η at x¯ on D . Moreover, suppose that _Ggj (0)=0 for j∈J and _Ght (0)=0 for t∈^T+ (x¯)∪^T- (x¯) . Then x¯ is a weak Pareto optimal point in problem (NMP).

Proof.

Suppose that x¯ is not a weak Pareto optimal point in problem (NMP). Then there exists ^x* ∈D such that _GFi (_fi (^x* )+s(^x* |"_Ci ))<_GFi (_fi (x¯)+s(x¯|"_Ci )), i=1,...,k . Since ..._wi ,x¯...=s(x¯|"_Ci ), i=1,...,k, [figure omitted; refer to PDF] Thus we get [figure omitted; refer to PDF] By assumption, F(=f(·)+(·^)T w) is _GF -invex with respect to η at x¯ on D . Then by Definition 1.3, for any i∈I , [figure omitted; refer to PDF] Hence by (2.16) and (2.17), we obtain [figure omitted; refer to PDF] Since (x¯,λ,ξ,μ,w) satisfy the G-Fritz John conditions, by λ[>, double =]0 , [figure omitted; refer to PDF] Since g is strictly _Gg -invex with respect to η at x¯ on D , [figure omitted; refer to PDF] Thus, by ξ[>, double =]0 , [figure omitted; refer to PDF] Then, (2.12) implies [figure omitted; refer to PDF] By assumption, _ht , t∈^T+ (x¯) , is _Ght -invex with respect to η at x¯ on D , and _ht ,t∈^T- (x¯) , is _Ght -incave with respect to η at x¯ on D . Then, by Definition 1.3, we have, [figure omitted; refer to PDF] Thus, for any t∈^T+ , [figure omitted; refer to PDF] Since ^x* ∈D and x¯∈D , then the inequality above implies [figure omitted; refer to PDF] Adding both sides of inequalities (2.19), (2.22), (2.25), and by (2.14), [figure omitted; refer to PDF] which contradicts (2.11). Hence, x¯ is a weak Pareto optimal for (NMP).

Theorem 2.4 (G-Karush-Kuhn-Tucker Sufficient Optimality Conditions).

Let (x¯,λ,ξ,μ,w) satisfy the G-Karush-Kuhn-Tucker conditions as follow: [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Further, assume that F(=f(·)+(·^)T w) is vector _GF -invex with respect to η at x¯ on D , g is strictly _Gg -invex with respect to η at x¯ on D , _ht ,t∈^T+ (x¯),is _Ght -invex with respect to η at x¯ on D , and _ht ,t∈^T- (x¯),is _Ght -incave with respect to η at x¯ on D . Moreover, suppose that _Ggj (0)=0 for j∈J and _Ght (0)=0 for t∈^T+ (x¯)∪^T- (x¯) . Then x¯ is a weak Pareto optimal point in problem (NMP).

Proof.

Suppose that x¯ is not a weak Pareto optimal point in problem (NMP). Then there exists ^x* ∈D such that _GFi (_fi (^x* )+s(^x* |"_Ci ))<_GFi (_fi (x¯)+s(x¯|"_Ci )), i=1,...,k . Since ..._wi ,x¯...=s(x¯|"_Ci ), i=1,...,k, [figure omitted; refer to PDF] Thus we get [figure omitted; refer to PDF] By assumption, F(=f(·)+(·^)T w) is _GF -invex with respect to η at x¯ on D . Then by Definition 1.3, for any i∈I , [figure omitted; refer to PDF] Hence by (2.32) and (2.33), we obtain [figure omitted; refer to PDF] Since (x¯,λ,ξ,μ,w) satisfy the G-Karush-Kuhn-Tucker conditions, by λ≥0 , [figure omitted; refer to PDF] Since g is strictly _Gg -invex with respect to η at x¯ on D , [figure omitted; refer to PDF] Thus, by ξ[>, double =]0 , [figure omitted; refer to PDF] Then, (2.28),(2.30) imply [figure omitted; refer to PDF] By assumption, _ht , t∈^T+ (x¯) , is _Ght -invex with respect to η at x¯ on D , and _ht , t∈^T- (x¯) , is _Ght -incave with respect to η at x¯ on D . Then, by Definition 1.3, we have, [figure omitted; refer to PDF] Thus, for any t∈^T+ , [figure omitted; refer to PDF] Since ^x* ∈D and x¯∈D , then the inequality above implies [figure omitted; refer to PDF] Adding both sides of inequalities (2.35), (2.38) and (2.41), [figure omitted; refer to PDF] which contradicts (2.27). Hence, x¯ is a weak Pareto optimal for (NMP).

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