Optimality Conditions and Duality in Nonsmooth Multiobjective Programs

Abstract

We study nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. Two types of Karush-Kuhn-Tucker optimality conditions with support functions are introduced. Sufficient optimality conditions are presented by using generalized convexity and certain regularity conditions. We formulate Wolfe-type dual and Mond-Weir-type dual problems for our nonsmooth multiobjective problems and establish duality theorems for (weak) Pareto-optimal solutions under generalized convexity assumptions and regularity conditions.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Jong Kyu Kim

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

Received 29 October 2009; Accepted 14 March 2010

1. Introduction

Multiobjective programming problems arise when more than one objective function is to be optimized over a given feasible region. Pareto optimum is the optimality concept that appears to be the natural extension of the optimization of a single objective to the consideration of multiple objectives.

In 1961, Wolfe [1] obtained a duality theorem for differentiable convex programming. Afterwards, a number of different duals distinct from the Wolfe dual are proposed for the nonlinear programs by Mond and Weir [2]. Duality relations for multiobjective programming problems with generalized convexity conditions were given by several authors [3-10]. Majumdar [11] gave sufficient optimality conditions for differentiable multiobjective programming which modified those given in Singh [12] under the assumption of convexity, pseudoconvexity, and quasiconvexity of the functions involved at the Pareto-optimal solution. Subsequently, Kim et al. [13] gave a counterexample showing that some theorems of Majumdar [11] are incorrect and establish sufficient optimality theorems for (weak) Pareto-optimal solutions by using modified conditions. Later on, Kim and Schaible [6] introduced nonsmooth multiobjective programming problems involving locally Lipschitz functions for inequality and equality constraints. They extended sufficient optimality conditions in Kim et al. [13] to the nonsmooth case and established duality theorems for nonsmooth multiobjective programming problems involving locally Lipschitz functions.

In this paper, we apply the results in Kim and Schaible [6] for this problem to nonsmooth multiobjective programming problem involving support functions. We introduce nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions for inequality and equality constraints. Two kinds of sufficient optimality conditions under various convexity assumptions and certain regularity conditions are presented. We propose a Wolfe-type dual and a Mond-Weir-type dual for the primal problem and establish duality results between the primal problem and its dual problems under generalized convexity and regularity conditions.

2. Notation and Definitions

Let ^...n be the n -dimensional Euclidean space and ^...+n its nonnegative orthant.

We consider the following nonsmooth multiobjective programming problem involving locally Lipschitz functions:

[figure omitted; refer to PDF] where _fi :^...n [arrow right]... , i∈M={1,2,...,m} , _gj :^...n [arrow right]... , j∈P={1,2,...,p} , and _hk :^...n [arrow right]... , k∈Q={1,2,...,q} , are locally Lipschitz functions. Here, _Ci , i∈M , is compact convex sets in ^...n . We accept the formal writing C=(_C1 ,_C2 ,...,_Cm^)t with the convention that s(x|"C)=(s(x|"_C1 ),...,s(x|"_Cm )) , where s(x|"_Ci ) is the support function of _Ci (see Definition 2.2).

Throughout the article the following notation for order relations in ^...n will be used:

[figure omitted; refer to PDF]

Definition 2.1.

(i) A real-valued function F:^...n [arrow right]... is said to be locally Lipschitz if for each z∈^...n , there exist a positive constant K and a neighborhood N of z such that, for every x,y∈N , [figure omitted; refer to PDF] where ||·|| denotes any norm in ^...n .

(ii) The Clarke generalized directional derivative [14] of a locally Lipschitz function F at x in the direction d∈^...n , denoted by ^F0 (x;d) , is defined as follows: [figure omitted; refer to PDF] where y is a vector in ^...n .

(iii) The Clarke generalized subgradient [14] of F at x is denoted by [figure omitted; refer to PDF]

(iv) F is said to be regular at x if for all d∈^...n the one-sided directional derivative ^{F[variant prime]} (x;d) exists and ^{F[variant prime]} (x;d)=^F0 (x;d) .

Definition 2.2 (see [10]).

Let C be a compact convex set in ^...n . The support function s(x|"C) of 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, for every x∈^...n there exists z such that

[figure omitted; refer to PDF] Equivalently,

[figure omitted; refer to PDF] The subdifferential of s(x|"C) is given by

[figure omitted; refer to PDF] For any set S⊂^...n , the normal cone to S at a point x∈S is defined by

[figure omitted; refer to PDF] It is readily verified that for a compact convex set C , y is in _NC (x) if and only if s(y|"C)=^xt y , or equivalently, x is in the subdifferential of s at y .

In the notation of the problem (MP), we recall the definitions of convexity, affine, pseudoconvexity, and quasiconvexity for locally Lipschitz functions.

Definition 2.3.

(i) f=(_f1 ,_f2 ,...,_fm ) is convex (strictly convex) at ^x0 ∈X if for each x∈X and any _ξi ∈∂_fi (^x0 ) , _fi (x)-_fi (^x0 )[>, double =](>)^ξit (x-^x0 ) , for all i∈M .

(ii) _gA is convex at ^x0 ∈X if for each x∈X and any _ζj ∈∂_gj (^x0 ) , _gj (x)-_gj (^x0 )[>, double =]^ζjt (x-^x0 ) , where j∈A , and _gA denotes the active constraints at ^x0 .

(iii) h=(_h1 ,_h2 ,...,_hk ) is convex at ^x0 ∈X if for each x∈X and any _αk ∈∂_hk (^x0 ) , _hk (x)-_hk (^x0 )[>, double =]^αkt (x-^x0 ) , for all k∈Q .

(iv) h is affine at ^x0 ∈X if for each x∈X and any _αk ∈∂_hk (^x0 ) , _hk (x)-_hk (^x0 )=^αkt (x-^x0 ) , for all k∈Q .

(v) f is pseudoconvex at ^x0 ∈X if for each x∈X and any _ξi ∈∂_fi (^x0 ) , ^ξit (x-^x0 )[>, double =]0 implies _fi (x)[>, double =]_fi (^x0 ) , for all i∈M .

(vi) f is strictly pseudoconvex at ^x0 ∈X if for each x∈X with x≠^x0 and any _ξi ∈∂_fi (^x0 ) , ^ξit (x-^x0 )[>, double =]0 implies _fi (x)>_fi (^x0 ) , for all i∈M .

(vii) f is quasiconvex at ^x0 ∈X if for each x∈X with x≠^x0 and any _ξi ∈∂_fi (^x0 ) , _fi (x)[<, double =]_fi (^x0 ) implies ^ξit (x-^x0 )[<, double =]0 , for all i∈M .

Finally, we recall the definition of Pareto-optimal (efficient, nondominated) and weak Pareto-optimal solutions of (MP).

Definition 2.4.

(i) A point ^x0 ∈X is said to be a Pareto-optimal solution of (MP) if there exists no other x∈X with f(x)≤f(^x0 ) .

(ii) A point ^x0 ∈X is said to be a weak Pareto-optimal solution of (MP) if there exists no other x∈X with f(x)<f(^x0 ) .

3. Sufficient Optimality Conditions

In this section, we introduce the following two types of (KKT ) conditions which differ only in the nonnegativity of the multipliers for the equality constraint and neither of which includes a complementary slackness condition, common in necessary optimality conditions [14].

[figure omitted; refer to PDF] [figure omitted; refer to PDF] In Theorems 3.1 and 3.2 and Corollaries 3.3 and 3.4 below we present new versions including support functions of the results by Kim and Schaible in [6] for smooth problems (MP) involving (KKT ).

Theorem 3.1.

Let (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (KKT ). If f(·)+^zt (·) is pseudoconvex at ^x0 , _gA and h are quasiconvex at ^x0 , and f is regular at ^x0 , then ^x0 is a weak Pareto-optimal solution of (MP).

Proof.

Let (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (KKT ). Then g(^x0 )[<, double =]0 , h(^x0 )=0 , [figure omitted; refer to PDF] From (3.1), there exist _ξi ∈∂_fi (^x0 ) , _ζj ∈∂_gj (^x0 ) , and _αk ∈∂_hk (^x0 ) such that [figure omitted; refer to PDF] Suppose that ^x0 ∈X is not a weak Pareto-optimal solution of (MP). Then there exists x¯∈X such that f(x¯)+s(x¯|"C)<f(^x0 )+s(^x0 |"C) that implies f(x¯)+^zt x¯<f(^x0 )+^ztx0 because of ^zt x[<, double =]s(x|"C) and the assumption ^ztx0 =s(^x0 |"C) which means that this function s(x|"C) is subdifferentiable and regular at ^x0 . By pseudoconvexity of f(·)+^zt (·) at ^x0 , we have [figure omitted; refer to PDF] Since _gj (x¯)[<, double =]0=_gj (^x0 ) , j∈A , we obtain the following inequality with the help of quasiconvexity of _gA at ^x0 : [figure omitted; refer to PDF] Also, since _hk (x¯)=0=_hk (^x0 ) , k∈Q , it follows from quasiconvexity of h at ^x0 that [figure omitted; refer to PDF] From (3.3)-(3.5), we obtain [figure omitted; refer to PDF] which contradicts (3.2). Hence ^x0 is a weak Pareto-optimal solution of (MP).

Theorem 3.2.

Let (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (KKT ). If f(·)+^zt (·) is strictly pseudoconvex at ^x0 , _gA and h are quasiconvex at ^x0 , and f is regular at ^x0 , then ^x0 is a Pareto-optimal solution of (MP).

Corollary 3.3.

Let (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (KKT ). If f(·)+^zt (·) , _gA and h are convex at ^x0 , and f is regular at ^x0 , then ^x0 is a weak Pareto-optimal solution of (MP).

Corollary 3.4.

Let (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (KKT ). If f(·)+^zt (·) is strictly convex at ^x0 , _gA and h are convex at ^x0 , and f is regular at ^x0 , then ^x0 is a Pareto-optimal solution of (MP).

Theorem 3.5.

Let (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (^{KKT[variant prime]} ). If f(·)+^zt (·) is quasiconvex at ^x0 , (^v0)t_gA +(^w0)t h is strictly pseudoconvex at ^x0 , and f , _gA , and h are regular at ^x0 , then ^x0 is a Pareto-optimal solution of (MP).

Proof.

Suppose that ^x0 ∈X is not a Pareto-optimal solution of (MP). Then there exists x¯∈X such that f(x¯)+s(x¯|"C)≤f(^x0 )+s(^x0 |"C) , that implies f(x¯)+^zt x¯≤f(^x0 )+^ztx0 because of ^zt x[<, double =]s(x|"C) and the assumption ^ztx0 =s(^x0 |"C) . By quasiconvexity of f(·)+^zt (·) at ^x0 , we have [figure omitted; refer to PDF] Since (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (^{KKT[variant prime]} ), we obtain [_∑j∈A^vj0_ζj +_∑k∈Q^wk0_αk^]t (x¯-^x0 )[>, double =]0 for some _ζj ∈∂_gj (^x0 ) , j∈A , and _αk ∈∂_hk (^x0 ) , k∈Q . By regularity of _gA and h at ^x0 , there exists β∈∂(^v0)t_gA +(^w0)t h such that ^βt (x¯-^x0 )[>, double =]0 . With the help of a strict pseudoconvexity of (^v0)t_gA +(^w0)t h , we have [figure omitted; refer to PDF] Since x¯∈X , we obtain [figure omitted; refer to PDF] Since _gA (^x0 )=h(^x0 )=0 , we obtain [figure omitted; refer to PDF] Substituting (3.9) and (3.10) for (3.8), we arrive at a contradiction. Hence ^x0 is a Pareto-optimal solution of (MP).

Now we present a result for nonsmooth problems (MP) which in the smooth case is similar to Singh's earlier result in [12] under generalized convexity.

Theorem 3.6.

Let (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (^{KKT[variant prime]} ). If (^u0)t (f(·)+^zt (·))+(^v0)t_gA +(^w0)t h is pseudoconvex at ^x0 , and f , _gA , and h are regular at ^x0 , then ^x0 is a weak Pareto-optimal solution of (MP).

Proof.

Suppose that ^x0 ∈X is not a weak Pareto-optimal solution of (MP). Then there exists x¯∈X such that f(x¯)+s(x¯|"C)<f(^x0 )+s(^x0 |"C) , that implies f(x¯)+^zt x¯<f(^x0 )+^ztx0 because of ^zt x[<, double =]s(x|"C) and the assumption ^ztx0 =s(^x0 |"C) . Since x¯∈X , we have _gA (x¯)[<, double =]0=_gA (^x0 ) and h(x¯)=0=h(^x0 ) . Therefore, f(x¯)+s(x¯|"C)-f(^x0 )+s(^x0 |"C)<0 , _gA (x¯)-_gA (^x0 )[<, double =]0 , and h(x¯)-h(^x0 )=0 . Hence (^u0)t (f(x¯)+^zt (x¯))+(^v0)t_gA (x¯)+(^w0)t h(x¯)<(^u0)t (f(^x0 )+^zt (^x0 ))+(^v0)t_gA (^x0 )+(^w0)t h(^x0 ) . Since f , _gA , and h are regular at ^x0 , we obtain [figure omitted; refer to PDF] By pseudoconvexity of (^u0)t (f(·)+^zt (·))+(^v0)t_gA +(^w0)t h , we have ^βt (x¯-^x0 )<0 for any β∈∂(_∑i∈M^ui0 (_fi (^x0 )+^zitx0 )+_∑j∈A^vj0_gj (^x0 )+_∑k∈Q^wk0_hk (^x0 )) . We easily see that this contradicts 0∈_∑i∈M^ui0 (∂_fi (^x0 )+_zi )+_∑j∈A^vj0 ∂_gj (^x0 )+_∑k∈Q^wk0 ∂_hk (^x0 ) . Hence ^x0 is a weak Pareto-optimal solution of (MP).

Theorem 3.7.

Let (^x0 ,^u0 ,^v0 ,^w0 ) satisfy (^{KKT[variant prime]} ). If (^u0)t (f(·)+^zt (·))+(^v0)t_gA +(^w0)t h is strictly pseudoconvex at ^x0 , and f , _gA , and h are regular at ^x0 , then ^x0 is a Pareto-optimal solution of (MP).

The proof is similar to the one used for the previous theorem.

4. Duality

Following Mond and Weir [2], in this section we formulate a Wolfe-type dual problem (WD) and a Mond-Weir-type dual problem (MD) of the nonsmooth problem (MP) and establish duality theorems. We begin with a Wolfe-type dual problem:

[figure omitted; refer to PDF] Here e=(1,...,1^)t ∈^...m .

We now derive duality relations.

Theorem 4.1.

Let x be feasible for (MP) and (y,u,v,w) feasible for (WD). If f(·)+^zt (·) , g and ^wt h are convex, and f is a regular function, then f(x)+s(x|"C) [not <] f(y)+^zt y+^vt g(y)e+^wt h(y)e .

Proof.

Let x be feasible for (MP) and (y,u,v,w) feasible for (WD). Then g(x)[<, double =]0 , h(x)=0 , [figure omitted; refer to PDF] According to (4.1), there exist _ξi ∈∂_fi (y) , _ζj ∈∂_gj (y) , and _αk ∈∂_hk (y) such that [figure omitted; refer to PDF] Assume that [figure omitted; refer to PDF] Multiplying (4.3) by u and using the equality ^zt x=s(x|"C) , we have [figure omitted; refer to PDF] since u≥0 and ^ut e=1 . Now by convexity of f(·)+^zt (·) , g and ^wt h , we obtain [figure omitted; refer to PDF] Since ^vt g(x)[<, double =]0 and ^wt h(x)=0 , we obtain the following inequality from (4.2), (4.5): [figure omitted; refer to PDF] which contradicts (4.4).

Hence the weak duality theorem holds.

Now we derive a strong duality theorem.

Theorem 4.2.

Let x¯ be a weak Pareto-optimal solution of (MP) at which a constraint qualification holds [14]. Then there exist u¯∈^...m , v¯∈^...p , and w¯∈^...q such that (x¯,u¯,v¯,w¯) is feasible for (WD) and ^zt x¯=s(x¯|"C) . In addition, if f(·)+^zt (·) , g and ^wt h are convex, and f is a regular function, then (x¯,u¯,v¯,w¯) is a weak Pareto-optimal solution of (WD) and the optimal values of (MP) and (WD) are equal.

Proof.

From the (KKT ) necessary optimality theorem [14], there exist u∈^...m , v∈^...p , and w∈^...q such that [figure omitted; refer to PDF] Since u≥0 , we can scale the ^{ui[variant prime]} s , ^{vj[variant prime]} s and ^{wk[variant prime]} s as follows: [figure omitted; refer to PDF] Then (x¯,u¯,v¯,w¯) is feasible for (WD). Since x¯ is feasible for (MP), it follows from Theorem 4.1 that [figure omitted; refer to PDF] for any feasible solution (x,u,v,w) of (WD). Hence (x¯,u¯,v¯,w¯) is a weak Pareto-optimal solution of (WD) and the optimal values of (MP) and (WD) are equal.

Remark 4.3.

If we replace the convexity hypothesis of f(·)+^zt (·) by strict convexity in Theorems 4.1 and 4.2, then these theorems hold for the case of a Pareto-optimal solution.

Remark 4.4.

If we replace the convexity hypothesis of ^wt h by affineness of h in Theorems 4.1 and 4.2, then these theorems are also valid.

Theorem 4.5.

Let x be feasible for (MP) and (y,u,v,w) feasible for (WD). If ^ut (f(·)+^zt (·))+^vt g+^wt h is pseudoconvex and f , g , and h are regular functions, then f(x)+s(x|"C)[neither < nor =]f(y)+^zt y+^vt g(y)e+^wt h(y)e .

Proof.

Suppose to the contrary that f(x)+s(x|"C)≤f(y)+^zt y+^vt g(y)e+^wt h(y)e . By feasibility of x , we obtain [figure omitted; refer to PDF] Since f , g , and h are regular functions, we have ^βt (x-y)<0 by the pseudoconvexity of ^ut (f(·)+^zt (·))+^vt g+^wt h for any β∈∂(_∑i∈Mui (_fi (y)+^zit y)+_∑j∈Pvjgj (y)+_∑k∈Qwkhk (y)) . This contradicts the feasibility of (y,u,v,w) . Hence the weak duality theorem holds.

Theorem 4.6.

Let x¯ be a weak Pareto-optimal solution of (MP) at which a constraint qualification holds [14]. Then there exist u¯∈^...m , v¯∈^...p , and w¯∈^...q such that (x¯,u¯,v¯,w¯) is feasible for (WD) and ^zt x¯=s(x¯|"C) . If in addition ^ut (f(·)+^zt (·))+^vt g+^wt h is pseudoconvex and f , g , and h are regular functions, then (x¯,u¯,v¯,w¯) is a weak Pareto-optimal solution of (WD) and the optimal values of (MP) and (WD) are equal.

The proof is similar to the one used for Theorem 4.2.

Remark 4.7.

If we replace the pseudoconvexity hypothesis of ^ut (f(·)+^zt (·))+^vt g+^wt h by strictly pseudoconvexity in Theorems 4.5 and 4.6, then these results hold for the case of a Pareto-optimal solution.

We now prove duality relations between (MP) and the following Mond-Weir-type dual problem:

[figure omitted; refer to PDF]

Theorem 4.8.

Let x be feasible for (MP) and (y,u,v,w) feasible for (MD). If f(·)+^zt (·) , g and ^wt h are convex, and f is a regular function, then f(x)+s(x|"C)[not <]f(y)+^zt y .

Proof.

Let x be feasible for (MP) and (y,u,v,w) feasible for (MD). Then g(x)[<, double =]0 , h(x)=0 , [figure omitted; refer to PDF] [figure omitted; refer to PDF] According to (4.11), there exist _ξi ∈∂_fi (y) , _ζj ∈∂_gj (y) , and _αk ∈∂_hk (y) such that [figure omitted; refer to PDF] Assume that [figure omitted; refer to PDF] Multiplying (4.14) by u and using the inequality ^zt x[<, double =]s(x|"C) , we have [figure omitted; refer to PDF] By convexity of f(·)+^zt (·) , g and ^wt h , we obtain [figure omitted; refer to PDF] From (4.12) and (4.16), we obtain [figure omitted; refer to PDF] since ^vt g(x)[<, double =]0 and ^wt h(x)=0 . However, (4.17) contradicts (4.15). Hence the proof is complete.

Theorem 4.9.

Let x¯ be a weak Pareto-optimal solution of (MP) at which a constraint qualification holds. Then there exist u¯∈^...m , v¯∈^...p , and w¯∈^...q such that (x¯,u¯,v¯,w¯) is feasible for (MD) and ^zt x¯=s(x¯|"C) . If in addition f(·)+^zt (·) , g and ^wt h are convex, and f is a regular function, then (x¯,u¯,v¯,w¯) is a weak Pareto-optimal solution of (MD) and the optimal values of (MP) and (MD) are equal.

Proof.

Let x¯ be a weak Pareto-optimal solution of (MP) such that a constraint qualification is satisfied at x¯ . According to the (KKT ) necessary optimality theorem, there exist u∈^...m , v∈^...p , and w∈^...q such that [figure omitted; refer to PDF] Since u≥0 , we can scale the ^{ui[variant prime]} s , ^{vj[variant prime]} s , and ^{wk[variant prime]} s as in the proof of Theorem 4.2 such that (x¯,u¯,v¯,w¯) is feasible for (MD), it follows from Theorem 4.8 that f(x¯)+s(x¯|"C)[not <]f(y)+^zt y for any feasible solution (y,u,v,w) of (MD). Hence (x¯,u¯,v¯,w¯) is a weak Pareto-optimal solution of (MD) and the optimal values of (MP) and (MD) are equal.

Remark 4.10.

If we replace the convexity hypothesis of f(·)+^zt (·) by strict convexity in Theorems 4.8 and 4.9, then these theorems hold in the sense of a Pareto-optimal solution.

Remark 4.11.

If we replace the convexity hypothesis of ^wt h by affineness of h in Theorems 4.8 and 4.9, then these theorems are also valid.

Theorem 4.12.

Let x be feasible for (MP) and (y,u,v,w) feasible for (MD). If ^ut (f(·)+^zt (·))+^vt g+^wt h is pseudoconvex and f , g , and h are regular functions, then f(x)+s(x|"C)[neither < nor =]f(y)+^zt y .

Proof.

Suppose that f(x)+s(x|"C)≤f(y)+^zt y . By using the feasibility assumptions and ^zt x[<, double =]s(x|"C) , we obtain [figure omitted; refer to PDF] By the same argument as in the proof of Theorem 4.5, we arrive at a contradiction.

Theorem 4.13.

Let x¯ be a weak Pareto-optimal solution of (MP) at which a constraint qualification holds. Then there exist u¯∈^...m , v¯∈^...p , and w¯∈^...q such that (x¯,u¯,v¯,w¯) is feasible for (MD) and ^zt x¯=s(x¯|"C) . If in addition ^ut (f(·)+^zt (·))+^vt g+^wt h is pseudoconvex and f , g , and h are regular functions, then (x¯,u¯,v¯,w¯) is a weak Pareto-optimal solution of (MD) and the optimal values of (MP) and (MD) are equal.

The proof is similar to the one used for the previous theorem.

Remark 4.14.

If we replace the pseudoconvexity hypothesis of ^ut (f(·)+^zt (·))+^vt g+^wt h by strict pseudoconvexity in Theorems 4.12 and 4.13, then these results hold for the case of a Pareto-optimal solution.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2009-0074482).

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