Higher-order symmetric duality for a class of multiobjective fractional programming problems

Abstract

In this paper, a pair of nondifferentiable multiobjective fractional programming problems is formulated. For a differentiable function, we introduce the definition of higher-order (F, α, ρ, d)-convexity, which extends some kinds of generalized convexity, such as second order F-convexity and higher-order F -convexity. Under the higher-order (F, α, ρ, d)-convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems.

Mathematics Subject Classification (2010) 90C29; 90C30; 90C46.

Introduction

Symmetric duality in nonlinear programming in which the dual of the dual is the primal was introduced by Dorn [1]. The notion of symmetric duality was developed significantly by Dantzig et al. [2], and the Wolfe dual models presented in [2]. Mond [3] presented a slightly different pair of symmetric dual nonlinear programs and obtained more generalized duality results than that of Dantzig et al. [2]. Mond and Weir [4] then gave another pair of symmetric dual nonlinear programs in which a weaker convexity assumption was imposed on involved functions. Later, Mond and Weir [5], Weir and Mond [6] as well as Gulati et al. [7] generalized single objective symmetric duality to multiobjective case.

Chandra et al. [8] first formulated a pair of symmetric dual fractional programs with certain convexity hypothesis. Pandey [9] introduced second-order η-invex function for multiobjective fractional programming problem and established weak and strong duality theorems. Yang et al. [10] discussed a class of nondifferentiable multiobjective fractional programming problems, and proved duality theorems under the assumptions of invex (pseudoinvex, pseudoincave) functions. Higher-order duality in nonlinear programs have been studied by some researchers. Mangasarian [11] formulated a class of higher-order dual problems for the nonlinear programming problem by introducing twice differentiable functions. Mond and Zhang [12] obtained duality results for various higher-order dual programming problems under higher-order invexity assumptions. Under invexity-type conditions, such as higher-order type I, higher-order pseudo-type I, and higher-order quasi-type I conditions, Mishra and Rueda [13] gave various duality results. Recently, Chen [14] also discussed the duality theorems under higher-order F-convexity (F-pseudo-convexity, F-quasi-convexity) for a pair of multiobjective nondifferentiable program. But, up to now, there is not sufficient literatures dealing with higher-order fractional symmetric duality.

In this paper, we first formulate a pair of nondifferentiable multiobjective fractional pro-gramming problems. For a differentiable function h: Rⁿ ×Rⁿ → R, we introduce the definition of higher-order (F, α, ρ, d)-convexity, which extends some kinds of generalized convexity, such as second order F-convexity in [15] and higher-order F -convexity in [14]. Under the higher-order (F, α, ρ, d)- convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems.

Preliminaries

Let Rⁿ be the n-dimensional Euclidean space and let $R_{+}^n$ be its non-negative orthant. The following conventions for vectors in Rⁿ will be used:

$$\begin{array}{c}x<y\text{if and only if}y-x\in \text{int }{R}^n;\\ x\le y\text{if and only if}y-x\in R_{+}^n\backslash \left\{0\right\};\\ x\leqq y\text{if and only if}y-x\in R_{+}^n;\\ x\nleqq y\text{is the negation of}x\le y.\end{array}$$

For a real-valued twice differentiable function h(x, y) defined on an open set in Rⁿ × R^m, denote by ${\nabla }_{x}h\left(\overline{x},\overline{y}\right)$ the gradient vector of h with respect to x at $\left(\overline{x},\overline{y}\right),{\nabla }_{xx}h\left(\overline{x},\overline{y}\right)$ the hessian matrix with respect to x at $\left(\overline{x},\overline{y}\right)$. Similarly, ${\nabla }_{y}h\left(\overline{x},\overline{y}\right),{\nabla }_{xy}h\left(\overline{x},\overline{y}\right)\text{and}{\nabla }_{yy}h\left(\overline{x},\overline{y}\right)$ are also defined.

Let C be a compact convex set in Rⁿ. The support function of C is defined by

$$s\left(x|C\right)=\text{max}\left\{x^{T}y:y\in C\right\}.$$

A support function, being convex and everywhere finite, has a subdifferential, that is, there exists a z ∈ Rⁿ such that

$$s\left(y|C\right)\geqq s\left(x|C\right)+z^T\left(y-x\right),\forall x\in C.$$

The subdifferential of s(x|C) is given by

$$\partial s\left(x|C\right)=\left\{z\in C:z^{T}x=s\left(x|C\right)\right\}.$$

For a convex set D ⊂ Rⁿ, the normal cone to D at a point x ∈ D is defined by

$$N_D\left(x\right)=\left\{y\in R^n:y^T\left(z-x\right)\leqq 0,\forall z\in D\right\}.$$

When C is a compact convex set, y ∈ N _C (x) if and only if s(y|C) = x^T y, or equivalently, x ∈ ∂s(y|C).

Consider the following multiobjective programming problem (P):

$$\text{Minimize}f\left(x\right)\text{subject to }g\left(x\right)\leqq 0,x\in X,$$

where f: Rⁿ → R^m, g: Rⁿ → R^l and X ⊂ Rⁿ. Denote by S the set of feasible solutions of (P).

Definition 2.1. (a) A feasible solution x₀ is said to be an efficient solution of (P) if there is no other x ∈ S such that f(x) ≤ f(x₀).

• (b) A feasible solution x₀ is said to be a properly efficient solution of (P) if it is an efficient solution of (P), and there exists a real number M > 0 such that for all i ∈ {1, ..., m}, x ∈ S, and f _i (x) < f _i (x₀),

  $$f_i\left(x_0\right)-f_i\left(x\right)\leqq M\left(f_j\left(x\right)-f_j\left(x_0\right)\right)$$

for some j ∈ {1, ..., m} such that f _j (x) > f _j (x₀).

Definition 2.2. A functional F: X × X × Rⁿ → R (where X ⊂ Rⁿ) is sublinear in its third component if for all (x, u) ∈ X × X,

$$\begin{array}{c}F\left(x,u;a_1+a_2\right)\leqq F\left(x,u;a_1\right)+F\left(x,u;a_2\right)\text{for all }{a}_1,a_2\in R^n;\\ F\left(x,u;\alpha a\right)=\alpha F\left(x,u;a\right)\text{for all}\alpha \in R_{+}\text{and for all }a\in R^n.\end{array}$$

For convenience, we write F _{x, u} (a) = F (x, u, a).

We now introduce higher-order (F, α, ρ, d)-convex function. Where, F: X × X × Rⁿ → R is a sublinear functional, α: X × X → R₊ \ {0}, ρ ∈ R and d: X × X → R. Let Φ: X → R and h: X × Rⁿ → R be differentiable real valued functions.

Definition 2.3. Φ is said to be higher-order (F, α, ρ, d)-convex at u ∈ X with respect to h if, ∀(x, p) ∈ X × Rⁿ,

$$\Phi \left(x\right)-\Phi \left(u\right)\geqq F_{x,u}\left(\alpha \left({\nabla }_x\Phi \left(u\right)+{\nabla }_{p}h\left(u,p\right)\right)\right)+h\left(u,p\right)-p^T{\nabla }_{p}h\left(u,p\right)+\rho d^2\left(x,u\right).$$

Remark 2.1. (1) When α = 1, and ρ = 0 or d = 0, the higher-order (F, α, ρ, d)-convexity reduces to higher-order F-convexity in [14].

• (2) When α = 1, ρ = 0 or d = 0, and $h\left(u,p\right)=\frac{1}{2}{p}^T{\nabla }_{xx}\Phi \left(u\right)p$, the higher-order (F, α, ρ, d)-convexity reduces to second order F-convexity in [15].

we now give an example of higher-order (F, α, ρ, d)-convex function with respect to h(u, p), which is not higher-order F -convex and second order F-convex.

Example 2.1. Let X ⊂ R, X = {x: x ≧ 1}, f: X → R, F: X × X × R → R, h: X × R → R and d: X × X → R given as follows

$$f\left(x\right)=x+\frac{2}{x+1},F_{x,u}\left(a\right)=\left|a\right|{\left(x-u\right)}^2,h\left(u,p\right)=\frac{p}{u+1},d\left(x,u\right)=x-u.$$

And let u = 1, ρ = -1, $\alpha =\frac{3}{4}$. Then for all (x, p) ∈ X × R

$$\begin{array}{cc}\hfill f\left(x\right)-f\left(u\right)& =\frac{x^2-x}{x+1}\geqq F_{x,u}\left(\frac{3}{4}\left({\nabla }_{x}f\left(u\right)+{\nabla }_{p}h\left(u,p\right)\right)\right)\hfill \\ +h\left(u,p\right)-p^T{\nabla }_{p}h\left(u,p\right)-d^2\left(x,u\right)=-\frac{1}{4}{\left(x-1\right)}^2.\hfill \end{array}$$

This implies f(x) is a higher-order (F, α, ρ, d)-convex function with respect to h at u. But when we let x = 2, p = 3 and x = 6, p = 3 respectively, we have

$$\begin{array}{c}f\left(2\right)-f\left(1\right)=\frac{2}{3}<F_{x,u}\left({\nabla }_{x}f\left(u\right)+{\nabla }_{p}h\left(u,p\right)\right)+h\left(u,p\right)-p^T{\nabla }_{p}h\left(u,p\right)=\frac{3}{4},\\ f\left(6\right)-f\left(1\right)=\frac{30}{7}<F_{x,u}\left({\nabla }_{x}f\left(u\right)+{\nabla }_{xx}f\left(u\right)\right)-\frac{1}{2}{p}^T{\nabla }_{xx}f\left(u\right)p=\frac{66}{4}.\end{array}$$

Hence, f is neither a higher-order F-convex function nor a second order F-convex function. From now on, suppose that the sublinear functional F satisfies the following condition:

$$F_{x,y}\left(a\right)+a^{T}y\geqq 0,\forall a\in R_{+}^n.$$

(1)

Higher-order symmetric duality

In the section, we consider the following multiobjective fractional symmetric dual problems: (MFP) Minimize L(x, y, p) = (L₁(x, y, p₁), ..., L _k (x, y, p _k ))^T subject to

$$\begin{array}{c}\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_y{f}_i\left(x,y\right)-z_i+{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)\right)\right.\hfill \\ \left.-L_i\left(x,y,p_i\right)\left({\nabla }_y{g}_i\left(x,y\right)+r_i+{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)\right)\right]\leqq 0,\hfill \\ y^T\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_y{f}_i\left(x,y\right)-z_i+{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)\right)\right.\hfill \\ \left.-L_i\left(x,y,p_i\right)\left({\nabla }_y{g}_i\left(x,y\right)+r_i+{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)\right)\right]\geqq 0,\hfill \\ \lambda >0,{\lambda }^{T}e=1,z_i\in D_i,r_i\in F_i,i=1\dots ,k.\hfill \end{array}$$

(MFD) Maximize M(u, v, q) = (M₁(u, v, q₁),..., M _k (u, v, q _k ))^T subject to

$$\begin{array}{c}\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_x{f}_i\left(u,v\right)+w_i+{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)\right)\right.\hfill \\ \left.-M_i\left(u,v,q_i\right)\left({\nabla }_x{g}_i\left(u,v\right)-t_i+{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)\right)\right]\geqq 0,\hfill \\ u^T\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_x{f}_i\left(u,v\right)+w_i+{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)\right)\right.\hfill \\ \left.-M_i\left(u,v,q_i\right)\left({\nabla }_x{g}_i\left(u,v\right)-t_i+{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)\right)\right]\leqq 0,\hfill \\ \lambda >0,{\lambda }^{T}e=1,w_i\in C_i,t_i\in E_i,i=1\dots ,k.\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill L_i\left(x,y,p_i\right)& =\frac{f_i\left(x,y\right)+s\left(x|C_i\right)-y^T{z}_i+H_i\left(x,y,p_i\right)-p_i^T{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)}{g_i\left(x,y\right)-s\left(x|E_i\right)+y^T{r}_i+G_i\left(x,y,p_i\right)-p_i^T{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)},\hfill \\ \hfill M_i\left(u,v,q_i\right)& =\frac{f_i\left(u,v\right)-s\left(v|D_i\right)+u^T{w}_i+{\Phi }_i\left(u,v,q_i\right)-q_i^T{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)}{g_i\left(u,v\right)+s\left(v|F_i\right)-u^T{t}_i+{\Psi }_i\left(u,v,q_i\right)-q_i^T{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)},\hfill \end{array}$$

f _i : R _n × R _m → R; g _i : Rⁿ × R^m → R; H _i , G _i : Rⁿ × R^m → R and Φ _i , Ψ _i : R _n × R _m × R _n → R are twice differentiable functions for all i = 1 ..., k. C _i , E _i are compact convex sets in Rⁿ, and D _i , F _i are compact convex sets in R^m, i = 1, ..., k. e = (1, ..., 1)^T ∈ R^k. p _i ∈ R^m, q _i ∈ Rⁿ, i = 1, ..., k, p = (p₁, ..., p _k ), q = (q₁, ..., q _k ). It is assumed that in the feasible regions the numerators are nonnegative and denominators are positive.

We let S = (S₁, ..., S _k )^T , W = (W₁, ..., W _k )^T ∈ R^k. Then we can express the programs (MFP) and (MFD) equivalently as:

(MFP) _S Minimize S subject to

$$\begin{array}{c}\left(f_i\left(x,y\right)+s\left(x|C_i\right)-y^T{z}_i+H_i\left(x,y,p_i\right)-p_i^T{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)\right)\hfill \\ -S_i\left(g_i\left(x,y\right)-s\left(x|E_i\right)+y^T{r}_i+G_i\left(x,y,p_i\right)-p_i^T{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)\right)=0,i=1,\dots ,k,\end{array}$$

(2)

$$\begin{array}{c}\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_y{f}_i\left(x,y\right)-z_i+{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)\right)\right.\hfill \\ \left.-S_i\left({\nabla }_y{g}_i\left(x,y\right)+r_i+{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)\right)\right]\leqq 0,\hfill \end{array}$$

(3)

$$\begin{array}{c}{y}^T\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_y{f}_i\left(x,y\right)-z_i+{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)\right)\right.\hfill \\ \left.-S_i\left({\nabla }_y{g}_i\left(x,y\right)+r_i+{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)\right)\right]\geqq 0,\hfill \\ \lambda >0,{\lambda }^{T}e=1,z_i\in D_i,r_i\in F_i,i=1\dots ,k.\hfill \end{array}$$

(4)

(MFD) _W Maximize W subject to

$$\begin{array}{c}\left(f_i\left(u,v\right)-s\left(v|D_i\right)+u^T{w}_i+{\Phi }_i\left(u,v,q_i\right)-q_i^T{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)\right)\hfill \\ -W_i\left(g_i\left(u,v\right)+s\left(v|F_i\right)-u^T{t}_i+{\Psi }_i\left(u,v,q_i\right)-q_i^T{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)\right)=0,i=1,\dots ,k,\hfill \end{array}$$

(5)

$$\begin{array}{c}\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_x{f}_i\left(u,v\right)+w_i+{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)\right)\right.\hfill \\ \left.-W_i\left({\nabla }_x{g}_i\left(u,v\right)-t_i+{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)\right)\right]\geqq 0,\hfill \end{array}$$

(6)

$$\begin{array}{c}{u}^T\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_x{f}_i\left(u,v\right)+w_i+{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)\right)\right.\hfill \\ \left.-W_i\left({\nabla }_x{g}_i\left(u,v\right)-t_i+{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)\right)\right]\leqq 0,\hfill \\ \lambda >0,{\lambda }^{T}e=1,w_i\in C_i,t_i\in E_i,i=1\dots ,k.\hfill \end{array}$$

(7)

Now we can prove weak, strong and converse duality theorems for (MFP) _S and (MFD) _W , but equally apply to (MFP) and (MFD).

Theorem 3.1 (Weak duality). Let (x, y, S, z₁, ..., z _k , r₁, ..., r _k , λ, p) be feasible for (MFD) _S and let (u, v, W, w₁, ..., w _k , t₁ ..., t _k , λ, q) be feasible for (MFD) _W . Let ∀i ∈ {1, ..., k}, f _i (., v) + (.)^T w _i be higher-order (F, α, ρ _i , d _i )-convex at u with respect to Φ _i (u, v, q _i ), - (g _i (., v) - (.)^T t _i ) be higher-order (F, α, ρ, d _i )-convex at u with respect to -Ψ _i (u, v, q _i ), - (f _i (x, .) - (.)^Tz _i ) be higher-order $\left(K,\overline{\alpha },{\overline{\rho }}_i,{\overline{d}}_i\right)$-convex at y with respect to -H _i (x, y, p _i ), g _i (x, .) + (.)^T r _i be higher-order $\left(K,\overline{\alpha },{\overline{\rho }}_i,{\overline{d}}_i\right)$-convex at y with respect to G _i (x, y, p _i ), where sublinear functional F: Rⁿ × Rⁿ × Rⁿ → R and K: R^m × R^m × R^m → R satisfy the condition (1). If the following conditions hold:

$$g_i\left(x,v\right)+v^T{r}_i-s\left(x|E_i\right)>0,i=1,\dots ,k,$$

(8)

$$\sum _{i=1}^k{\lambda }_i\left(\left(1+W_i\right){\rho }_i{d}_i^2\left(x,u\right)+\left(1+S_i\right){\overline{\rho }}_i{\overline{d}}_i^2\left(v,y\right)\right)\geqq 0.$$

(9)

Then S ≰ W.

Proof. Since (u, v, W, w₁, ..., w _k , t₁ ..., t _k , λ, q) is feasible for (MFD) _W , from (6), (7) and F satisfies condition (1), it follows that

$$F_{x,u}\left(\sum _{i=1}^k{\lambda }_i\left[\left({\nabla }_x{f}_i\left(u,v\right)+w_i+{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)\right)-W_i\left({\nabla }_x{g}_i\left(u,v\right)-t_i+{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)\right)\right]\right)\geqq 0.$$

(10)

Using the convexity assumptions of f _i (., v) + (.)^T w _i and -(g _i (., v) - (.)^T t _i ) at u, we have

$$\begin{array}{c}{f}_i\left(x,v\right)+x^T{w}_i-f_i\left(u,v\right)-u^T{w}_i\hfill \\ \geqq F_{x,u}\left(\alpha \left({\nabla }_x{f}_i\left(u,v\right)+w_i+{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)\right)\right)+{\Phi }_i\left(u,v,q_i\right)-q_i^T{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)+{\rho }_i{d}_i^2\left(x,u\right),\hfill \\ -g_i\left(x,v\right)+x^T{t}_i+g_i\left(u,v\right)-u^T{t}_i\hfill \\ \geqq F_{x,u}\left(\alpha \left(-{\nabla }_x{g}_i\left(u,v\right)+t_i-{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)\right)\right)-{\Psi }_i\left(u,v,q_i\right)+q_i^T{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)+{\rho }_i{d}_i^2\left(x,u\right).\hfill \end{array}$$

Since F is a sublinear functional and λ > 0, W ≧ 0, α > 0, from (10) and the above two inequalities, we have

$$\begin{array}{c}\sum _{i=1}^k{\lambda }_i\left(f_i\left(x,v\right)+x^T{w}_i-f_i\left(u,v\right)-u^T{w}_i-{\Phi }_i\left(u,v,q_i\right)+q_i^T{\nabla }_{q_i}{\Phi }_i\left(u,v,q_i\right)\right)\hfill \\ +\sum _{i=1}^k{\lambda }_i{W}_i\left(g_i\left(u,v\right)+v^T{r}_i-u^T{t}_i+{\Psi }_i\left(u,v,q_i\right)-q_i^T{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)\right)\hfill \\ +\sum _{i=1}^k{\lambda }_i{W}_i\left(x^T{t}_i-g_i\left(x,v\right)-v^T{r}_i\right)\geqq \sum _{i=1}^k{\lambda }_i\left(1+W_i\right){\rho }_i{d}_i^2\left(x,u\right).\hfill \end{array}$$

(11)

Since v^T r _i ≦ s(v|F _i ), from (5) and (11), we have

$$\sum _{i=1}^k{\lambda }_i\left[\left(f_i\left(x,v\right)+x^T{w}_i-s\left(v|D_i\right)\right)+W_i\left(x^T{t}_i-v^T{r}_i-g_i\left(x,v\right)\right)\right]\geqq \sum _{i=1}^k{\lambda }_i\left(1+W_i\right){\rho }_i{d}_i^2\left(x,u\right).$$

(12)

On the other hand, from (3), (4) and sublinear functional K satisfies condition (1), we obtain

$$\begin{array}{cc}\hfill K_{v,y}& \left(-\sum _{i=1}^k{\lambda }_i\left(\left({\nabla }_y{f}_i\left(x,y\right)-z_i+{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)\right)\right.\right.\hfill \\ \left.\left.-S_i\left({\nabla }_y{g}_i\left(x,y\right)+r_i+{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)\right)\right)\right)\geqq 0.\hfill \end{array}$$

(13)

Using the convexity assumptions of -f _i (x, .) + (.)^T z _i and g _i (x, .) + (.)^T r _i at y, we have

$$\begin{array}{cc}\hfill -f_i\left(x,v\right)+v^T{z}_i+f_i\left(x,y\right)-y^T{z}_i& \geqq K_{v,y}\left(\overline{\alpha }\left(-{\nabla }_y{f}_i\left(x,y\right)+z_i-{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)\right)\right)\hfill \\ -H_i\left(x,y,p_i\right)+p_i^T{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)+{\overline{\rho }}_i{\overline{d}}_i^2\left(v,y\right),\hfill \\ \hfill g_i\left(x,v\right)+v^T{r}_i-g_i\left(x,y\right)-y^T{r}_i& \geqq K_{v,y}\left(\overline{\alpha }\left({\nabla }_y{g}_i\left(x,y\right)+r_i+{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)\right)\right)\hfill \\ +G_i\left(x,y,p_i\right)-p_i^T{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)+{\overline{\rho }}_i{\overline{d}}_i^2\left(v,y\right).\hfill \end{array}$$

Since K is a sublinear functional, and λ > 0, S ≧ 0, $\overline{\alpha }>0$, from (13) and the above two inequalities, it holds

$$\begin{array}{c}\sum _{i=1}^k{\lambda }_i\left(-f_i\left(x,v\right)+v^T{z}_i+f_i\left(x,y\right)-y^T{z}_i+H_i\left(x,y,p_i\right)-p_i^T{\nabla }_{p_i}{H}_i\left(x,y,p_i\right)\right)\hfill \\ +\sum _{i=1}^k{\lambda }_i{S}_i\left(-g_i\left(x,y\right)+x^T{t}_i-y^T{r}_i-G_i\left(x,y,p_i\right)+p_i^T{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)\right)\hfill \\ +\sum _{i=1}^k{\lambda }_i{S}_i\left(g_i\left(x,v\right)+v^T{r}_i-x^T{t}_i\right)\geqq \sum _{i=1}^k{\lambda }_i\left(1+S_i\right){\overline{\rho }}_i{\overline{d}}_i^2\left(v,y\right).\hfill \end{array}$$

(14)

Since x^T t _i ≦ s(x|E _i ), from (2) and (14) we have

$$\sum _{i=1}^k{\lambda }_i\left[\left(-f_i\left(x,v\right)+v^T{z}_i-s\left(x|C_i\right)\right)+S_i\left(g_i\left(x,v\right)+v^T{r}_i-x^T{t}_i\right)\right]\geqq \sum _{i=1}^k{\lambda }_i\left(1+S_i\right){\overline{\rho }}_i{\overline{d}}_i^2\left(v,y\right).$$

Adding the above inequality and (12), we get

$$\begin{array}{c}\sum _{i=1}^k{\lambda }_i\left(v^T{z}_i-s\left(v|D_i\right)+x^T{w}_i-s\left(x|C_i\right)\right)+\sum _{i=1}^k{\lambda }_i\left(S_i-W_i\right)\left(g_i\left(x,v\right)+v^T{r}_i-x^T{t}_i\right)\hfill \\ \geqq \sum _{i=1}^k{\lambda }_i\left({\rho }_i{d}_i^2\left(x,u\right)\left(1+W_i\right)+{\overline{\rho }}_i{\overline{d}}_i^2\left(v,y\right)\left(1+S_i\right)\right).\hfill \end{array}$$

Since λ _i > 0, v^T z _i - s(v|D _i ) + x^T w _i - s(x|C _i ) ≦ 0, i = 1, ..., k, by (9) it yields

$$\sum _{i=1}^k{\lambda }_i\left(S_i-W_i\right)\left(g_i\left(x,v\right)+v^T{r}_i-x^T{t}_i\right)\geqq 0.$$

By assumptions (8), we have g _i (x, v)+v^T r _i -x^T t _i > 0, i = 1, ..., k. Since λ > 0, it follows that S ≰ W. □

Theorem 3.2 (Strong duality). Let $\left(\overline{x},\overline{y},\overline{S},{\overline{z}}_1,\dots ,{\overline{z}}_k,{\overline{r}}_1,\dots ,{\overline{r}}_k,\overline{\lambda },\overline{p}\right)$ be a properly efficient solution of (MFP) _S , and fix $\lambda =\overline{\lambda }$ in (MFD) _W . Suppose that

(a)

$\begin{array}{c}{\nabla }_x{H}_i\left(\overline{x},\overline{y},0\right)={\nabla }_x{G}_i\left(\overline{x},\overline{y},0\right)=0,{\nabla }_{q_i}{\Phi }_i\left(\overline{x},\overline{y},0\right)={\nabla }_{q_i}{\Psi }_i\left(\overline{x},\overline{y},0\right)=0,\hfill \\ H_i\left(\overline{x},\overline{y},0\right)=G_i\left(\overline{x},\overline{y},0\right)=0,{\Phi }_i\left(\overline{x},\overline{y},0\right)={\Psi }_i\left(\overline{x},\overline{y},0\right)=0,{\nabla }_y{H}_i\left(\overline{x},\overline{y},0\right)={\nabla }_y{G}_i\left(\overline{x},\overline{y},0\right)=0,\hfill \\ {\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},0\right)={\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},0\right)=0,i=1,\dots ,k.\hfill \end{array}$

(b) For all i ∈ {1, ..., k},

$$f_i\left(\overline{x},\overline{y}\right)+s\left(\overline{x}|C_i\right)-{\overline{y}}^T{\overline{z}}_i+H_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{p}}_i^T{\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)>0.$$

(c) (i) ${\nabla }_{p_i{p}_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i{p}_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\ne 0$ for ${\overline{p}}_i=0$, i = 1, ..., k and ${\nabla }_{p_i{p}_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i{p}_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)$ is nonsingular for all i = 1, ..., k,

• (ii) ${\sum }_{i=1}^k{\overline{\lambda }}_i\left({\nabla }_{yy}{f}_i\left(\overline{x},\overline{y}\right)-{\overline{S}}_i{\nabla }_{yy}{g}_i\left(\overline{x},\overline{y}\right)\right)$ is positive definite and ${\overline{p}}_i^T\left(\left({\nabla }_y{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_y{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)-\left({\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)\geqq 0$ for all i = 1, ..., k, or ${\sum }_{i=1}^k{\overline{\lambda }}_i\left({\nabla }_{yy}{f}_i\left(\overline{x},\overline{y}\right)-{\overline{S}}_i{\nabla }_{yy}{g}_i\left(\overline{x},\overline{y}\right)\right)$ is negative definite and ${\overline{p}}_i^T\left(\left({\nabla }_y{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_y{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)-\left({\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)\leqq 0$ for all i = 1, ..., k.

• (iii) $\left\{{\nabla }_y{f}_i\left(\overline{x},\overline{y}\right)-{\overline{z}}_i+{\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i\left({\nabla }_y{g}_i\left(\overline{x},\overline{y}\right)+{\overline{r}}_i+{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right):i=1,\dots ,k\right\}$ is linearly independent.

Then $\overline{p}=0$, and there exist ${\overline{w}}_i\in C_i$ and ${\overline{t}}_i\in E_i$, i = 1, ..., k such that $\left(\overline{x},\overline{y},\overline{S},{\overline{w}}_1,...,{\overline{w}}_k,{\overline{t}}_1,\dots ,{\overline{t}}_k,\overline{\lambda },\overline{q}=0\right)$ is a feasible solution of (MFD) _W . Furthermore, if the hypotheses in Theorem 3.1 are satisfied, then $\left(\overline{x},\overline{y},\overline{S},{\overline{w}}_1,...,{\overline{w}}_k,{\overline{t}}_1,\dots ,{\overline{t}}_k,\overline{\lambda },\overline{q}=0\right)$ is a properly efficient solution of (MFD) _W , and the two objective values are equal.

Proof. Since $\left(\overline{x},\overline{y},\overline{S},{\overline{z}}_1,\dots ,{\overline{z}}_k,{\overline{r}}_1,\dots ,{\overline{r}}_k,\overline{\lambda },\overline{p}\right)$ is a properly efficient solution of (MFP) _S , by the Fritz John type necessary optimality conditions [16], there exist α ∈ R^k, β ∈ R^k, γ ∈ R^m, δ ∈ R, μ ∈ R^k and ${\overline{w}}_i\in R^n,{\overline{t}}_i\in R^n$, i = 1, ..., k such that

$$\begin{array}{c}\sum _{i=1}^k{\beta }_i\left(\left({\nabla }_x{f}_i\left(\overline{x},\overline{y}\right)+{\overline{w}}_i+{\nabla }_x{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)-{\overline{S}}_i\left({\nabla }_x{g}_i\left(\overline{x},\overline{y}\right)-{\overline{t}}_i+{\nabla }_x{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)\hfill \\ +{\left(\gamma -\delta \overline{y}\right)}^T\sum _{i=1}^k{\overline{\lambda }}_i\left({\nabla }_{yx}{f}_i\left(\overline{x},\overline{y}\right)-{\overline{S}}_i{\nabla }_{yx}{g}_i\left(\overline{x},\overline{y}\right)\right)\hfill \\ +\sum _{i=1}^k{\left({{\nabla }_{p_i}}_x{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_{i}x}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)}^T\left(\left(\gamma -\delta \overline{y}\right){\overline{\lambda }}_i-{\beta }_i{\overline{p}}_i\right)=0,\hfill \end{array}$$

(15)

$$\begin{array}{c}\sum _{i=1}^k\left({\beta }_i-\delta {\overline{\lambda }}_i\right)\left(\left({\nabla }_y{f}_i\left(\overline{x},\overline{y}\right)-z_i+{\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right.-\left.{\overline{S}}_i\left({\nabla }_y{g}_i\left(\overline{x},\overline{y}\right)+{\overline{r}}_i+{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)\hfill \\ +\sum _{i=1}^k{\beta }_i\left(\left({\nabla }_y{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_y{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)-\left({\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)\hfill \\ +\sum _{i=1}^k{\overline{\lambda }}_i\left({\left({\nabla }_{yy}{f}_i\left(\overline{x},\overline{y}\right)-{\overline{S}}_i{\nabla }_{yy}{g}_i\left(\overline{x},\overline{y}\right)\right)}^T\left(\gamma -\delta \overline{y}\right)\right)\hfill \\ +\sum _{i=1}^k{\left({\nabla }_{p_{i}y}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_{i}y}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)}^T\left(-{\beta }_i{\overline{p}}_i+\left(\gamma -\delta \overline{y}\right){\overline{\lambda }}_i\right)=0,\hfill \end{array}$$

(16)

$$\begin{array}{c}{\alpha }_i-{\beta }_i\left(g_i\left(\overline{x},\overline{y}\right)-s\left(\overline{x}|E_i\right)+{\overline{y}}^T{\overline{r}}_i+G_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{p}}_i^T{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\hfill \\ -{\left(\gamma -\delta \overline{y}\right)}^T\left({\overline{\lambda }}_i\left({\nabla }_y{g}_i\left(\overline{x},\overline{y}\right)+{\overline{r}}_i+{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)=0,i=1,\dots ,k,\hfill \end{array}$$

(17)

$$\begin{array}{c}{\left(\gamma -\delta \overline{y}\right)}^T\left(\left({\nabla }_y{f}_i\left(\overline{x},\overline{y}\right)-{\overline{z}}_i+{\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i\left({\nabla }_y{g}_i\left(\overline{x},\overline{y}\right)+{\overline{r}}_i+{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)\right.\hfill \\ -{\mu }_i=0,i=1,\dots ,k,\hfill \end{array}$$

(18)

$${\left({\overline{\lambda }}_i\left(\gamma -\delta \overline{y}\right)-{\beta }_i{\overline{p}}_i\right)}^T\left({\nabla }_{p_i{p}_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i{p}_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)=0,i=1,\dots ,k,$$

(19)

$${\beta }_i\overline{y}+\left(\gamma -\delta \overline{y}\right){\overline{\lambda }}_i\in N_{D_i}\left({\overline{z}}_i\right),i=1,\dots ,k,$$

(20)

$${\beta }_i{\overline{S}}_i\overline{y}+{\overline{\lambda }}_i{\overline{S}}_i\left(\gamma -\delta \overline{y}\right)\in N_{F_i}\left({\overline{r}}_i\right),i=1,\dots ,k,$$

(21)

$$\begin{array}{c}{\gamma }^T\sum _{i=1}^k{\overline{\lambda }}_i\left(\left({\nabla }_y{f}_i\left(\overline{x},\overline{y}\right)-{\overline{z}}_i+{\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right.\hfill \\ \left.-{\overline{S}}_i\left({\nabla }_y{g}_i\left(\overline{x},\overline{y}\right)+{\overline{r}}_i+{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)=0,\hfill \end{array}$$

(22)

$$\begin{array}{c}\delta {\overline{y}}^T\sum _{i=1}^k{\overline{\lambda }}_i\left(\left({\nabla }_y{f}_i\left(\overline{x},\overline{y}\right)-{\overline{z}}_i+{\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right.\hfill \\ \left.-{\overline{S}}_i\left({\nabla }_y{g}_i\left(\overline{x},\overline{y}\right)+{\overline{r}}_i+{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)=0,\hfill \end{array}$$

(23)

$${\mu }^T\overline{\lambda }=0,$$

(24)

$${\overline{w}}_i\in C_i,{\overline{t}}_i\in E_i,{\overline{x}}^T{\overline{t}}_i=s\left(\overline{x}|E_i\right),{\overline{x}}^T{\overline{w}}_i=s\left(\overline{x}|C_i\right),i=1,\dots ,k,$$

(25)

$$\left(\alpha ,\beta ,\gamma ,\delta ,\mu \right)\ne 0,\left(\alpha ,\gamma ,\delta ,\mu \right)\geqq 0.$$

(26)

Since $\overline{\lambda }>0$, and μ ≧ 0, (24) implies μ = 0. Consequently, (18) yields

$$\begin{array}{c}{\left(\gamma -\delta \overline{y}\right)}^T\left(\left({\nabla }_y{f}_i\left(\overline{x},\overline{y}\right)-{\overline{z}}_i+{\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right.\right.\hfill \\ \left.-{\overline{S}}_i\left({\nabla }_y{g}_i\left(\overline{x},\overline{y}\right)+{\overline{r}}_i+{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)=0,i=1,\dots ,k.\hfill \end{array}$$

(27)

By assumption (i) and (19), we have

$${\overline{\lambda }}_i\left(\gamma -\delta \overline{y}\right)={\beta }_i{\overline{p}}_i,i=1,...,k.$$

(28)

Multiplying (16) $\left(\gamma -\delta \overline{y}\right)$ by left, from (27) and (28) we have

$$\begin{array}{c}{\left(\gamma -\delta \overline{y}\right)}^T\sum _{i=1}^k{\beta }_i\left(\left({\nabla }_y{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_y{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)-\left({\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)\hfill \\ +{\left(\gamma -\delta \overline{y}\right)}^T\sum _{i=1}^k{\overline{\lambda }}_i\left({\nabla }_{yy}{f}_i\left(\overline{x},\overline{y}\right)-{\overline{S}}_i{\nabla }_{yy}{g}_i\left(\overline{x},\overline{y}\right)\right)\left(\gamma -\delta \overline{y}\right)=0.\hfill \end{array}$$

Since $\overline{\lambda }>0$, from (28) and the above equation, we have

$$\begin{array}{c}\sum _{i=1}^k\frac{{\beta }_i^2}{{\overline{\lambda }}_i}{\overline{p}}_i^T\left(\left({\nabla }_y{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_y{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)-\left({\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)\hfill \\ +{\left(\gamma -\delta \overline{y}\right)}^T\sum _{i=1}^k{\overline{\lambda }}_i\left({\nabla }_{yy}{f}_i\left(\overline{x},\overline{y}\right)-{\overline{S}}_i{\nabla }_{yy}{g}_i\left(\overline{x},\overline{y}\right)\right)\left(\gamma -\delta \overline{y}\right)=0.\hfill \end{array}$$

Which by assumption (ii), we can obtain

$$\gamma -\delta \overline{y}=0.$$

(29)

Using (29) in (28), we have ${\beta }_i{\overline{p}}_i=0$, i = 1, ..., k. This implies that ${\overline{p}}_i=0$ when β _i ≠ 0, for all i ∈ {1, ..., k}. Hence, by assumption (1), we get

$$\sum _{i=1}^k{\beta }_i\left(\left({\nabla }_y{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_y{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)-\left({\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)=0.$$

Combining this with (16), (28) and (29), it follows that

$$\sum _{i=1}^k\left({\beta }_i-\delta {\overline{\lambda }}_i\right)\left({\nabla }_y{f}_i\left(\overline{x},\overline{y}\right)-{\overline{z}}_i+{\nabla }_{p_i}{H}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)-{\overline{S}}_i\left({\nabla }_y{g}_i\left(\overline{x},\overline{y}\right)+{\overline{r}}_i+{\nabla }_{p_i}{G}_i\left(\overline{x},\overline{y},{\overline{p}}_i\right)\right)\right)=0,$$

which by assumption (iii), it yields

$${\beta }_i-\delta {\overline{\lambda }}_i=0,i=1,\dots ,k.$$

(30)

We claim that δ ≠ 0, otherwise, from (29) and (30) we get β = 0, γ = 0. Using (29) in (17), we get α = 0. This contradicts with (26). Hence δ = 0. Since $\overline{\lambda }>0$, from (30) we get β > 0. Hence ${\beta }_i{\overline{p}}_i=0$, i = 1, ..., k implies ${\overline{p}}_i=0$, i = 1, ..., k. Using (28), (29) and the fact ${\overline{p}}_i=0$, i = 1, ..., k in (15), by assumption (a), we get

$$\sum _{i=1}^k{\beta }_i\left(\left({\nabla }_x{f}_i\left(\overline{x},\overline{y}\right)+{\overline{w}}_i\right)-{\overline{S}}_i\left({\nabla }_x{g}_i\left(\overline{x},\overline{y}\right)-{\overline{t}}_i\right)\right)=0,$$

combining this with (30) and δ > 0, $\overline{\lambda }>0$, it holds

$$\sum _{i=1}^k{\overline{\lambda }}_i\left(\left({\nabla }_x{f}_i\left(\overline{x},\overline{y}\right)+{\overline{w}}_i\right)-{\overline{S}}_i\left({\nabla }_x{g}_i\left(\overline{x},\overline{y}\right)-{\overline{t}}_i\right)\right)=0,$$

(31)

which yields

$${\overline{x}}^T\sum _{i=1}^k{\overline{\lambda }}_i\left(\left({\nabla }_x{f}_i\left(\overline{x},\overline{y}\right)+{\overline{w}}_i\right)-{\overline{S}}_i\left({\nabla }_x{g}_i\left(\overline{x},\overline{y}\right)-{\overline{t}}_i\right)\right)=0.$$

(32)

On the other hand, by assumption (a) and (2) we get

$$\left(f_i\left(\overline{x},\overline{y}\right)+s\left(\overline{x}|C_i\right)-{\overline{y}}^T{\overline{z}}_i\right)-{\overline{S}}_i\left(g_i\left(\overline{x},\overline{y}\right)-s\left(\overline{x}|E_i\right)+{\overline{y}}^T{\overline{r}}_i\right)=0,i=1,\dots ,k.$$

(33)

Since β > 0, by (20) and (29) we get $\overline{y}\in N_{D_i}\left({\overline{z}}_i\right)$, i = 1, ..., k. This implies

$${\overline{y}}^T{\overline{z}}_i=s\left(\overline{y}|D_i\right),i=1,\dots ,k.$$

(34)

Assumption (b) implies $\overline{S}>0$. By (21), we similarly have $\overline{y}\in N_{F_i}\left({\overline{r}}_i\right)$, i = 1, ..., k. This implies

$${\overline{y}}^T{\overline{r}}_i=s\left(\overline{y}|F_i\right),i=1,\dots ,k.$$

(35)

Combining (25), (33), (34) and (35), we get

$$\left(f_i\left(\overline{x},\overline{y}\right)+{\overline{x}}^T{\overline{w}}_i-s\left(\overline{y}|D_i\right)\right)-{\overline{S}}_i\left(g_i\left(\overline{x},\overline{y}\right)\right.-{\overline{x}}^T{\overline{t}}_i+s\left(\overline{y}|F_i\right)=0,i=1,\dots ,k,$$

combining this with (31) and (32), by assumption (a), $\left(\overline{x},\overline{y},\overline{S},{\overline{w}}_1,...,{\overline{w}}_k,{\overline{t}}_1,\dots ,{\overline{t}}_k,\overline{\lambda },\overline{q}=0\right)$ is a feasible solution of (MFD) _W .

Under the assumptions of Theorem 3.1, if $\left(\overline{x},\overline{y},\overline{S},{\overline{w}}_1,...,{\overline{w}}_k,{\overline{t}}_1,\dots ,{\overline{t}}_k,\overline{\lambda },\overline{q}=0\right)$ is not an efficient solution of (MFD) _W , then there exists other feasible solution $\left(u,v,W,w_1,\dots ,w_k,t_1,\dots ,t_k,\overline{\lambda },q\right)$, of (MFD) _W such that $\overline{S}\le W$. Since $\left(\overline{x},\overline{y},\overline{S},{\overline{z}}_1,\dots ,{\overline{z}}_k,{\overline{r}}_1,\dots ,{\overline{r}}_k,\overline{\lambda },\overline{p}\right)$ is a feasible solution of (MFP) _S , by Theorem 3.1, we have $\overline{S}\nleqq W$, hence the contradiction implies $\left(\overline{x},\overline{y},\overline{S},{\overline{w}}_1,...,{\overline{w}}_k,{\overline{t}}_1,\dots ,{\overline{t}}_k,\overline{\lambda },\overline{q}=0\right)$ is an efficient solution of (MFD) _W .

If $\left(\overline{x},\overline{y},\overline{S},{\overline{w}}_1,...,{\overline{w}}_k,{\overline{t}}_1,\dots ,{\overline{t}}_k,\overline{\lambda },\overline{q}=0\right)$ is not a properly efficient solution of (MFD) _W , then there exists other feasible solution $\left(u,v,W,w_1,\dots ,w_k,t_1,\dots ,t_k,\overline{\lambda },q\right)$ of (MFD) _W such that for an index i ∈ {1, ..., k} and any real number M > 0, $W_i-{\overline{S}}_i>M\left({\overline{S}}_j-W_j\right)$ for j satisfying ${\overline{S}}_j>W_j$ whenever $W_i>{\overline{S}}_i$ This implies $W_i>{\overline{S}}_i$ can be made arbitrarily large and this contradicts with Theorem 3.1. And it is easy to find that the two objective values are equal. □

Theorem 3.3 (Strict converse duality). Let $\left(\overline{u},\overline{v},\overline{W},{\overline{w}}_1,\dots ,{\overline{w}}_k,{\overline{t}}_1,\dots ,{\overline{t}}_k,\overline{\lambda },\overline{q}\right)$ be a properly efficient solution of (MFD) _W , and fix $\lambda =\overline{\lambda }$ in (MFP) _S . Suppose that

• (a) $\begin{array}{c}{\nabla }_x{\Phi }_i\left(\overline{u},\overline{v},0\right)={\nabla }_x{\Psi }_i\left(\overline{u},\overline{v},0\right)=0,{\nabla }_{q_i}{\Phi }_i\left(\overline{u},\overline{v},0\right)={\nabla }_{q_i}{\Psi }_i\left(\overline{u},\overline{v},0\right)=0,\hfill \\ H_i\left(\overline{u},\overline{v},0\right)=G_i\left(\overline{u},\overline{v},0\right)=0,{\Phi }_i\left(\overline{u},\overline{v},0\right)={\Psi }_i\left(\overline{u},\overline{v},0\right)=0,{\nabla }_y{\Phi }_i\left(\overline{u},\overline{v},0\right)={\nabla }_y{\Psi }_i\left(\overline{u},\overline{v},0\right)=0,\hfill \\ {\nabla }_{p_i}{H}_i\left(\overline{u},\overline{v},0\right)={\nabla }_{p_i}{G}_i\left(\overline{u},\overline{v},0\right)=0,i=1,\dots ,k.\hfill \end{array}$

(b) For all i ∈ {1, ..., k},

$$f_i\left(\overline{u},\overline{v}\right)-s\left(\overline{v}|D_i\right)+{\overline{u}}^T{\overline{w}}_i+{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)-{\overline{q}}_i^T{\nabla }_{q_i}{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)>0.$$

(c) (i) ${\nabla }_{q_i{q}_i}{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)-{\overline{W}}_i{\nabla }_{q_i{q}_i}{\Psi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)\ne 0$, for ${\overline{q}}_i=0$, i = 1, ..., k, and ${\nabla }_{q_i{q}_i}{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)-{\overline{W}}_i{\nabla }_{q_i{q}_i}{\Psi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)$ is nonsingular for all i = 1, ..., k, and

• (ii) $\sum _{i=1}^k{\overline{\lambda }}_i\left({\nabla }_{xx}{f}_i\left(\overline{u},\overline{v}\right)-{\overline{W}}_i{\nabla }_{xx}{g}_i\left(\overline{u},\overline{v}\right)\right)$ is positive definite and ${\overline{q}}_i^T\left(\left({\nabla }_x{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)-{\overline{W}}_i{\nabla }_x{\Psi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)\right)-\left({\nabla }_{q_i}{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)-{\overline{W}}_i{\nabla }_{q_i}{\Psi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)\right)\right)\geqq 0$ for all i = 1, ..., k, or $\sum _{i=1}^k{\overline{\lambda }}_i\left({\nabla }_{xx}{f}_i\left(\overline{u},\overline{v}\right)-{\overline{W}}_i{\nabla }_{xx}{g}_i\left(\overline{u},\overline{v}\right)\right)$ is negative definite and ${\overline{q}}_i^T\left(\left({\nabla }_x{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)-{\overline{W}}_i{\nabla }_x{\Psi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)\right)-\left({\nabla }_{q_i}{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)-{\overline{W}}_i{\nabla }_{q_i}{\Psi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)\right)\right)\leqq 0$ for all i = 1, ..., k.

• (iii) $\left\{{\nabla }_x{f}_i\left(\overline{u},\overline{v}\right)+{\overline{w}}_i+{\nabla }_{q_i}{\Phi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)-{\overline{W}}_i\left({\nabla }_x{g}_i\left(\overline{u},\overline{v}\right)-{\overline{t}}_i+{\nabla }_{q_i}{\Psi }_i\left(\overline{u},\overline{v},{\overline{q}}_i\right)\right):i=1,\dots ,k\right\}$ is linearly independent.

Then $\overline{q}=0$, and there exist ${\overline{z}}_i\in D_i$ and ${\overline{r}}_i\in F_i$, i = 1, ..., k such that $\left(\overline{u},\overline{v},\overline{W},{\overline{z}}_1,...,{\overline{z}}_k,{\overline{r}}_1,\dots ,{\overline{r}}_k,\overline{\lambda },\overline{p}=0\right)$ is a feasible solution of (MFP) _S . Furthermore, if the hypotheses in Theorem 3.1 are satisfied, then $\left(\overline{u},\overline{v},\overline{W},{\overline{z}}_1,...,{\overline{z}}_k,{\overline{r}}_1,\dots ,{\overline{r}}_k,\overline{\lambda },\overline{p}=0\right)$ is a properly efficient solution of (MFP) _S , and the two objective values are equal. □

Remark 3.1.(1) If k = 1, $H_1\left(x,y,p_1\right)=\frac{1}{2}{p}_1^T{\nabla }_{yy}{f}_1\left(x,y\right)p_1$, $g_1\left(x,y\right)-s\left(x|E_1\right)+y^T{r}_1+G_1\left(x,y,p_1\right)-p_1^T{\nabla }_{p_1}{G}_1\left(x,y,p_1\right)=1$, ${\Phi }_1\left(u,v,q_1\right)=\frac{1}{2}{q}_1^T{\nabla }_{xx}{f}_1\left(u,v\right)q_1$, and $g_1\left(u,v\right)+s\left(v|F_1\right)-u^T{t}_1+{\Psi }_1\left(u,v,q_1\right)-q_1^T{\nabla }_{q1}{\Psi }_1\left(u,v,q_1\right)=1$, then (MFP) _S and (MFD) _W becomes the problems considered by Hou and Yang [17].

• (2) If k = 1, $g_1\left(x,y\right)-s\left(x|E_1\right)+y^T{r}_1+G_1\left(x,y,p_1\right)-p_1^T{\nabla }_{p_1}{G}_1\left(x,y,p_1\right)=1$, and $g_1\left(u,v\right)+s\left(v|F_1\right)-u^T{t}_1+{\Psi }_1\left(u,v,q_1\right)-q_1^T{\nabla }_{q1}{\Psi }_1\left(u,v,q_1\right)=1$, then (MFP) _S and (MFD) _W becomes the problems considered by Mishra [18].

• (3) If $g_i\left(x,y\right)-s\left(x|E_i\right)+y^T{r}_i+G_i\left(x,y,p_i\right)-p_i^T{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)=1$, and $g_i\left(u,v\right)+s\left(v|F_i\right)-u^T{t}_i+{\Psi }_i\left(u,v,q_i\right)-q_i^T{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)=1$ for all i {1, ..., k}, then (MFP) _S and (MFD) _W becomes the problems considered by Chen [14].

• (4) If $g_i\left(x,y\right)-s\left(x|E_i\right)+y^T{r}_i+G_i\left(x,y,p_i\right)-p_i^T{\nabla }_{p_i}{G}_i\left(x,y,p_i\right)=1$, $g_i\left(u,v\right)+s\left(v|F_i\right)-u^T{t}_i+{\Psi }_i\left(u,v,q_i\right)-q_i^T{\nabla }_{q_i}{\Psi }_i\left(u,v,q_i\right)=1$, $H_i\left(x,y,p_i\right)=\frac{1}{2}{p}_i^T{\nabla }_{yy}{f}_i\left(x,y\right)p_i,{\Phi }_i\left(u,v,q_i\right)=\frac{1}{2}{q}_i^T{\nabla }_{xx}{f}_i\left(u,v\right)q_i$, for all i ∈ {1, ..., k}, and there is not the condition λ^T e = 1 in (MFP) _S and (MFD) _W , then the two problems reduce to the problems considered by Yang et al. [19].