ABSTRACT

In this article, for a differentiable function $H\mathrm{<mo>\; :\; </mo>}{R}^n\times R^n\to R$ , we introduce the definition of the higher-order $\left(V\mathrm{,}\alpha \mathrm{,}\beta \mathrm{,}\rho \mathrm{,}d\right)$ -invexity. Three duality models for a multiobjective fractional programming problem involving nondifferen- tiability in terms of support functions have been formulated and usual duality relations have been established under the higher-order $\left(V\mathrm{,}\alpha \mathrm{,}\beta \mathrm{,}\rho \mathrm{,}d\right)$ -invex assumptions.

Keywords:

Efficient Solution, Support Function, Multiobjective Fractional Programming, Generalized Invexity

1. Introduction

Consider the following nonlinear programming problem (P) Minimize $f\left(x\right)$ subject to $g\left(x\right)\le 0$ , where $f\mathrm{<mo>\; :\; </mo>}{R}^n\to R$ and $g\mathrm{<mo>\; :\; </mo>}{R}^n\to R$ are twice differen- tiable functions. The Mangasarian [1] second-order dualof (P) is (DP) Maximize

$f\left(u\right)-y^{\text{T}}g\left(u\right)-\frac{1}{2}{p}^{\text{T}}{\nabla }^2\left[f\left(u\right)-y^{\text{T}}g\left(u\right)\right]p$

such that $\nabla \left[f\left(u\right)-y^{\text{T}}g\left(u\right)\right]+{\nabla }^2\left[f\left(u\right)-y^{\text{T}}g\left(u\right)\right]p=0$

By introducing two differentiable functions $H\mathrm{<mo>\; :\; </mo>}{R}^n\times R^n\to R$ and $K\mathrm{<mo>\; :\; </mo>}{R}^n\times R^n\to R^m$ , Mangasarian [1] formulated the following higher-order dual of (P): (DP)₁ Maximize

$f\left(u\right)-y^{\text{T}}g\left(u\right)+H\left(u\mathrm{,}p\right)-y^{\text{T}}K\left(u\mathrm{,}p\right)$

such that ${\nabla }_{p}H\left(u\mathrm{,}p\right)-{\nabla }_p\left[y^{\text{T}}K\left(u\mathrm{,}p\right)\right]=0,$ $y\ge \mathrm{0,}$where ${\nabla }_{p}H\left(u\mathrm{,}p\right)$ denotes the $n\times 1$ gradient of $H\left(u\mathrm{,}p\right)$ with respect to p and ${\nabla }_p\left(y^{\text{T}}K\left(u\mathrm{,}p\right)\right)$ denotes the $n\times 1$ , gradient of $y^{\text{T}}K\left(u\mathrm{,}p\right)$ with respect to p.

Further, Egudo [2] studied the following multiobjective fractional program- ming problem: (MFPP) Minimize

$G\left(x\right)=\left(\frac{f_1\left(x\right)}{g_1\left(x\right)}\mathrm{,}\frac{f_2\left(x\right)}{g_2\left(x\right)}\mathrm{,}\cdots \mathrm{,}\frac{f_k\left(x\right)}{g_k(x))}\right)$

subject to

$x\in X^0=\left\{x\in X\subset R^n\mathrm{<mo>\; :\; </mo>}{h}_j\left(x\right)\le \mathrm{0,\; <mi>\; </mi>}j\in M\right\}\mathrm{,}$

where $f=\left(f_1,f_2,\cdots ,f_k\right):X\to R^k,\mathrm{<mi>\; </mi>}g=\left(g_1,g_2,\cdots ,g_k\right):X\to R^k$ and $h=\left(h_1\mathrm{,}{h}_2\mathrm{,}\cdots \mathrm{,}{h}_m\right)\mathrm{<mo>\; :\; </mo>}X\to R^m$ are differentiable on X. Also, he discussed duality results for Mond-Weir and Schaible type dual programs under generalized convexity.

For the nondifferentiable multiobjective programming problem: (MPP) Mini- mize

$G\left(x\right)=\left(f_1\left(x\right)+S\left(x|C_1\right),f_2\left(x\right)+S\left(x|C_2\right),\cdots ,f_k\left(x\right)+S\left(x|C_k\right)\right)$

subject to $x\in X^0=\left\{x\in X\subset R^n:g_j\left(x\right)+S\left(x|E_j\right)\le 0,j=1,2,\cdots ,m\right\},$ where $f_i:X\to R\left(i=1,2,\cdots ,k\right)$ and $g_j:X\to R\mathrm{<mi>\; </mi>}\left(j=1,2,\cdots ,m\right)$ are differentiable func- tions. $C_i$ and $E_j$ are compact convex sets in $R^n$ and $S\left(x\mathrm{<mo>\; |\; </mo>}{C}_i\right)\mathrm{<mi>\; </mi>}\left(i=1,2,\cdots ,k\right)\mathrm{<mi>\; </mi>}$ and $S\left(x\mathrm{<mo>\; |\; </mo>}{E}_j\right)\mathrm{<mi>\; </mi>}\left(j=1,2,\cdots ,m\right)$ denote the support func- tions of compact convex sets, various researchers have worked. Gulati and Agarwal [3] introduced the higher-order Wolfe-type dual model of (MPP) and proved duality theorems under higher-order $\left(F\mathrm{,}\rho \mathrm{,}\rho \mathrm{,}d\right)$ -type I-assump- tions.

In last several years, various optimality and duality results have been obtained for multiobjective fractional programming problems. In Chen [4] , multiobjective fractional problem and its duality theorems have been considered under higher- order $\left(F\mathrm{,}\alpha \mathrm{,}\rho \mathrm{,}d\right)$ -convexity. Later on, Suneja et al. [5] discussed higher-order Mond-Weir and Schaible type nondifferentiable dual programs and their duality theorems under higher-order $\left(F\mathrm{,}\rho \mathrm{,}\sigma \right)$ -type I-assumptions. Several researchers have also worked in this directions such as ( [6] [7] ).

In this paper, we first introduce the definition of higher-order $\left(V\mathrm{,}\alpha \mathrm{,}\beta \mathrm{,}\rho \mathrm{,}d\right)$ - invex with respect to differentiable function $H\mathrm{<mo>\; :\; </mo>}{R}^n\times R^n\to R$ . We also construct a nontrivial numerical example which illustrates the existence of such a function. We then formulate three higher-order dual problems corresponding to the multiobjective nondifferentiable fractional programming problem. Further, we establish usual duality relations for these primal-dual pairs under aforesaid assumptions.

2. Preliminaries

Let $X\subseteq R^n$ be an open set and $\varphi \mathrm{<mo>\; :\; </mo>}X\to R\mathrm{,\; <mi>\; </mi>}H\mathrm{<mo>\; :\; </mo>}X\times R^n\to R$ be differentiable functions. $\alpha \mathrm{,\; <mi>\; </mi>}\beta \mathrm{<mo>\; :\; </mo>}X\times X\to R_{+}\backslash \left\{0\right\}$ , $\eta \mathrm{<mo>\; :\; </mo>}X\times X\to R^n$ , $\rho \in R^n$ and $\theta \mathrm{<mo>\; :\; </mo>}X\times X\to R^n$ .

Definition 2.1. $\varphi$ is said to be (strictly) higher-order $\left(V\mathrm{,}\alpha \mathrm{,}\beta \mathrm{,}\rho \mathrm{,}\theta \right)$ -invex at u with respect to $H\left(u\mathrm{,}p\right)$ , if there exist $\eta \mathrm{,\; <mi>\; </mi>}\alpha \mathrm{,\; <mi>\; </mi>}\beta \mathrm{,\; <mi>\; </mi>}\rho$ and $\theta$ such that, for any $x\in X$ and $p\in R^n$ ,

$\begin{array}{l}\alpha \left(x\mathrm{,}u\right)\left[\varphi \left(x\right)-\varphi \left(u\right)\right]\mathrm{<mi>\; </mi>}(>)\ge \mathrm{<mi>\; </mi>}{\eta }^{\text{T}}\left(x\mathrm{,}u\right)\left(\nabla \varphi \left(u\right)+{\nabla }_{p}H\left(u\mathrm{,}p\right)\right)\\ +\beta \left(x\mathrm{,}u\right)\left[H\left(u\mathrm{,}p\right)-p^{\text{T}}{\nabla }_{p}H\left(u\mathrm{,}p\right)\right]+\rho {\Vert \theta \left(x\mathrm{,}u\right)\Vert }^2\mathrm{.}\end{array}$

Example 2.1. Let $\varphi \mathrm{<mo>\; :\; </mo>}R\to R$ be such that $\varphi \left(x\right)=x^4+x^2+1$ .

Let

$\eta \left(x,u\right)=\frac{1}{2}\left(x^2+u^2\right),\mathrm{<mi>\; </mi>}H\left(u,p\right)=-2p{\left(x+1\right)}^2$ .

Also, suppose

$\alpha \left(x,u\right)=1,\mathrm{<mi>\; </mi>}\beta \left(x,u\right)=2,\mathrm{<mi>\; </mi>}\rho =-1,\mathrm{<mi>\; </mi>}\Vert \theta \left(x,u\right)\Vert ={\left(x^2+u^2\right)}^{\frac{1}{2}}$ .

Now,

$\begin{array}{l}\xi =\alpha \left(x\mathrm{,}u\right)\left[\varphi \left(x\right)-\varphi \left(u\right)\right]-{\eta }^{\text{T}}\left(x\mathrm{,}u\right)\left(\nabla \varphi \left(u\right)+{\nabla }_{p}H\left(u\mathrm{,}p\right)\right)\\ -\beta \left(x\mathrm{,}u\right)\left[H\left(u\mathrm{,}p\right)-p^{\text{T}}{\nabla }_{p}H\left(u\mathrm{,}p\right)\right]-\rho {\Vert \theta \left(x\mathrm{,}u\right)\Vert }^2.\end{array}$

$\xi =\left(x^4+x^2-u^4-u^2\right)-\frac{1}{2}\left(x^2+u^2\right)\left[4u^3+2u-2{\left(u+1\right)}^2\right]-\left(x^2+u^2\right)$

$\xi =x^4+x^2$ (at $u=0$ ).

$\ge 0,\mathrm{<mi>\; </mi>}\forall \mathrm{<mi>\; </mi>}x\in R$ .

Hence, $\varphi$ is higher-order $\left(V\mathrm{,}\alpha \mathrm{,}\beta \mathrm{,}\rho \mathrm{,}\theta \right)$ -invex at $u=0$ with respect to $H\left(u\mathrm{,}p\right)$ .

Remark 2.1.

1) If $H\left(u,p\right)=0$ , then the Definition 2.1 reduces to $\left(V\mathrm{,}\rho \right)$ -invex function introduced by Kuk et al. [8] .

2) If $H\left(u,p\right)=0$ and $\rho =0$ , then the Definition 2.1 becomes that of V-invexity introduced by Jeyakumar and Mond [9] .

3) If $H\left(u,p\right)=\frac{1}{2}{p}^{\text{T}}{\nabla }^2\varphi \left(u\right)p,\mathrm{<mi>\; </mi>}\alpha \left(x,u\right)=0$ and $\rho =0$ , then above definition yields in η-bonvexity given by Pandey [10] .

4) If $\beta =1$ , then the Definition 2.1 reduced in $\left(V\mathrm{,}\alpha \mathrm{,}\rho \mathrm{,}\theta \right)$ -invex given by Gulati and Geeta [11] .

A differentiable function $f=\left(f_1\mathrm{,}{f}_2\mathrm{,}\cdots \mathrm{,}{f}_k\right)\mathrm{<mo>\; :\; </mo>}X\to R^k$ is $\left(V\mathrm{,}\alpha \mathrm{,}\beta \mathrm{,}\rho \mathrm{,}\theta \right)$ -invex if for all $i=1,2,\cdots ,k$ , $f_i$ is $\left(V\mathrm{,}{\alpha }_i\mathrm{,}{\beta }_i\mathrm{,}{\rho }_i\mathrm{,}{\theta }_i\right)$ -invex.

Definition 2.2. [12] . Let C be a compact convex set in $R^n$ . The support function of C is defined by

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

3. Problem Formulation

Consider the multiobjective programming problem with support function given as: (MFP) Minimize

$F\left(x\right)=\left\{\frac{f_1\left(x\right)+S\left(x|C_1\right)}{g_1\left(x\right)-S\left(x|D_1\right)},\frac{f_2\left(x\right)+S\left(x|C_2\right)}{g_2\left(x\right)-S\left(x|D_2\right)},\cdots ,\frac{f_k\left(x\right)+S\left(x|C_k\right)}{g_k\left(x\right)-S\left(x|D_k\right)}\right\}$

subject to $x\in X^0=\left\{x\in X\subset R^n:h_j\left(x\right)+S\left(x|E_j\right)\le 0,j=1,2,\cdots ,m\right\},$

where $f=\left(f_1,f_2,\cdots ,f_k\right):X\to R^k$ , $g=\left(g_1,g_2,\cdots ,g_k\right):X\to R^k$ and $h=\left(h_1\mathrm{,}{h}_2\mathrm{,}\cdots \mathrm{,}{h}_m\right)\mathrm{<mo>\; :\; </mo>}X\to R^m$ are differentiable on X, $f_i\left(\mathrm{.}\right)+S\left(\mathrm{.\; <mo>\; |\; </mo>}{C}_i\right)\ge 0$ and $g_i(.)-S\left(.|D_i\right)>0$ . Let $H_i\mathrm{<mo>\; :\; </mo>}X\times R^n\to R$ be differentiable functions, $C_i\mathrm{,\; <mi>\; </mi>}{D}_i$ and $E_j$ are compact convex sets in $R^n$ , for all $i=1,2,\cdots ,k,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m$ .

Definition 3.1. [3] . A point $x^0\in X^0$ is said to be an efficient solution (or Pareto optimal) of (MFP), if there exists no $x\in X^0$ such that for every

$i=1,2,\cdots ,k$ , $\frac{f_i\left(x\right)+S\left(x|C_i\right)}{g_i\left(x\right)-S\left(x|D_i\right)}\le \frac{f_i\left(x^0\right)+S\left(x^0|C_i\right)}{g_i\left(x^0\right)-S\left(x^0|D_i\right)}$

and for some $r=1,2,\cdots ,k$ ,

$\frac{f_r\left(x\right)+S\left(x|C_r\right)}{g_r\left(x\right)-S\left(x|D_r\right)}<\frac{f_r\left(x^0\right)+S\left(x^0|C_r\right)}{g_r\left(x^0\right)-S\left(x^0|D_r\right)}$ .

We now state theorems 3.1-3.2, whose proof follows on the lines [13] .

Theorem 3.1. For some t, if $f_t\left(\mathrm{.}\right)+{\left(\mathrm{.}\right)}^{\text{T}}{z}_t$ and $-\left(g_t\left(\mathrm{.}\right)-{\left(\mathrm{.}\right)}^{\text{T}}{v}_t\right)$ are higher- order $\left(V\mathrm{,}{\alpha }_t\mathrm{,}{\beta }_t\mathrm{,}{\rho }_t\mathrm{,}{\theta }_t\right)$ -invex at u with respect to $H_t\left(u\mathrm{,}p\right)$ for same $\eta \left(x\mathrm{,}u\right)$ . Then, the fractional function $\left(\frac{f_t\left(\mathrm{.}\right)+{\left(\mathrm{.}\right)}^{\text{T}}{z}_t}{g_t\left(\mathrm{.}\right)-{\left(\mathrm{.}\right)}^{\text{T}}{v}_t}\right)$ is higher-order $\left(V\mathrm{,}{\overline{\alpha }}_t\mathrm{,}{\overline{\beta }}_t\mathrm{,}{\overline{\rho }}_t\mathrm{,}{\overline{\theta }}_t\right)$ -invex at u with respect to ${\overline{H}}_t\left(u\mathrm{,}p\right)$ , where

${\overline{\alpha }}_t\left(x,u\right)=\left(\frac{g_t\left(x\right)-x^{\text{T}}{v}_t}{g_t\left(u\right)-u^{\text{T}}{v}_t}\right){\alpha }_t\left(x,u\right)$ , ${\overline{\beta }}_t\left(x,u\right)={\beta }_t\left(x,u\right)$ ,

${\overline{\theta }}_t\left(x,u\right)={\theta }_t\left(x,u\right){\left(\frac{1}{g_t\left(u\right)-u^{\text{T}}{v}_t}+\frac{f_t\left(u\right)+u^{\text{T}}{z}_t}{{\left(g_t\left(u\right)-u^{\text{T}}{v}_t\right)}^2}\right)}^{\frac{1}{2}}$ , ${\overline{\rho }}_t\left(x,u\right)={\rho }_t\left(x,u\right)$

and

${\overline{H}}_t\left(u\mathrm{,}p\right)=\left(\frac{1}{g_t\left(u\right)-u^{\text{T}}{v}_t}+\frac{f_t\left(u\right)+u^{\text{T}}{z}_t}{{\left(g_t\left(u\right)-u^{\text{T}}{v}_t\right)}^2}\right)H_t\left(u\mathrm{,}p\right)$ .

Theorem 3.2. In Theorem 3.1,if either $-\left(g_t\left(\mathrm{.}\right)-{\left(\mathrm{.}\right)}^{\text{T}}{v}_t\right)$ is strictly higher- order $\left(V\mathrm{,}{\alpha }_t\mathrm{,}{\beta }_t\mathrm{,}{\rho }_t\mathrm{,}{\theta }_t\right)$ -invex at u with respect to $H_t\left(u\mathrm{,}p\right)$ and $\left(f_t(.)-{(.)}^{\text{T}}{z}_t\right)>0$ or $\left(f_t(.)-{(.)}^{\text{T}}{z}_t\right)$ is strictly higher-order $\left(V\mathrm{,}{\alpha }_t\mathrm{,}{\beta }_t\mathrm{,}{\rho }_t\mathrm{,}{\theta }_t\right)$ - invex at u with respect to $H_t\left(u\mathrm{,}p\right)$ , then $\left(\frac{f_t(.)+{(.)}^{\text{T}}{z}_t}{g_t(.)-{(.)}^{\text{T}}{z}_t}\right)$ is strictly higher- order $\left(V\mathrm{,}{\overline{\alpha }}_t\mathrm{,}{\overline{\beta }}_t\mathrm{,}{\overline{\rho }}_t\mathrm{,}{\overline{\theta }}_t\right)$ -invex at $u\in X$ with respect to ${\overline{H}}_t\left(u\mathrm{,}p\right)$ .

Theorem 3.3 (Necessary Condition) [14] . Assume that $\overline{x}$ is an efficient solution of (MFP) and the Slater’s constraint qualification is satisfied on X. Then there exist ${\overline{\lambda }}_i>0,\mathrm{<mi>\; </mi>}{\overline{\mu }}_j\in R^m,\mathrm{<mi>\; </mi>}{\overline{z}}_i\in R^n,\mathrm{<mi>\; </mi>}{\overline{v}}_i\in R^n$ and ${\overline{w}}_j\in R^m,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m$ , such that

$\sum _{i=1}^k{\overline{\lambda }}_i\nabla \left(\frac{f_i\left(\overline{x}\right)+{\overline{x}}^{\text{T}}{\overline{z}}_i}{g_i\left(\overline{x}\right)-{\overline{x}}^{\text{T}}{\overline{v}}_i}\right)+\sum _{j=1}^m{\overline{\mu }}_j\nabla \left(h_j\left(\overline{x}\right)+{\overline{x}}^{\text{T}}{\overline{w}}_j\right)=0,$ (1)

$\sum _{j\mathrm{<mo>\; =\; </mo>1}}^m{\overline{\mu }}_j\left(h_j\left(\overline{x}\right)+{\overline{x}}^{\text{T}}{\overline{w}}_j\right)=\mathrm{0,}$ (2)

${\overline{x}}^{\text{T}}{\overline{z}}_i=S\left(\overline{x}|C_i\right),\mathrm{<mi>\; </mi>}{\overline{z}}_i\in C_i,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,$ (3)

${\overline{x}}^{\text{T}}{\overline{v}}_i=S\left(\overline{x}\mathrm{<mo>\; |\; </mo>}{D}_i\right)\mathrm{,\; <mi>\; </mi>}{\overline{v}}_i\in D_i\mathrm{,\; <mi>\; </mi>}i=\mathrm{1,2,}\cdots \mathrm{,}k\mathrm{,}$ (4)

${\overline{x}}^{\text{T}}{\overline{w}}_j=S\left(\overline{x}|E_j\right),\mathrm{<mi>\; </mi>}{\overline{w}}_j\in E_j,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m,$ (5)

${\overline{\lambda }}_i>0,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,\mathrm{<mi>\; </mi>}{\overline{\mu }}_j\ge 0,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m.$ (6)

Theorem 3.4. (Sufficient Condition). Let u be a feasible solution of (MFP). Then, there exist ${\lambda }_i>0,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k$ and ${\mu }_j\ge 0,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m$ , such that

$\sum _{i=1}^k{\lambda }_i\nabla \left(\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right)+\sum _{j=1}^m{\mu }_j\nabla \left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)=0,$ (7)

$\sum _{j\mathrm{<mo>\; =\; </mo>1}}^m{\mu }_j\left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)=\mathrm{0,}$ (8)

$u^{\text{T}}{z}_i=S\left(u|C_i\right),\mathrm{<mi>\; </mi>}{z}_i\in C_i,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,$ (9)

$u^{\text{T}}{v}_i=S\left(u|D_i\right),\mathrm{<mi>\; </mi>}{v}_i\in D_i,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,$ (10)

$u^{\text{T}}{w}_j=S\left(u|E_j\right),\mathrm{<mi>\; </mi>}{w}_j\in E_j,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m,$ (11)

${\overline{\lambda }}_i>0,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,\mathrm{<mi>\; </mi>}{\overline{\mu }}_j\ge 0,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m.$ (12)

Let, for $i=1,2,\cdots ,k,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m$ ,

1) $\left(f_i\left(\mathrm{.}\right)+{\left(\mathrm{.}\right)}^{\text{T}}{z}_i\right)$ and $-\left(g_i(.)-{(.)}^{\text{T}}{v}_i\right)$ be higher-order $\left(V\mathrm{,}{\alpha }_i^1\mathrm{,}{\beta }_i^1\mathrm{,}{\rho }_i^1\mathrm{,}{\theta }_i^1\right)$ - invex at u with respect to $H_i\left(u\mathrm{,}p\right)$ ,

2) $\left(h_j(.)+{(.)}^{\text{T}}{w}_j\right)$ be higher-order $\left(V\mathrm{,}{\alpha }_j^2\mathrm{,}{\beta }_j^2\mathrm{,}{\rho }_j^2\mathrm{,}{\theta }_j^2\right)$ -invex at u with res- pect to $G_j\left(u\mathrm{,}p\right)$ ,

3) $\sum _{i=1}^k{\lambda }_i{\overline{\rho }}_i^1{\Vert {\overline{\theta }}_i^1\left(x,u\right)\Vert }^2+\sum _{j=1}^m{\mu }_j{\rho }_j^2{\Vert {\theta }_j^2\left(x,u\right)\Vert }^2\ge 0,$

4) $\sum _{i=1}^k{\lambda }_i\left({\nabla }_p{\overline{H}}_i\left(u,p\right)\right)+\sum _{j=1}^m{\mu }_j\left({\nabla }_p{G}_j\left(u,p\right)\right)=0$ , $\sum _{i=1}^k{\lambda }_i\left({\overline{H}}_i\left(u,p\right)-p^{\text{T}}{\nabla }_p{\overline{H}}_i\left(u,p\right)\right)\ge 0$ and $\sum _{j=1}^m{\mu }_j\left(G_j\left(u,p\right)-p^{\text{T}}{\nabla }_p{G}_j\left(u,p\right)\right)\ge 0$ ,

5) ${\alpha }_i^1\left(x,u\right)={\alpha }_j^2\left(x,u\right)={\beta }_i^1\left(x,u\right)={\beta }_j^2\left(x,u\right)=\alpha \left(x,u\right),$

where

${\overline{\alpha }}_i\left(x,u\right)=\left(\frac{g_i\left(x\right)-x^{\text{T}}{v}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right){\alpha }_i\left(x,u\right)$ , ${\overline{\beta }}_i\left(x,u\right)={\beta }_i\left(x,u\right)$ ,

${\overline{\theta }}_i\left(x,u\right)={\theta }_i\left(x,u\right){\left(\frac{1}{g_i\left(u\right)-u^{\text{T}}{v}_i}+\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{{\left(g_i\left(u\right)-u^{\text{T}}{v}_i\right)}^2}\right)}^{\frac{1}{2}}$

and ${\overline{\rho }}_i\left(x,u\right)={\rho }_i\left(x,u\right)$ .

Then, u is an efficient solution of (MFP).

Proof. Suppose u is not an efficient solution of (MFP). Then there exists $x\in X^0$ such that

$\frac{f_i\left(x\right)+S\left(x|C_i\right)}{g_i\left(x\right)-S\left(x|D_i\right)}\le \frac{f_i\left(u\right)+S\left(u|C_i\right)}{g_i\left(u\right)-S\left(u|D_i\right)},\mathrm{<mi>\; </mi>}\text{forall}i=1,2,\cdots ,k$

and

$\frac{f_r\left(x\right)+S\left(x|C_r\right)}{g_r\left(x\right)-S\left(x|D_r\right)}<\frac{f_r\left(u\right)+S\left(u|C_r\right)}{g_r\left(u\right)-S\left(u|D_r\right)},\mathrm{<mi>\; </mi>}\text{for}\text{some}r=1,2,\cdots ,k,$

which implies

$\begin{array}{c}\frac{f_i\left(x\right)+x^{\text{T}}{z}_i}{g_i\left(x\right)-x^{\text{T}}{v}_i}\le \frac{f_i\left(x\right)+S\left(x|C_i\right)}{g_i\left(x\right)-S\left(x|D_i\right)}\le \frac{f_i\left(u\right)+S\left(u|C_i\right)}{g_i\left(u\right)-S\left(u|D_i\right)}\\ =\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i},\text{forall}\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k\end{array}$ (13)

and

$\begin{array}{c}\frac{f_r\left(x\right)+x^{\text{T}}{z}_r}{g_r\left(x\right)-x^{\text{T}}{v}_r}\le \frac{f_r\left(x\right)+S\left(x|C_r\right)}{g_r\left(x\right)-S\left(x|D_r\right)}<\frac{f_r\left(u\right)+S\left(u|C_r\right)}{g_r\left(u\right)-S\left(u|D_r\right)}\\ =\frac{f_r\left(u\right)+u^{\text{T}}{z}_r}{g_r\left(u\right)-u^{\text{T}}{v}_r},\mathrm{<mi>\; </mi>}\text{for}\text{some}\mathrm{<mi>\; </mi>}r=1,2,\cdots ,k.\end{array}$ (14)

Since ${\lambda }_i>0,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k$ , inequalities (13) and (14) gives

$\sum _{i=1}^k{\lambda }_i\left(\frac{f_i\left(x\right)+x^{\text{T}}{z}_i}{g_i\left(x\right)-x^{\text{T}}{v}_i}-\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right)<0.$ (15)

From Theorem 3.1, for each $i\mathrm{,\; <mi>\; </mi>1}\le i\le k$ , $\left(\frac{f_i(.)+{(.)}^{\text{T}}{z}_i}{g_i(.)-{(.)}^{\text{T}}{v}_i}\right)$

is higher-order $\left(V\mathrm{,}{\overline{\alpha }}_i^1\mathrm{,}{\overline{\beta }}_i^1\mathrm{,}{\overline{\rho }}_i^1\mathrm{,}{\overline{\theta }}_i^1\right)$ -invex at $u\in X^0$ with respect to ${\overline{H}}_i\left(u\mathrm{,}p\right)$ , we have

${\overline{\alpha }}_i^1\left(x,u\right)\left[\frac{f_i\left(x\right)+x^{\text{T}}{z}_i}{g_i\left(x\right)-x^{\text{T}}{v}_i}-\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right]$

$\begin{array}{l}\ge {\eta }^{\text{T}}\left(x,u\right)\left[\nabla \left(\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right)+{\nabla }_p{\overline{H}}_i\left(u,p\right)\right]\\ +{\overline{\beta }}_i^1\left(x,u\right)\left[{\overline{H}}_i\left(u,p\right)-p^{\text{T}}{\nabla }_p{\overline{H}}_i\left(u,p\right)\right]+{\overline{\rho }}_i^1{\Vert {\overline{\theta }}_i^1\left(x,u\right)\Vert }^2.\end{array}$ (16)

where

${\overline{\alpha }}_i\left(x,u\right)=\left(\frac{g_i\left(x\right)-x^{\text{T}}{v}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right){\alpha }_i\left(x,u\right)$ , ${\overline{\beta }}_i\left(x,u\right)={\beta }_i\left(x,u\right)$ ,

${\overline{\theta }}_i\left(x,u\right)={\theta }_i\left(x,u\right){\left(\frac{1}{g_i\left(u\right)-u^{\text{T}}{v}_i}+\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{{\left(g_i\left(u\right)-u^{\text{T}}{v}_i\right)}^2}\right)}^{\frac{1}{2}}$ , ${\overline{\rho }}_i\left(x,u\right)={\rho }_i\left(x,u\right)$

and ${\overline{H}}_i\left(u,p\right)=\left(\frac{1}{g_i\left(u\right)-u^{\text{T}}{v}_i}+\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{{\left(g_i\left(u\right)-u^{\text{T}}{v}_i\right)}^2}\right)H_i\left(u,p\right)$ .

By hypothesis 2), we get

$\begin{array}{l}{\alpha }_j^2\left(x,u\right)\left[h_j\left(x\right)+x^{\text{T}}{w}_j-\left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)\right]\\ \ge {\eta }^{\text{T}}\left(x,u\right)\left[\nabla \left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)+{\nabla }_p{G}_j\left(u,p\right)\right]\\ +{\beta }_j^2\left(x,u\right)\left[G_j\left(u,p\right)-p^{\text{T}}{\nabla }_p{G}_j\left(u,p\right)\right]+{\rho }_j^2{\Vert {\theta }_j^2\left(x,u\right)\Vert }^2.\end{array}$ (17)

Adding the two inequalities after multiplying (16) by ${\lambda }_i$ and (17) by ${\mu }_j$ , we obtain

$\begin{array}{l}\sum _{i=1}^k{\lambda }_i{\overline{\alpha }}_i^1\left(x,u\right)\left[\frac{f_i\left(x\right)+x^{\text{T}}{z}_i}{g_i\left(x\right)-x^{\text{T}}{v}_i}-\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right]\\ +\sum _{j=1}^m{\mu }_j{\alpha }_j^2\left(x,u\right)\left[h_j\left(x\right)+x^{\text{T}}{w}_j-\left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)\right]\\ \ge {\eta }^{\text{T}}\left(x,u\right)\sum _{i=1}^k{\lambda }_i\left[\nabla \left(\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right)+{\nabla }_p{\overline{H}}_i\left(u,p\right)\right]\end{array}$

$\begin{array}{l}+{\eta }^{\text{T}}\left(x,u\right)\sum _{j=1}^m{\mu }_j\left[\nabla \left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)+{\nabla }_p{G}_j\left(u,p\right)\right]\\ +\sum _{i=1}^k{\lambda }_i{\overline{\beta }}_i\left(x,u\right)\left[{\overline{H}}_i\left(u,p\right)-p^{\text{T}}{\nabla }_p{\overline{H}}_i\left(u,p\right)\right]\\ +\sum _{j=1}^m{\mu }_j{\beta }_j^2\left(x,u\right)\left[G_j\left(u,p\right)-p^{\text{T}}{\nabla }_p{G}_j\left(u,p\right)\right]\\ +\sum _{i=1}^k{\lambda }_i{\overline{\rho }}_i^1{\Vert {\overline{\theta }}_i^1\left(x,u\right)\Vert }^2+\sum _{j=1}^m{\mu }_j{\rho }_j^2{\Vert {\theta }_j^2\left(x,u\right)\Vert }^2.\end{array}$ (18)

Using hypothesis 3)-4), we get

$\begin{array}{l}\sum _{i=1}^k{\lambda }_i\left[\frac{f_i\left(x\right)+x^{\text{T}}{z}_i}{g_i\left(x\right)-x^{\text{T}}{v}_i}-\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right]+\sum _{j=1}^m{\mu }_j\left[h_j\left(x\right)+x^{\text{T}}{w}_j-\left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)\right]\\ \ge {\eta }^{\text{T}}\left(x,u\right)\sum _{i=1}^k{\lambda }_i\nabla \left(\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right)+{\eta }^{\text{T}}\left(x,u\right)\sum _{j=1}^m{\mu }_j\nabla \left(h_j\left(u\right)+u^{\text{T}}{w}_j\right).\end{array}$ (19)

Further, using (7)-(8), therefore

$\sum _{i=1}^k{\lambda }_i\left[\frac{f_i\left(x\right)+x^{\text{T}}{z}_i}{g_i\left(x\right)-x^{\text{T}}{v}_i}-\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right]+\sum _{j=1}^m{\mu }_j\left[h_j\left(x\right)+x^{\text{T}}{w}_j\right]\ge 0.$ (20)

Since x is feasible solution for (MFP), it follows that

$\sum _{i=1}^k{\lambda }_i\left(\frac{f_i\left(x\right)+x^{\text{T}}{z}_i}{g_i\left(x\right)-x^{\text{T}}{v}_i}\right)\ge \sum _{i=1}^k{\lambda }_i\left(\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right).$

This contradicts (15). Therefore, u is an efficient solution of (MFP).

4. Duality Model-I

Consider the following dual (MFD)₁ of (MFP): (MFD)₁ Maximize

$\begin{array}{l}[\frac{f_1\left(u\right)+u^{\text{T}}{z}_1}{g_1\left(u\right)-u^{\text{T}}{v}_1}+\sum _{j=1}^m{\mu }_j\left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)+\left({\overline{H}}_1\left(u,p\right)-p^{\text{T}}{\nabla }_p{\overline{H}}_1\left(u,p\right)\right)\\ +\sum _{j=1}^m{\mu }_j\left(G_j\left(u,p\right)-p^{\text{T}}{\nabla }_p{G}_j\left(u,p\right)\right),\cdots ,\\ \frac{f_k\left(u\right)+u^{\text{T}}{z}_k}{g_k\left(u\right)-u^{\text{T}}{v}_k}+\sum _{j=1}^m{\mu }_j\left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)+\left({\overline{H}}_k\left(u,p\right)-p^{\text{T}}{\nabla }_p{\overline{H}}_k\left(u,p\right)\right)\\ +\sum _{j=1}^m{\mu }_j\left(G_j\left(u,p\right)-p^{\text{T}}{\nabla }_p{G}_j\left(u,p\right)\right)]\end{array}$

subject to

$\begin{array}{l}\sum _{i=1}^k{\lambda }_i\nabla \left(\frac{f_i\left(u\right)+u^{\text{T}}{z}_i}{g_i\left(u\right)-u^{\text{T}}{v}_i}\right)+\sum _{j=1}^m{\mu }_j\nabla \left(h_j\left(u\right)+u^{\text{T}}{w}_j\right)\\ +\sum _{i=1}^k{\lambda }_i{\nabla }_p{\overline{H}}_i\left(u,p\right)+\sum _{j=1}^m{\mu }_j{\nabla }_p{G}_j\left(u,p\right)=0,\end{array}$ (21)

$z_i\in C_i,\mathrm{<mi>\; </mi>}{v}_i\in D_i,\mathrm{<mi>\; </mi>}{w}_j\in E_j,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m,$

${\mu }_j\ge 0,\mathrm{<mi>\; </mi>}{\lambda }_i>0,\mathrm{<mi>\; </mi>}\sum _{i=1}^k{\lambda }_i=1,\mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m.$

Let $Z^0$ be feasible solution for (MFD)₁.

Theorem 4.1. (Weak duality theorem). Let $x\in X^0$ and $\left(u\mathrm{,}z\mathrm{,}v\mathrm{,}\mu \mathrm{,}\lambda \mathrm{,}w\mathrm{,}p\right)\in Z^0$ . Suppose that

1) for any $i=1,2,\cdots ,k$ , $\left(f_i\left(\mathrm{.}\right)+{\left(\mathrm{.}\right)}^{\text{T}}{z}_i\right)$ and $-\left(g_i(.)-{(.)}^{\text{T}}{v}_i\right)$ are higher- order $\left(V\mathrm{,}{\alpha }_i^1\mathrm{,}{\beta }_i^1\mathrm{,}{\rho }_i^1\mathrm{,}{\theta }_i^1\right)$ -invex at u with respect to $H_i\left(u\mathrm{,}p\right)$ ,

2) for any $j=1,2,\cdots ,m$ , $\left(h_j(.)+{(.)}^{\text{T}}{w}_j\right)$ is higher-order $\left(V\mathrm{,}{\alpha }_j^2\mathrm{,}{\beta }_j^2\mathrm{,}{\rho }_j^2\mathrm{,}{\theta }_j^2\right)$ -invex at u with respect to $G_j\left(u\mathrm{,}p\right)$ ,

3) $\sum _{i=1}^k{\lambda }_i{\overline{\rho }}_i^1{\Vert {\overline{\theta }}_i^1\left(x\mathrm{,}u\right)\Vert }^2+\sum _{j=1}^m{\mu }_j{\rho }_j^2{\Vert {\theta }_j^2\left(x\mathrm{,}u\right)\Vert }^2\ge 0.$

4) ${\overline{\alpha }}_i^1\left(x,u\right)={\alpha }_j^2\left(x,u\right)={\beta }_i^1\left(x,u\right)={\beta }_j^2\left(x,u\right)=\alpha \left(x,u\right),\mathrm{<mi>\; </mi>}\forall \mathrm{<mi>\; </mi>}i=1,2,\cdots ,k,$ $\mathrm{<mi>\; </mi>}j=1,2,\cdots ,m,$