1 Introduction

It is an undeniable fact that all of us are optimizers as we all make decisions for the sole purpose of maximizing our quality of life, productivity in time, and our welfare in some way or another. Since this is an ongoing struggle for creating the best possible among many inferior designs and is always the core requirement of human life, this fact yields the development of a massive number of techniques in this area, starting from the early ages of civilization until now. The efforts and lives behind this aim dedicated by many brilliant philosophers, mathematicians, scientists, and engineers have brought a high level of civilization we enjoy today. The decision process is relatively easier when there is a single criterion or object in mind. The process gets complicated when we have to make decisions in the presence of more than one criteria to judge the decisions. In such circumstances a single decision that optimizes all the criteria simultaneously may not exist. For handling such type of situations, we use multiobjective programming, also known as multiattribute optimization, which is the process of simultaneously optimizing two or more conflicting objectives subject to certain constraints. Multiobjective optimization problems can be found in various fields such as product and process design, finance, aircraft design, the oil and gas industry, automobile design, and other where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives.

Mangasarian [1] was the first who introduced the concept of second-order duality for nonlinear programming. Gulati and Gupta [2] introduced the concept of \(\eta_{1}\)-bonvexity/\(\eta _{2}\)-boncavity and derived duality results for a Wolfe-type model. The concept of G-invex function is given by Antczak [3] and derived some duality results for a constrained optimization problem. Later on, generalizing his earlier work, Antzcak [4] introduced \(G_{f}\)-invex functions for multivariate models and obtained optimality results for multiobjective programming problems. Liang et al. [5] discussed conditions for optimality and duality in a multiobjective programming problem. Bhatia and Garg [6] discussed the concept of \((V, p)\) invexity for nonsmooth vector functions and established duality results for multiobjective programs. Jayswal et al. [7] discussed multiobjective fractional programming problem involving an invex function. Stefaneseu and Ferrara [8] studied new invexities for multiobjective programming problem. Several researchers [9–21] have studied related areas.

This paper is organized as follows. In Sect. 2, we give some preliminaries and definitions used in this paper and also a nontrivial example for such type functions. In Sect. 3, we formulate second-order multiobjective symmetric G-Wolfe-type dual programs over arbitrary cones. We prove weak, strong, and converse duality theorems by using \(G_{f}\)-bonvexity/\(G_{f}\)-pseudobonvexity assumptions over arbitrary cones. Finally, we construct nontrivial numerical examples that are \(G_{f}\)-bonvex/\(G_{f}\)-pseudobonvex but neither η-bonvex/η-pseudobonvex nor η-invex/η-pseudoinvex functions.

2 Preliminaries and definitions

Let \(f = ( f_{1},f_{2},f_{3},\ldots, f_{k} ) : X\rightarrow R^{k}\) be a vector-valued differentiable function defined on a nonempty open set \(X\subseteq R^{n}\), and let \(I_{f_{i}} (X)\), \(i = 1,\ldots,k\), be the range of \(f_{i}\), that is, the image of X under \(f_{i}\). Let \(G_{f}=(G_{f_{1}},G_{f_{2}},\ldots,G_{f_{k}}):R\rightarrow R^{k}\) be a differentiable function such that every component \(G_{f_{i}}: I_{f_{i}}(X)\rightarrow R\) is strictly increasing on the range of \(I_{f_{i}}\), \(i=1,2,3,\ldots,k\).

Definition 2.1

The positive polar cone \(S^{*}\)of a cone \(S\subseteq R^{s}\) is defined by

$$S^{*}= \bigl\{ y\in R^{s}:x^{T}y\geq0 \bigr\} . $$

Consider the following vector minimization problem:

$$ \begin{aligned}[b] &\mbox{Min.}\quad f(y)= \bigl\{ f_{1}(y),f_{2}(y),\ldots,f_{k}(y) \bigr\} ^{T} \\ &\quad\mbox{subject to}\quad S^{0} = \bigl\{ y\in S \subset R^{n} : h_{j}(y)\leq0, j=1,2,3,\ldots,m \bigr\} , \end{aligned} $$
(MP)

where \(f=\{f_{1},f_{2},\ldots,f_{k}\}:S \rightarrow R^{k}\) and \(h=\{ h_{1},h_{2},\ldots,h_{m}\}:S \rightarrow R^{m}\) are differentiable functions on S.

Definition 2.2

\(\bar{y}\in S^{0}\) is an efficient solution of (MP) if there exists no other \(y\in S^{0}\) such that \(f_{r}(y)< f_{r}(\bar {y})\) for some \(r=1,2,3,\ldots,k\) and \(f_{i}(y)\leq f_{i}(\bar{y}) \) for all \(i=1,2,3,\ldots,k\).

Definition 2.3

If there exists a function \(\eta:S\times S \rightarrow R^{n}\) such that for all \(y\in S\),

$$f_{i}(y)-f_{i}(v)\geq\eta^{T}(y,v) \nabla_{y}f_{i}(v) \quad\mbox{for all } i=1,2,3,\ldots,k, $$

then f is called invex at \(v\in S\) with respect to η.

Definition 2.4

If there exist \(G_{f_{i}}: I_{f_{i}}(S)\rightarrow R\) and \(\eta: S\times S \rightarrow R^{n}\) such that for all \(y\in S\),

$$\begin{aligned} \eta^{T}(y,v)G_{f_{i}}^{\prime} \bigl(f_{i}(v) \bigr)\nabla_{y}f_{i}(v)\geq 0\quad\Rightarrow\quad G_{f_{i}} \bigl(f_{i}(x) \bigr)-G_{f_{i}} \bigl(f_{i}(v) \bigr) \geq0 \quad\mbox{for all } i=1,2,3,\ldots,k, \end{aligned}$$

then \(f_{i}\) is called \(G_{f_{i}}\)-pseudoinvex at \(u\in S\) with respect to η.

Definition 2.5

If there exist \(G_{f_{i}}: I_{f_{i}}(S)\rightarrow R\) and \(\eta: S\times S\rightarrow R^{n}\) such that for all \(y\in S\) and \(p\in R^{n}\),

$$\begin{aligned}G_{f_{i}}\bigl(f_{i}(y)\bigr)-G_{f_{i}} \bigl(f_{i}(v)\bigr)&\geq\eta ^{T}(y,v)\bigl[G_{f_{i}}^{\prime} \bigl(f_{i}(v)\bigr)\nabla_{y}f_{i}(v) \\ &\quad+ \bigl\{ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(v)\bigr) \nabla_{y}f_{i}(v) \bigl(\nabla _{y}f_{i}(v) \bigr)^{T} + G_{f_{i}}^{\prime}\bigl(f_{i}(v) \bigr)\nabla_{yy}f_{i}(v)\bigr\} p\bigr] \\ &\quad-\frac{1}{2}p^{T} \bigl[G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v) \bigr)\nabla _{y}f_{i}(v) \bigl( \nabla_{y}f_{i}(v) \bigr)^{T} +G_{f_{i}}^{\prime} \bigl(f_{i}(v) \bigr)\nabla _{yy}f_{i}(v) \bigr]p \\ &\quad\mbox{for all } i=1,2,3,\ldots,k,\end{aligned} $$

then \(f_{i}\) is called \(G_{f_{i}}\)-bonvex at \(v\in S\) with respect to η.

Definition 2.6

If there exist functions \(G_{f}\) and \(\eta: S\times S\rightarrow R^{n}\) such that for all \(y\in S\) and \(p\in R^{n}\),

$$\begin{gathered} \eta^{T}(y,v)\bigl[G_{f_{i}}^{\prime} \bigl(f_{i}(v)\bigr)\nabla_{y}f_{i}(v) \\ \quad{}+ \bigl\{ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(v)\bigr) \nabla_{y}f_{i}(v) \bigl(\nabla _{y}f_{i}(v) \bigr)^{T} + G_{f_{i}}^{\prime}\bigl(f_{i}(v) \bigr)\nabla_{yy}f_{i}(v)\bigr\} p\bigr]\geq0 \\ \quad\Rightarrow\quad G_{f_{i}} \bigl(f_{i}(y) \bigr)-G_{f_{i}} \bigl(f_{i}(v) \bigr) \\ \phantom{\quad\Rightarrow\quad}\qquad{}+ \frac {1}{2}p^{T} \bigl[G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v) \bigr) \nabla_{y}f_{i}(v) \bigl( \nabla _{y}f_{i}(v) \bigr)^{T} +G_{f_{i}}^{\prime} \bigl(f_{i}(v) \bigr) \nabla_{yy}f_{i}(v) \bigr]p\geq 0 \\ \phantom{\quad\Rightarrow\quad}\quad \quad\mbox{for all } i=1,2,3,\ldots,k,\end{gathered} $$

then \(f_{i}\) is called \(G_{f_{i}}\)-pseudobonvex at \(v\in S\) with respect to η.

Now, we discuss nontrivial numerical examples that are \(G_{f}\)-bonvex/\(G_{f}\)-pseudobonvex but neither η-bonvex/η-pseudobonvex nor η-invex/η-pseudoinvex functions.

Example 2.1

Let \(f:[-1,1]\rightarrow R^{4}\) be defined as

$$f(y)= \bigl\{ f_{1}(y), f_{2}(y),f_{3}(y),f_{4}(y) \bigr\} , $$

where \(f_{1}(y)=y^{10}\), \(f_{2}(y)=\arcsin y\), \(f_{3}(y)=\arctan y\), \(f_{4}(y)=\operatorname{arccot} y\), and let \(G_{f}= \{G_{f_{1}}, G_{f_{2}}, G_{f_{3}},G_{f_{4}}\}: R\rightarrow R^{4}\) be defined as

$$G_{f_{1}}(t)= t^{5}+5,\qquad G_{f_{2}}(t)=\sin t+2, \qquad G_{f_{3}}(t)= \tan t+9,\qquad G_{f_{4}}(t)= \cot t+2. $$

Let \(\eta:[-1,1]\times[-1,1]\rightarrow R\) be given as

$$\eta(y,v)= -\frac{1}{9}y^{14}+y+\frac{1}{99}y^{17}v^{5}- \frac{1}{7}y^{4}v^{3}+v^{3}. $$

To show that f is \(G_{f}\)-bonvex at \(v = 0\) with respect to η, we have to claim that

$$\begin{aligned} \pi_{i} &= G_{f_{i}}\bigl(f_{i}(y) \bigr)-G_{f_{i}}\bigl(f_{i}(v)\bigr) \\ &\quad-\eta ^{T}(y,v)\bigl[G_{f_{i}}^{\prime} \bigl(f_{i}(v)\bigr)\nabla_{y}f_{i}(v)+ \bigl\{ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(v)\bigr) \nabla_{y}f_{i}(v) \bigl(\nabla_{y}f_{i}(v) \bigr)^{T} + G_{f_{i}}^{\prime}\bigl(f_{i}(v) \bigr)\nabla _{yy}f_{i}(v)\bigr\} p\bigr] \\ &\quad+ \frac{1}{2}p^{T}\bigl[G_{f_{i}}^{\prime\prime } \bigl(f_{i}(v)\bigr)\nabla_{y}f_{i}(v) \bigl( \nabla_{y}f_{i}(v)\bigr)^{T} +G_{f_{i}}^{\prime } \bigl(f_{i}(v)\bigr)\nabla_{yy}f_{i}(v)\bigr]p \\ &\geq0,\quad i=1,2,3,4.\end{aligned} $$

Putting the values of \(f_{i}\), \(G_{f_{i}}\), \(i=1,2,3,4\), into the last expression, after simplifying at the point \(v=0\in[-1,1]\), we clearly see from Fig. 1 that \(\pi_{i} \geq0\), \(i=1,2,3, 4\), for all \(y\in[-1,1]\). Therefore f is \(G_{f}\)-bonvex at \(v=0\in[-1,1]\) with respect to η and p.

Figure 1
figure 1

\(\pi_{i}\geq0\) (\(i=1,2,3,4\)), \(\forall p, \forall y\in [-1,1]\)

Full size image

Now, suppose

$$\xi=f_{3}(y)-f_{3}(v)-\eta^{T}(y,v) \bigl[ \nabla_{y}f_{3}(v) - \nabla _{yy}f_{3}(v)p \bigr] + \frac{1}{2}p^{T} \bigl[\nabla_{yy}f_{3}(v) \bigr]p $$

or

$$\begin{aligned}& \xi= \arctan y-\arctan v \\& \phantom{\xi=}{}- \biggl( -\frac {1}{9}y^{14}+y+ \frac{1}{99}y^{17}v^{5}-\frac{1}{7}y^{4}v^{3}+v^{3} \biggr) \biggl[\frac{1}{1+v^{2}} -\frac{2vp}{(1+v^{2})^{2}} \biggr]-\frac{vp^{2}}{(1+v^{2})^{2}}, \\& \xi= \arctan y+\frac{1}{9}y^{14}-y \quad\mbox{at } v=0, \\& \xi\ngeq0 \quad\mbox{(from Fig. 2)}. \end{aligned}$$

Therefore \(f_{3}\) is not η-bonvex at \(v=0\) with respect to p. Hence f is not η-bonvex at \(v=0\) with respect to p.

Figure 2
figure 2

\(\xi= \arctan y+\frac{1}{9}y^{14}-y\ngeq0\), at \(v=0\), \(\forall p, \forall y\in[-1,1]\)

Full size image

Next,

$$\delta=f_{3}(y)-f_{3}(v) - \eta^{T}(y,v) \nabla_{y}f_{3}(v) $$

or

$$\begin{aligned}& \delta= \arctan y-\arctan v- \biggl( -\frac {1}{9}y^{14}+y+ \frac{1}{99}y^{17}v^{5}-\frac{1}{7}y^{4}v^{3}+v^{3} \biggr) \frac {1}{1+v^{2}}, \\& \delta= \arctan y+\frac{1}{9}y^{14}-y \quad\mbox{at } v=0, \\& \delta= \frac{\pi}{4}+\frac{1}{9}-1 < 0 \quad\mbox{at } y=1\in[-1,1]. \end{aligned}$$

Therefore \(f_{3}\) is not η-invex at \(v=0\). Hence f is not η-invex at \(v=0\).

Example 2.2

Let \(f:[-2,2]\rightarrow R^{2}\) be defined as

$$f(y)= \bigl\{ f_{1}(y), f_{2}(y) \bigr\} , $$

where \(f_{1}(y)= (\frac{e^{2y}-1}{e^{y}} )\), \(f_{2}(y)= y^{3}\), and \(G_{f}= \{G_{f_{1}}, G_{f_{2}}\}: R\rightarrow R^{2}\) is defined as

$$G_{f_{1}}(t)= t^{2}+1,\qquad G_{f_{2}}(t)= t^{2}+3. $$

Let \(\eta:[-2,2]\times[-2,2]\rightarrow R\) be given as

$$\eta(y,v)= y^{6}+v^{9}y^{4}+ v^{5} y+v+3. $$

To show that f is \(G_{f}\)-pseudobonvex at \(v = 0\) with respect to η, we have to claim that, for \(i=1,2\),

$$\begin{gathered} \zeta_{i} = \eta^{T}(y,v) \bigl[G_{f_{i}}^{\prime}\bigl(f_{i}(v)\bigr)\nabla _{y}f_{i}(v)+ \bigl\{ G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v)\bigr)\nabla _{y}f_{i}(v) \bigl( \nabla_{y}f_{i}(v)\bigr)^{T}+ G_{f_{i}}^{\prime} \bigl(f_{i}(v)\bigr)\nabla _{yy}f_{i}(v)\bigr\} p \bigr]\geq0 \\ \quad\Rightarrow\quad G_{f_{i}} \bigl(f_{i}(y) \bigr)-G_{f_{i}} \bigl(f_{i}(v) \bigr) \\ \phantom{\quad\Rightarrow\quad}\quad{}+ \frac {1}{2}p^{T} \bigl[G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v) \bigr) \nabla_{y}f_{i}(v) \bigl( \nabla _{y}f_{i}(v) \bigr)^{T} +G_{f_{i}}^{\prime} \bigl(f_{i}(v) \bigr) \nabla_{yy}f_{i}(v) \bigr]p\geq0.\end{gathered} $$

Let

$$\begin{aligned} \phi_{1}=\eta^{T}(y,v) \bigl[G_{f_{1}}^{\prime} \bigl(f_{1}(v) \bigr)\nabla _{y}f_{1}(v)+ \bigl\{ G_{f_{1}}^{\prime\prime} \bigl(f_{1}(v) \bigr)\nabla _{y}f_{1}(v) \bigl(\nabla_{y}f_{1}(v) \bigr)^{T}+ G_{f_{1}}^{\prime} \bigl(f_{1}(v) \bigr)\nabla _{xx}f_{1}(u) \bigr\} p \bigr]. \end{aligned}$$

Substituting the values of η and \(f_{1}\) at the point \(v=0\), we get

$$\begin{aligned} \phi_{1}\geq0\quad\mbox{for all } y\in[-2,2] \mbox{ and } p. \end{aligned}$$

Next, consider

$$\begin{aligned} \varphi_{1}=G_{f_{1}} \bigl(f_{1}(y) \bigr)-G_{f_{1}} \bigl(f_{1}(v) \bigr)+ \frac {1}{2}p^{T} \bigl[G_{f_{1}}^{\prime\prime} \bigl(f_{1}(v) \bigr) \nabla_{y}f_{1}(v) \bigl(\nabla _{y}f_{1}(v) \bigr)^{T} +G_{f_{1}}^{\prime} \bigl(f_{1}(v) \bigr)\nabla_{xx}f_{1}(u) \bigr]p_{1}. \end{aligned}$$

At \(v=0\), we get \(\varphi_{1}\geq0\) for all \(y\in[-1,1]\) and p (from Fig. 3);

$$\begin{gathered} \phi_{2}= \eta^{T}(y,v) \bigl[G_{f_{2}}^{\prime} \bigl(f_{2}(v) \bigr) \nabla_{y}f_{2}(v)+ \bigl\{ G_{f_{2}}^{\prime\prime} \bigl(f_{2}(v) \bigr) \nabla_{y}f_{2}(v) \bigl( \nabla _{y}f_{2}(v) \bigr)^{T}+ G_{f_{2}}^{\prime} \bigl(f_{2}(v) \bigr) \nabla_{yy}f_{2}(v) \bigr\} p \bigr], \\ \phi_{2}= \bigl( y^{6}+v^{9}y^{4}+ v^{5} y+v+3 \bigr) \bigl(6v^{5}+30v^{5}p \bigr). \end{gathered} $$

At the point \(v=0\), we have

$$\begin{aligned} \phi_{2}\geq0 \quad\mbox{for all } y\in[-2,2] \mbox{ and } p. \end{aligned}$$

Also,

$$ \begin{gathered} \varphi_{2}= G_{f_{2}} \bigl(f_{2}(y) \bigr)-G_{f_{2}} \bigl(f_{2}(v) \bigr)+ \frac {1}{2}p^{T} \bigl[G_{f_{2}}^{\prime\prime} \bigl(f_{2}(v) \bigr) \nabla_{y}f_{2}(v) \bigl(\nabla _{y}f_{2}(v) \bigr)^{T} +G_{f_{2}}^{\prime} \bigl(f_{2}(v) \bigr)\nabla_{yy}f_{2}(v) \bigr]p, \\ \varphi_{2}= y^{6}-v^{6}+ 15p^{2} v^{4}.\end{gathered} $$

At the point \(v=0\), we obtain

$$\begin{aligned} \varphi_{2}\geq0 \quad\mbox{for all } y\in[-2,2] \mbox{ and } p. \end{aligned}$$

Hence from the expressions \(\phi_{i}\) and \(\varphi_{i}\), \(i=1, 2\), we get that f is \(G_{f}\)-pseudobonvex at \(v=0\) with respect to η.

Figure 3
figure 3

\(\varphi_{1}= (\frac{e^{2y}-1}{e^{y}} )^{2}\geq0\) for all \(y\in[-2,2]\) and p

Full size image

Next, let

$$\begin{gathered} \phi_{3}= \eta^{T}(y,v) \bigl[ \nabla_{y}f_{2}(v)+ \nabla_{yy}f_{2}(v)p \bigr], \\ \phi_{3}= \bigl( y^{6}+v^{9}y^{4}+ v^{5} y+v+3 \bigr) \bigl[3v^{2}+6vp \bigr].\end{gathered} $$

At the point \(v=0\), we have

$$\begin{aligned} \phi_{3}\geq0 \quad\mbox{for all } y\in[-2,2] \mbox{ and } p. \end{aligned}$$

Further, consider

$$ \begin{gathered} \varphi_{3}=f_{2}(y)-f_{2}(v)+ \frac{1}{2}p^{2}\nabla_{yy}f_{2}(v), \\ \varphi_{3}=y^{3}-v^{3}+3p^{2}v. \end{gathered} $$

At the point \(v=0\), we obtain

$$\begin{aligned} \varphi_{3}\ngeq0 \quad\mbox{for all } y\in[-2,2] \mbox{ and } p \mbox{ (from Fig. 4)}. \end{aligned}$$

Hence \(f_{2}\) is not η-pseudobonvex at \(v=0\in[-2,2]\). Therefore \(f=(f_{1}, f_{2})\) is not η-pseudobonvex at \(v=0\in[-2,2]\).

Figure 4
figure 4

\(\varphi_{3}\geq0\) for all \(y\in[-2,2]\) and p

Full size image

Finally,

$$\begin{gathered} \phi_{4}= \eta^{T}(y,v) \nabla_{y}f_{2}(v), \\ \phi_{4}= 3 \bigl( y^{6}+v^{9}y^{4}+ v^{5} y+v+3 \bigr)v^{2}.\end{gathered} $$

At the point \(v=0\), we have

$$\begin{aligned} \phi_{4}\geq0 \quad\mbox{for all } y\in[-2,2] \mbox{ and } p. \end{aligned}$$

Also,

$$ \begin{gathered} \varphi_{4}=f_{2}(y)-f_{2}(v), \\ \varphi_{3}=y^{3}-v^{3}.\end{gathered} $$

At the point \(v=0\), we obtain

$$\begin{aligned} \varphi_{4}\ngeq0 \quad\mbox{for all } y\in[-2,2]. \end{aligned}$$

Hence \(f_{2}\) is not η-pseudoinvex at \(v=0\in[-2,2]\). Hence \(f=(f_{1}, f_{2})\) is not η-pseudoinvex at \(v=0\in[-2,2]\).

3 Second-order multiobjective G-Wolfe-type symmetric dual program

Consider the following pair of second-order multiobjective G-Wolfe-type dual programs over arbitrary cones.

Primal problem

(GWP)

Minimize

$$R(y,z,\lambda,p) = \bigl(R_{1}(y,z,\lambda _{1},p),R_{2}(y,z, \lambda_{2},p),\ldots, R_{k}(y,z,\lambda_{k},p) \bigr)^{T} $$

subject to

$$\begin{aligned}& \begin{aligned}[b]&{-} \sum_{i=1}^{k} \lambda_{i} \bigl[G_{f_{i}}^{\prime}\bigl(f_{i}(y,z) \bigr)\nabla _{y}f_{i}(y,z) \\ &\quad+ \bigl\{ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(y,z)\bigr) \nabla_{y}f_{i}(y,z) \bigl(\nabla_{y}f_{i}(y,z) \bigr)^{T}+ G_{f_{i}}^{\prime} \bigl(f_{i}(y,z) \bigr)\nabla _{yy}f_{i}(y,z)\bigr\} p \bigr]\in C_{2}^{*}, \end{aligned} \end{aligned}$$
(1)
$$\begin{aligned}& \lambda_{i} > 0,\qquad \lambda^{T} e_{k}=1,\qquad x\in C_{1},\qquad i=1,2,3,\ldots,k. \end{aligned}$$
(2)

Dual problem

(GWD)

Maximize

$$S(v,w,\lambda,q) = \bigl(S_{1}(v,w,\lambda _{1},q),S_{2}(v,w, \lambda_{2},q),\ldots,S_{k}(v,w,\lambda_{k},q) \bigr)^{T} $$

subject to

$$\begin{aligned}& \begin{aligned}[b]&\sum_{i=1}^{k} \lambda_{i} \bigl[G_{f_{i}}^{\prime}\bigl(f_{i}(v,w) \bigr)\nabla _{z}f_{i}(v,w) \\ &\quad+ \bigl\{ G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v,w)\bigr)\nabla_{z}f_{i}(v,w) \bigl( \nabla_{z}f_{i}(v,w) \bigr)^{T}+ G_{f_{i}}^{\prime} \bigl(f_{i}(v,w) \bigr)\nabla _{zz}f_{i}(v,w))\bigr\} q \bigr]\in C_{1}^{*}, \end{aligned} \end{aligned}$$
(3)
$$\begin{aligned}& \ \lambda_{i} > 0,\qquad \lambda^{T} e_{k}=1, \qquad v\in C_{2},\qquad i=1,2,3,\ldots,k, \end{aligned}$$
(4)

where for all \(i=1,2,3,\ldots,k\),

$$\begin{aligned}& \begin{aligned}R_{i}(y,z,\lambda,p)&= G_{f_{i}} \bigl(f_{i}(y,z)\bigr)-z^{T} \sum _{i=1}^{k}\lambda _{i} \bigl(G_{f_{i}}^{\prime}\bigl(f_{i}(y,z)\bigr) \nabla_{y}f_{i}(y,z) \\ &\quad +\bigl[ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(y,z)\bigr) \nabla_{y}f_{i}(y,z) \bigl(\nabla _{y}f_{i}(y,z) \bigr)^{T}+ G_{f_{i}}^{\prime}\bigl(f_{i}(y,z) \bigr)\nabla _{yy}f_{i}(y,z)\bigr]p \bigr) \\ &\quad- \frac{1}{2} \sum_{i=1}^{k} \lambda_{i}p^{T} \bigl(G_{f_{i}}^{\prime\prime} \bigl(f_{i}(y,z)\bigr)\nabla_{y}f_{i}(y,z) \bigl( \nabla _{y}f_{i}(y,z)\bigr)^{T} + G_{f_{i}}^{\prime}\bigl(f_{i}(y,z)\bigr)\nabla _{yy}f_{i}(y,z) \bigr)p,\end{aligned} \\& \begin{aligned}S_{i}(v,w,\lambda,q)&= G_{f_{i}} \bigl(f_{i}(v,w)\bigr)-v^{T} \sum _{i=1}^{k}\lambda _{i} \bigl(G_{f_{i}}^{\prime}\bigl(f_{i}(v,w)\bigr)\nabla _{z}f_{i}(v,w) \\ &\quad+\bigl[G_{f_{i}}^{\prime\prime}\bigl(f_{i}(v,w)\bigr) \nabla _{z}f_{i}(v,w) \bigl(\nabla_{z}f_{i}(v,w) \bigr)^{T} + G_{f_{i}}^{\prime }\bigl(f_{i}(v,w) \bigr)\nabla_{zz}f_{i}(v,w)\bigr]q \bigr) \\ &\quad- \frac{1}{2} \sum_{i=1}^{k} \lambda_{i}q^{T} \bigl(G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v,w)\bigr) \nabla_{z}f_{i}(v,w) \bigl(\nabla_{z}f_{i}(v,w)\bigr)^{T} \\ &\quad+G_{f_{i}}^{\prime }\bigl(f_{i}(v,w)\bigr) \nabla_{zz}f_{i}(v,w) \bigr)q,\end{aligned} \end{aligned}$$

and

  1. (i)

    \(e_{k} =(1,1,\ldots,1)\in R^{k}\) and \(\lambda\in R^{k}\).

  2. (ii)

    q and p are vectors in \(R^{n}\) and \(R^{m}\), respectively.

Let \(Y^{0}\) and \(Z^{0}\) be the sets of feasible solutions of (GWP) and (GWD), respectively.

Theorem 3.1

(Weak duality)

Let\((y,z,\lambda,p)\in Y^{0}\)and\((v,w,\lambda,q)\in Z^{0}\). Suppose that for all\(i=1,2,3,\ldots,k\),

  1. (i)

    \(f_{i}(\cdot,v)\)is\(G_{f_{i}}\)-bonvex atvwith respectη,

  2. (ii)

    \(f_{i}(x,\cdot)\)be\(G_{f_{i}}\)-boncave atywith respectη,

  3. (iii)

    \(\eta_{1}(y,v)+u \in C_{1}\)and\(\eta_{2}(w,z) + y \in C_{2}\).

Then the following inequalities cannot hold together:

$$\begin{aligned} R_{i}(y, z, \lambda, p) \leq S_{i}(v,w, \lambda, q) \quad\textit{for all } i=1,2,3,\ldots,k, \end{aligned}$$
(5)

and

$$\begin{aligned} R_{r}(y, z, \lambda, p) < S_{r}(v,w, \lambda, q) \quad\textit{for at least one } r \in K. \end{aligned}$$
(6)

Proof

If possible, then suppose inequalities (5) and (6) hold. For \(\lambda> 0\), we obtain

$$ \begin{aligned}[b] &\sum_{i=1}^{k} \lambda_{i} \Biggl[G_{f_{i}}\bigl(f_{i}(y,z) \bigr)-z^{T} \sum_{i=1}^{k} \lambda_{i} \bigl(G_{f_{i}}^{\prime}\bigl(f_{i}(y,z) \bigr)\nabla _{y}f_{i}(y,z) \\ &\qquad+ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(y,z)\bigr) \nabla_{y}f_{i}(y,z) \bigl(\nabla_{y}f_{i}(y,z) \bigr)^{T}+ G_{f_{i}}^{\prime} \bigl(f_{i}(y,z) \bigr) \nabla_{yy}f_{i}(y,z)\bigr)p\Biggr] \\ &\qquad- \frac {1}{2} \sum_{i=1}^{k} \lambda_{i} \bigl(p^{T} \bigl[G_{f_{i}}^{\prime\prime } \bigl(f_{i}(y,z) \bigr)\nabla_{y}f_{i}(y,z) \bigl(\nabla _{y}f_{i}(y,z) \bigr)^{T}+G_{f_{i}}^{\prime} \bigl(f_{i}(y,z) \bigr)\nabla _{yy}f_{i}(y,z) \bigr]p \bigr) \\ &\quad< \sum_{i=1}^{k} \lambda_{i} \Biggl[G_{f_{i}}\bigl(f_{i}(v,w) \bigr)-v^{T} \sum_{i=1}^{k} \lambda_{i} \bigl(G_{f_{i}}^{\prime}\bigl(f_{i}(v,w) \bigr)\nabla _{z}f_{i}(v,w)+G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v,w)\bigr)\nabla _{z}f_{i}(v,w) \\ &\qquad+\bigl(\nabla_{z}f_{i}(v,w)\bigr)^{T}+ G_{f_{i}}^{\prime} \bigl(f_{i}(v,w) \bigr) \nabla_{zz}f_{i}(v,w) \bigr)q\Biggr] \\ &\qquad- \frac {1}{2} \sum_{i=1}^{k} \lambda_{i} \bigl(q^{T} \bigl[G_{f_{i}}^{\prime\prime } \bigl(f_{i}(v,w) \bigr)\nabla_{z}f_{i}(v,w) \bigl(\nabla _{z}f_{i}(v,w) \bigr)^{T}+G_{f_{i}}^{\prime} \bigl(f_{i}(v,w) \bigr)\nabla _{zz}f_{i}(v,w) \bigr]q \bigr) ].\hspace{-12pt}\end{aligned} $$
(7)

From assumption (i) we get

$$\begin{gathered} G_{f_{i}}\bigl(f_{i}(y,w)\bigr)-G_{f_{i}} \bigl(f_{i}(v,w)\bigr) \\ \quad \geq\eta _{1}{x,u}^{T}\bigl[G_{f_{i}}^{\prime} \bigl(f_{i}(v,w)\bigr)\nabla_{z}f_{i}(v,w) \\ \qquad{}+\bigl\{ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(v,w)\bigr) \nabla_{z}f_{i}(v,w) \bigl(\nabla _{z}f_{i}(v,w) \bigr)^{T}+G_{f_{i}}^{\prime}\bigl(f_{i}(v,w) \bigr)\nabla_{zz}f_{i}(v,w)\bigr\} q\bigr] \\ \qquad{}- \frac{1}{2}q^{T} \bigl[G_{f_{i}}^{\prime\prime } \bigl(f_{i}(v,w) \bigr) \nabla_{z}f_{i}(v,w) \bigl(\nabla _{z}f_{i}(v,w) \bigr)^{T}+G_{f_{i}}^{\prime} \bigl(f_{i}(v,w) \bigr)\nabla_{zz}f_{i}(v,w) \bigr]q.\end{gathered} $$

Since \(\lambda> 0\), this inequality yields

$$ \begin{aligned}[b] & \sum_{i=1}^{k} \lambda_{i} \bigl[G_{f_{i}}\bigl(f_{i}(y,w) \bigr)-G_{f_{i}}\bigl(f_{i}(v,w)\bigr) \bigr] \\ &\quad\geq\eta _{1}^{T}(x,u) \Biggl\{ \sum _{i=1}^{k}\lambda_{i} \bigl[G_{f_{i}}^{\prime }\bigl(f_{i}(v,w)\bigr) \nabla_{z}f_{i}(v,w) \\ &\qquad+ \bigl\{ G_{f_{i}}^{\prime\prime }\bigl(f_{i}(v,w)\bigr) \nabla_{z}f_{i}(v,w) \bigl(\nabla_{z}f_{i}(v,w) \bigr)^{T}+ G_{f_{i}}^{\prime} \bigl(f_{i}(v,w) \bigr) \nabla_{zz}f_{i}(v,w)\bigr\} q \bigr] \Biggr\} \\ &\qquad - \frac{1}{2} \sum_{i=1}^{k} \lambda_{i} q^{T} \bigl[G_{f_{i}}^{\prime\prime } \bigl(f_{i}(v,w) \bigr)\nabla_{z}f_{i}(v,w) \bigl(\nabla _{z}f_{i}(v,w) \bigr)^{T}+G_{f_{i}}^{\prime} \bigl(f_{i}(v,w) \bigr)\nabla_{zz}f_{i}(v,w) \bigr]q.\end{aligned} $$
(8)

From the dual constraint (3) and assumption (iii) it follows that

$$\begin{aligned}& \bigl[\eta_{1}(y,v)+v \bigr]^{T} \Biggl\{ \sum _{i=1}^{k}\lambda_{i} \bigl[G_{f_{i}}^{\prime } \bigl(f_{i}(v,w) \bigr) \nabla_{z}f_{i}(v,w) \\ & \quad{}+ \bigl\{ G_{f_{i}}^{\prime\prime } \bigl(f_{i}(v,w) \bigr)\nabla_{z}f_{i}(v,w) \bigl( \nabla_{z}f_{i}(v,w) \bigr)^{T}+ G_{f_{i}}^{\prime} \bigl(f_{i}(v,w) \bigr) \nabla_{zz}f_{i}(v,w) \bigr\} q \bigr] \Biggr\} \geq0, \end{aligned}$$

which implies

$$\begin{gathered}\eta_{1}(y,v)^{T} \Biggl\{ \sum _{i=1}^{k}\lambda_{i} \bigl[G_{f_{i}}^{\prime }\bigl(f_{i}(v,w)\bigr) \nabla_{z}f_{i}(v,w) \\ \qquad{}+ \bigl\{ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(v,w) \bigr)\nabla_{z}f_{i}(v,w) \bigl(\nabla _{z}f_{i}(v,w) \bigr)^{T}+ G_{f_{i}}^{\prime}\bigl(f_{i}(v,w) \bigr)\nabla_{zz}f_{i}(v,w)\bigr\} q \bigr] \Biggr\} \\ \quad\geq-v^{T} \Biggl\{ \sum_{i=1}^{k} \lambda_{i} \bigl[G_{f_{i}}^{\prime}\bigl(f_{i}(v,w) \bigr)\nabla_{z}f_{i}(v,w) \\ \qquad{}+ \bigl\{ G_{f_{i}}^{\prime\prime}\bigl(f_{i}(v,w)\bigr) \nabla_{z}f_{i}(v,w) \bigl(\nabla _{z}f_{i}(v,w) \bigr)^{T}+ G_{f_{i}}^{\prime}\bigl(f_{i}(v,w) \bigr)\nabla_{zz}f_{i}(v,w)\bigr\} q \bigr] \Biggr\} . \end{gathered} $$

Using inequalities (3) and (8), we obtain

$$\begin{aligned} & \sum_{i=1}^{k} \lambda_{i} \Biggl[G_{f_{i}}\bigl(f_{i}(y,w) \bigr)-G_{f_{i}}\bigl(f_{i}(v,w)\bigr) \\ &\qquad+ \frac{1}{2} \sum_{i=1}^{k} \lambda_{i} q^{T}\bigl[G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v,w)\bigr)\nabla _{z}f_{i}(v,w) \bigl(\nabla_{z}f_{i}(v,w)\bigr)^{T} +G_{f_{i}}^{\prime}\bigl(f_{i}(v,w)\bigr) \nabla_{zz}f_{i}(v,w)\bigr]q \Biggr] \\ &\quad \geq-v^{T} \Biggl\{ \sum _{i=1}^{k} \lambda_{i} \bigl[G_{f_{i}}^{\prime } \bigl(f_{i}(v,w) \bigr) \nabla_{z}f_{i}(v,w) \\ &\qquad+ \bigl\{ G_{f_{i}}^{\prime\prime } \bigl(f_{i}(v,w) \bigr)\nabla_{z}f_{i}(v,w) \bigl( \nabla _{z}f_{i}(v,w) \bigr)^{T}+G_{f_{i}}^{\prime} \bigl(f_{i}(v,w) \bigr)\nabla_{zz}f_{i}(v,w) \bigr\} q \bigr] \Biggr\} . \end{aligned}$$
(9)

Using assumption (iv) and primal constraint (1), we get

$$ \begin{aligned}[b] &\sum_{i=1}^{k} \lambda_{i} \biggl[-G_{f_{i}}\bigl(f(y,w)\bigr) + G_{f_{i}}\bigl(f_{i}(y,z)\bigr) \\ &\qquad-\frac{1}{2}p^{T}\bigl[G_{f_{i}}^{\prime\prime } \bigl(f_{i}(y,z)\bigr)\nabla_{y}f_{i}(y,z) \bigl( \nabla_{y}f_{i}(y,z)\bigr)^{T} +G_{f_{i}}^{\prime}\bigl(f_{i}(y,z)\bigr) \nabla_{yy}f_{i}(y,z)\bigr]p \biggr] \\ &\quad\geq z^{T} \sum_{i=1}^{k} \lambda_{i} \bigl[G_{f_{i}}^{\prime } \bigl(f_{i}(y,z) \bigr)\nabla_{y}f_{i}(y,z) \\ &\qquad+ \bigl\{ G_{f_{i}}^{\prime\prime } \bigl(f_{i}(y,z) \bigr)\nabla_{y}f_{i}(y,z) \bigl( \nabla_{y}f_{i}(y,z) \bigr)^{T} + G_{f_{i}}^{\prime} \bigl(f_{i}(y,z) \bigr) \nabla_{yy}f_{i}(y,z) \bigr\} p \bigr].\end{aligned} $$
(10)

Finally, adding inequalities (9) and (10) and using \(\lambda^{T}e_{k} =1\), we obtain

$$\begin{gathered} \sum_{i=1}^{k} \lambda_{i} \Biggl[G_{f_{i}}\bigl(f_{i}(y,z) \bigr)-z^{T} \sum_{i=1}^{k} \lambda_{i} \bigl(G_{f_{i}}^{\prime}\bigl(f_{i}(y,z) \bigr)\nabla _{y}f_{i}(y,z) \\ \qquad{} +G_{f_{i}}^{\prime}\bigl(f_{i}(y,z)\bigr) \nabla_{y}f_{i}(y,z) \bigl(\nabla _{y}f_{i}(y,z) \bigr)^{T}+ G_{f_{i}}^{\prime}\bigl(f_{i}(y,z) \bigr)\nabla _{yy}f_{i}(y,z) \bigr)p\Biggr] \\ \qquad{}- \frac{1}{2} \sum_{i=1}^{k} \lambda_{i} \bigl(p^{T}\bigl[G_{f_{i}}^{\prime\prime} \bigl(f_{i}(y,z)\bigr)\nabla_{y}f_{i}(y,z) \bigl( \nabla_{y}f_{i}(y,z)\bigr)^{T}+ G_{f_{i}}^{\prime}\bigl(f_{i}(y,z)\bigr)\nabla _{yy}f_{i}(y,z)\bigr]p \bigr) \\ \quad\geq \sum_{i=1}^{k} \lambda_{i} \Biggl[G_{f_{i}}\bigl(f_{i}(v,w) \bigr)-v^{T} \sum_{i=1}^{k} \lambda_{i} \bigl(G_{f_{i}}^{\prime}\bigl(f_{i}(v,w) \bigr)\nabla_{z}f_{i}(v,w) \\ \qquad{}+ G_{f_{i}}^{\prime}\bigl(f_{i}(v,w)\bigr) \nabla_{z}f_{i}(v,w) \bigl(\nabla _{z}f_{i}(v,w) \bigr)^{T} + G_{f_{i}}^{\prime}\bigl(f_{i}(v,w) \bigr) \nabla _{zz}f_{i}(v,w)\bigr)q\Biggr] \\ \qquad{}- \frac{1}{2} \sum_{i=1}^{k} \lambda_{i} \bigl(q^{T}\bigl[G_{f_{i}}^{\prime\prime} \bigl(f_{i}(v,w)\bigr)\nabla_{z}f_{i}(v,w) \bigl( \nabla _{z}f_{i}(v,w)\bigr)^{T}+ G_{f_{i}}^{\prime}\bigl(f_{i}(v,w)\bigr) \nabla_{yy} f_{i}(v,w)\bigr]q \bigr) .\end{gathered} $$

This contradicts (7). Hence the result. □

Remark 3.1

Since every \(G_{f}\)-bonvex function is \(G_{f}\)-pseudobonvex, Theorem 3.1 can also be obtained under \(G_{f}\)-pseudobonvexity assumptions.

Remark 3.2

A vector space V over field K, the span of a set S, may be defined as the set of all finite linear combinations of elements (vectors) of S:

$$\operatorname{span}(S)= \Biggl\{ \sum_{i=1}^{k} \lambda_{i}v_{i} : k\in N, u_{i}\in S, \lambda_{i}=1,2,3,\ldots,k \Biggr\} . $$

Theorem 3.2

(Strong duality)

Let\((\bar{y},\bar{z},\bar {\lambda},\bar{p})\)be an efficient solution of (GWP); fix\(\lambda= \bar{\lambda}\)in (GWD) such that

  1. (i)

    for all\(i=1,2,3,\ldots,k\), \([G_{f_{i}}^{\prime\prime }(f_{i}(\bar{y},\bar{z}))\nabla_{z}f_{i}(\bar{y},\bar{z})(\nabla _{z}f_{i}(\bar{y},\bar{z}))^{T}+ G_{f_{i}}^{\prime}(f_{i}(\bar{y},\bar {z}))\nabla_{zz}f_{i}(\bar{y},\bar{z})]\)is nonsingular,

  2. (ii)

    \(\sum_{i=1}^{k}\bar{\lambda}_{i}\nabla_{z} (\{ G_{f_{i}}^{\prime\prime}(f_{i}(\bar{y},\bar{z})) \nabla_{z}f_{i}(\bar {y},\bar{z})(\nabla_{z}f_{i}(\bar{y},\bar{z}))^{T}+ G_{f_{i}}^{\prime }(f_{i}(\bar{y},\bar{z}))\nabla_{zz}f_{i}(\bar{y},\bar{z})\}\bar{p} )\bar{p} \notin \operatorname{span} \{G_{f_{1}}^{\prime}(f_{1}(\bar {y},\bar{z}))\nabla_{z}f_{1}(\bar{z},\bar{x}),\ldots, G_{f_{k}}^{\prime }(f_{k}(\bar{y},\bar{z}))\nabla_{z}f_{k}(\bar{y},\bar{z}) \} \setminus\{0\}\),

  3. (iii)

    the vectors\(\{G_{f_{1}}^{\prime}(f_{1}(\bar{y},\bar {z}))\nabla_{z}f_{1}(\bar{z},\bar{x}),G_{f_{2}}^{\prime}(f_{2}(\bar {y},\bar{z}))\nabla_{z}f_{2}(\bar{y},\bar{z}),\ldots,G_{f_{k}}^{\prime }(f_{k}(\bar{y},\bar{z}))\nabla_{z}f_{k}(\bar{y},\bar{z}) \}\)are linearly independent,

  4. (iv)

    \(\sum_{i=1}^{k}\bar{\lambda}_{i}\nabla_{y} (\{ G_{f_{i}}^{\prime\prime}(f_{i}(\bar{y},\bar{z})) \nabla_{z}f_{i}(\bar {y},\bar{z})(\nabla_{z}f_{i}(\bar{y},\bar{z}))^{T} + G_{f_{i}}^{\prime }(f_{i}(\bar{y},\bar{z}))\nabla_{zz}f_{i}(\bar{y},\bar{z})\}\bar{p} )\bar{p} =0\)implies that\(\bar{p}=0\).

Then for\(\bar{q}=0\), we have\((\bar{v},\bar{w},\bar{\lambda},\bar {p}=0)\in Z^{0}\)and\(R(\bar{y},\bar{z},\bar{\lambda},\bar{q})=S(\bar {y},\bar{z},\bar{\lambda},\bar{q})\). Also, from Theorem 3.1it follows that\((\bar{v},\bar{w},\bar{\lambda},\bar{p}=0)\)is an efficient solution for (GWD).

Proof

By the Fritz–John necessary conditions [22] there exist \(\alpha\in R^{k}\), \(\beta\in R^{m}\), and \(\eta\in R\) such that

$$\begin{aligned}& \begin{aligned}[b] &(y-\bar{y})^{T} \Biggl\{ \sum _{i=1}^{k}\alpha_{i} \bigl[G_{f_{i}}^{\prime }\bigl(f_{i}(\bar{y},\bar{z})\bigr) \nabla_{y}f_{i}(\bar{y},\bar{z})\bigr] \\ &\quad+ \sum_{i=1}^{k}\bar{ \lambda}_{i} \bigl[G_{f_{i}}^{\prime\prime}\bigl(f_{i}( \bar {y},\bar{z})\bigr) \nabla_{y}f_{i}(\bar{y},\bar{z}) \nabla_{y}f_{i}(\bar {y},\bar{z})+G_{f_{i}}^{\prime} \bigl(f_{i}(\bar{y},\bar{z})\bigr)\nabla _{xy}f_{i}( \bar{y},\bar{z}) \bigr] \bigl(\beta-\bigl(\alpha^{T} e_{k} \bigr)\bar {y} \bigr) \\ &\quad+ \sum_{i=1}^{k}\bar{ \lambda}_{i}\nabla_{y} \bigl[\bigl(G_{f_{i}}^{\prime\prime} \bigl(f_{i}(\bar{y},\bar{z})\bigr)\nabla_{y}f_{i}( \bar {y},\bar{z}) \bigl(\nabla_{y}f_{i}(\bar{y},\bar{z}) \bigr)^{T}+G_{f_{i}}^{\prime} \bigl(f_{i}(\bar {y},\bar{z}) \bigr)\nabla_{yy}f_{i}(\bar{y},\bar{z})\bigr) \bar{p} \bigr] \\ &\quad\times \biggl(\beta - \bigl(\alpha^{T} e_{k} \bigr) \biggl(\bar{y}+ \frac{1}{2}\bar{p} \biggr) \biggr) \Biggr\} =0\quad \mbox{for all } y\in C_{1}, \end{aligned} \end{aligned}$$
(11)
$$\begin{aligned}& \begin{aligned}[b] &\sum_{i=1}^{k} \bigl(\alpha_{i}-\bigl(\alpha^{T} e_{k}\bigr) \bar{\lambda}_{i} \bigr) \bigl[G_{f_{i}}^{\prime} \bigl(f_{i}(\bar{y},\bar{z})\bigr)\nabla_{z}f_{i}( \bar {y},\bar{z}) \bigr] \\ &\quad+ \sum_{i=1}^{k}\bar{ \lambda}_{i} \bigl[ \bigl\{ G_{f_{i}}^{\prime\prime} \bigl(f_{i}(\bar{y},\bar{z})\bigr) \nabla_{z}f_{i}( \bar {y},\bar{z}) \bigl(\nabla_{z}f_{i}(\bar{y},\bar{z}) \bigr)^{T} + G_{f_{i}}^{\prime }\bigl(f_{i}( \bar{y},\bar{z})\bigr)\nabla_{zz}f_{i}(\bar{y},\bar{z}) \bigr\} \\ &\quad\times \bigl(\beta- \bigl(\alpha^{T} e_{k}\bigr) ( \bar{y}+ \bar{p}) \bigr)\nabla_{z}f_{i}(\bar{y},\bar{z}) \bigl( \nabla_{z}f_{i}(\bar{y},\bar{z}) \bigr)^{T} \\ &\quad+ G_{f_{i}}^{\prime} \bigl(f_{i}(\bar{y},\bar{z}) \bigr) \nabla_{zz}f_{i}(\bar{z},\bar {x}))\bar{p}\}\bigr] \biggl(\beta- \bigl(\alpha^{T} e_{k} \bigr) \biggl(\bar{y}+ \frac{1}{2} \bar {p} \biggr) \biggr) =0, \end{aligned} \end{aligned}$$
(12)
$$\begin{aligned}& \begin{aligned}[b] &\bigl[G_{f_{i}}^{\prime\prime} \bigl(f_{i}(\bar{y},\bar{z})\bigr)\nabla _{z}f_{i}( \bar{y},\bar{z}) \bigl(\nabla_{z}f_{i}(\bar{y},\bar{z}) \bigr)^{T}+G_{f_{i}}^{\prime} \bigl(f_{i}( \bar{y}, \bar{z}) \bigr) \nabla_{zz}f_{i}(\bar {y},\bar{z}) \bigr] \\ &\quad\times \bigl[ \bigl(\beta- \bigl(\alpha^{T}e_{k} \bigr) (\bar{p}+\bar {y}) \bigr)\bar{\lambda}_{i} \bigr]=0,\quad i=1,2,3,\ldots,k,\end{aligned} \end{aligned}$$
(13)
$$\begin{aligned}& G_{f_{i}}^{\prime} \bigl(f_{i}(\bar{y},\bar{z})\bigr)\nabla_{z}f_{i}( \bar{y},\bar {z}) \bigl(\beta-\bigl(\alpha^{T} e_{k}\bar{y} \bigr) \bigr) + \eta e_{k} \\& \quad{}+ \biggl\{ \biggl(\beta-\bigl(\alpha^{T} e_{k}\bigr) \biggl(\bar{y}+ \frac{1}{2}\bar{p}_{1}\biggr) \biggr)^{T} \bigl(G_{f_{1}}^{\prime\prime}\bigl(f_{1}( \bar{y},\bar{z})\bigr)\nabla _{y}f_{1}(\bar{z},\bar{x}) \bigl(\nabla_{y}f_{1}(\bar{z},\bar{x})\bigr)^{T} \\& \quad{} + G_{f_{1}}^{\prime}\bigl(f_{1}(\bar{y},\bar{z}) \bigr) \nabla_{zz}f_{1}(\bar{z},\bar {x})\bigr) \bar{p}_{1} ),\ldots, \\& \quad \biggl(\beta-\bigl(\alpha^{T} e_{k}\bigr) \biggl( \bar{y}+ \frac {1}{2}\bar{p}_{k}\biggr) \biggr)^{T} \bigl(G_{f_{k}}^{\prime\prime}\bigl(f_{k}(\bar {y},\bar{z}) \bigr)\nabla_{z}f_{k}(\bar{y},\bar{z}) \bigl( \nabla_{z}f_{k}(\bar {y},\bar{z})\bigr)^{T} \\& \quad{}+ G_{f_{k}}^{\prime} \bigl(f_{k}(\bar{y}, \bar{z}) \bigr) \nabla_{zz}f_{k}(\bar {y},\bar{z})\bigr)\bar{p} \big) \biggr\} = 0, \end{aligned}$$
(14)
$$\begin{aligned}& \begin{aligned} [b]&\beta^{T} \sum _{i=1}^{k}\bar{\lambda}_{i} \bigl[G_{f_{i}}^{\prime}\bigl(f_{i}(\bar {y},\bar{z}) \bigr)\nabla_{z}f_{i}(\bar{y},\bar{z}) \\ &\quad+ \bigl\{ G_{f_{i}}^{\prime\prime} \bigl(f_{i}( \bar{y}, \bar{z}) \bigr)\nabla_{z}f_{i}(\bar {y},\bar{z}) \bigl( \nabla_{z}f_{i}(\bar{y},\bar{z}) \bigr)^{T} +G_{f_{i}}^{\prime } \bigl(f_{i}(\bar{y},\bar{z}) \bigr) \nabla_{zz}f_{i}(\bar{y},\bar{z}) \bigr\} \bar{p}\bigr]=0, \end{aligned} \end{aligned}$$
(15)
$$\begin{aligned}& \eta^{T} \bigl[\bar{\lambda}^{T} e_{k}-1 \bigr]= 0, \end{aligned}$$
(16)
$$\begin{aligned}& (\alpha,\beta)\geq0,\qquad ( \alpha,\beta,\eta)\neq0. \end{aligned}$$
(17)

Equation (14) can be rewritten as

$$ \begin{aligned}[b] &G_{f_{i}}^{\prime} \bigl(f_{i}(\bar{y},\bar{z})\bigr)\nabla_{z}f_{i}( \bar{y},\bar {z}) \bigl(\beta-\bigl(\alpha^{T} e_{k}\bigr) \bar{y} \bigr) \\ &\quad+ \biggl(\beta-\bigl(\alpha^{T} e_{k}\bigr) \biggl( \bar{y}+ \frac{1}{2}\bar{p}\biggr) \biggr)^{T} \bigl( \bigl(G_{f_{i}}^{\prime \prime}\bigl(f_{i}(\bar{y},\bar{z})\bigr) \nabla_{z}f_{i}(\bar{y},\bar{z}) \bigl(\nabla _{z}f_{i}(\bar{y},\bar{z})\bigr)^{T} \\ &\quad+ G_{f_{i}}^{\prime} \bigl(f_{i}(\bar{y}, \bar{z}) \bigr) \nabla_{zz}f_{i}(\bar {z},\bar{x})\bigr)\bar{p} \bigr)+ \eta= 0,\quad i=1,2,3,\ldots,k.\end{aligned} $$
(18)

By assumption (i), since \(\bar{\lambda}_{i} > 0\) for \(i=1,2,3,\ldots,k\), (18) gives

$$\begin{aligned} \beta= \bigl(\alpha^{T}e_{k} \bigr) (\bar{p} + \bar{y}),\quad i=1,2,3,\ldots,k. \end{aligned}$$
(19)

If \(\alpha=0\), then (19) implies that \(\beta=0\). Further, equation (18) gives \(\eta=0\). Consequently, \((\alpha,\beta,\eta )=0\), which contradicts (17). Hence \(\alpha\neq0\), or \(\alpha^{T} e_{k} > 0\).

Using (19) and \(\alpha^{T}e_{k}> 0\) in (12), we get

$$ \begin{aligned}[b] &\sum_{i=1}^{k} \bar{\lambda}_{i} \bigl[ \bigl(\nabla_{z} \bigl\{ \bigl(G_{f_{i}}^{\prime\prime}\bigl(f_{i}(\bar{y},\bar{z})\bigr) \nabla_{z}f_{i}(\bar {y},\bar{z}) \bigl( \nabla_{z}f_{i}(\bar{y},\bar{z})\bigr)^{T} + G_{f_{i}}^{\prime }\bigl(f_{i}(\bar{y},\bar{z})\bigr) \nabla_{zz}f_{i}(\bar{z},\bar{x})\bigr)\bar{p} \bigr\} \bar{p} \bigr) \bigr] \\ &\quad= -\frac{2}{\alpha^{T} e_{k}}\sum_{i=1}^{k} \bigl[G_{f_{i}}^{\prime} \bigl(f_{i}(\bar {y},\bar{z}) \bigr)\nabla_{z}f_{i}(\bar{y},\bar{z}) \bigr] \bigl( \alpha_{i}- \bigl(\alpha ^{T} e_{k} \bigr)\bar{ \lambda}_{i} \bigr).\end{aligned} $$
(20)

It follows from assumption (ii) that

$$\begin{aligned} \sum_{i=1}^{k}\bar{ \lambda}_{i} \bigl[ \bigl(\nabla_{y} \bigl\{ \bigl(G_{f_{i}}^{\prime\prime} \bigl(f_{i}(\bar{y},\bar{z}) \bigr) \nabla_{z}f_{i}(\bar {y},\bar{z}) \bigl( \nabla_{z}f_{i}(\bar{y},\bar{z}) \bigr)^{T}+ G_{f_{i}}^{\prime } \bigl(f_{i}(\bar{y},\bar{z}) \bigr) \nabla_{z}f_{i}(\bar{y},\bar{z}) \bigr)\bar{p} \bigr\} \bar{p} \bigr) \bigr]=0. \end{aligned}$$
(21)

Hence by assumption (iv) we get \(\bar{p}=0\), and therefore inequality (19) implies

$$\begin{aligned} \beta= \bigl(\alpha^{T} e_{k} \bigr) \bar{y}. \end{aligned}$$
(22)

Now, using \(\bar{p}=0\) and (20), we obtain

$$\begin{aligned} \sum_{i=1}^{k} \bigl(\alpha_{i}- \bigl(\alpha^{T} e_{k} \bigr)\bar{\lambda}_{i} \bigr) \bigl[G_{f_{i}}^{\prime} \bigl(f_{i}(\bar{y}, \bar{z}) \bigr)\nabla_{z}f_{i}(\bar{y},\bar{z}) \bigr]=0. \end{aligned}$$

Assumption (iii) yields

$$\begin{aligned} \alpha_{i}= \bigl(\alpha^{T} e_{k} \bigr)\bar{\lambda}_{i},\quad i=1,2,3,\ldots,k. \end{aligned}$$
(23)

Using \(\alpha^{T}e_{k}> 0\) and (21)–(23) in (11), we get

$$\begin{aligned} (y-\bar{y})^{T} \sum_{i=1}^{k} \bar{\lambda}_{i} \bigl[G_{f_{i}}^{\prime } \bigl(f_{i}(\bar{y},\bar{z}) \bigr) \nabla_{y}f_{i}( \bar{y},\bar{z}) \bigr]\geq0 \quad\mbox{for all } y\in C_{1}. \end{aligned}$$
(24)

Let \(y\in C_{1}\). Then, \(y +\bar{y}\in C_{1}\), and it follows that

$$\begin{aligned} y^{T} \sum_{i=1}^{k} \bar{\lambda}_{i} \bigl[G_{f_{i}}^{\prime} \bigl(f_{i}(\bar {y},\bar{z}) \bigr) \nabla_{y}f_{i}( \bar{y},\bar{z}) \bigr]\geq0 \quad\mbox{for all } y\in C_{1}. \end{aligned}$$
(25)

Therefore

$$\begin{aligned} \sum_{i=1}^{k}\bar{ \lambda}_{i} \bigl[G_{f_{i}}^{\prime} \bigl(f_{i}( \bar {y},\bar{z}) \bigr) \nabla_{y}f_{i}( \bar{y},\bar{z}) \bigr]\in C_{1}^{*}. \end{aligned}$$
(26)

Also, from (22) we have

$$\begin{aligned} \bar{y}= \frac{\bar{\beta}}{\bar{\alpha}^{T}e_{k}}\in C_{2}. \end{aligned}$$
(27)

Hence \((\bar{v},\bar{w},\bar{\lambda},\bar{p}=0)\) satisfies the dual constraints and \(Z^{0}\).

Now, letting \(y=0\) and \(y=2\bar{y}\) in (24), we get

$$\begin{aligned} \bar{y}^{T} \sum_{i=1}^{k} \bar{\lambda}_{i} \bigl[G_{f_{i}}^{\prime } \bigl(f_{i}(\bar{y},\bar{z}) \bigr) \nabla_{y}f_{i}( \bar{y},\bar{z}) \bigr]=0. \end{aligned}$$
(28)

Using (28) and \(\bar{q}=\bar{p}=0\) completes the proof. □

Theorem 3.3

(Converse duality)

Let\((\bar{v},\bar{w},\bar {\lambda},\bar{q})\)be an efficient solution of (GWD). Fix\(\lambda= \bar{\lambda}\)in (GWP) such that

  1. (i)

    for all\(i=1,2,3,\ldots,k\), \([G_{f_{i}}^{\prime\prime }(f_{i}(\bar{v},\bar{w}))\nabla_{z}f_{i}(\bar{v},\bar{w})(\nabla _{z}f_{i}(\bar{v},\bar{w}))^{T} + G_{f_{i}}^{\prime}(f_{i}(\bar{v},\bar {w}))\nabla_{zz} f_{i}(\bar{v},\bar{w})]\)is nonsingular,

  2. (ii)

    \(\sum_{i=1}^{k}\bar{\lambda}_{i}\nabla_{z} (\{ G_{f_{i}}^{\prime\prime}(f_{i}(\bar{v},\bar{w}))\nabla_{z}f_{i}(\bar {v},\bar{w})(\nabla_{z}f_{i}(\bar{v},\bar{w}))^{T} + G_{f_{i}}^{\prime }(f_{i}(\bar{v},\bar{w}))\nabla_{zz}f_{i}(\bar{v},\bar{w})\} \bar{q} )\bar{q} \notin \operatorname{span} \{G_{f_{1}}^{\prime }(f_{1}(\bar{v},\bar{w}))\nabla_{z}f_{1}(\bar{v},\bar{w}),\ldots, G_{f_{k}}^{\prime}(f_{k}(\bar{u},\bar{v}))\nabla_{z}f_{k}(\bar{u},\bar {v}) \}\setminus\{0\}\),

  3. (iii)

    the vectors\(\{G_{f_{1}}^{\prime}(f_{1}(\bar{v},\bar {w}))\nabla_{z}f_{1}(\bar{v},\bar{w}),G_{f_{2}}^{\prime}(f_{2}(\bar {v},\bar{w}))\nabla_{z}f_{2}(\bar{v},\bar{w}),\ldots,G_{f_{k}}^{\prime }(f_{k}(\bar{v},\bar{w}))\nabla_{z}f_{k}(\bar{v},\bar{w}) \}\)are linearly independent,

  4. (iv)

    \(\sum_{i=1}^{k}\bar{\lambda}_{i}\nabla_{z} (\{ G_{f_{i}}^{\prime\prime}(f_{i}(\bar{v},\bar{w}))\nabla_{z}f_{i}(\bar {v},\bar{w})(\nabla_{z}f_{i}(\bar{v},\bar{w}))^{T} + G_{f_{i}}^{\prime }(f_{i}(\bar{v},\bar{w}))\nabla_{zz}f_{i}(\bar{v},\bar{w})\}\bar{q} )\bar{q}=0\Rightarrow \bar{q}=0\).

Then, taking\(\bar{p}=0\), we have that\((\bar{v},\bar{w},\bar{\lambda },\bar{p}=0)\in Y^{0}\)and\(R(\bar{v},\bar{w},\bar{\lambda},\bar{p})= S(\bar{u},\bar{v},\bar{\lambda},\bar{p})\). Also, by Theorem3.1\((\bar{v},\bar{w},\bar{\lambda},\bar{p}=0)\)is an efficient solution for (GWP).

Proof

Proof follows the lines of Theorem 3.2. □

4 Concluding remarks

In this paper, we have formulated a second-order symmetric G-Wolfe-type dual problem for a nonlinear multiobjective optimization problem with cone constraints. A number of duality relations are further established under \(G_{f}\)-bonvexity/\(G_{f}\)-pseudobonvexity assumptions on the function f. We have discussed various numerical examples to show the existence of \(G_{f}\)-bonvex/\(G_{f}\)-pseudobonvex functions. The question arises whether the duality results developed in this paper hold for G-Wolfe- or mixed-type higher-order multiobjective optimization problems. This may be the future direction for the researchers working in this area.