Non-differentiable minimax fractional programming with higher-order type Ifunctions

Abstract

In this article, we are concerned with a class of non-differentiable minimaxfractional programming problems and their higher-order dual model. Weak, strongand converse duality theorems are discussed involving generalized higher-ordertype I functions. The presented results extend some previously known results onnon-differentiable minimax fractional programming.

1 Introduction

In nonlinear optimization, problems, where minimization and maximization process areperformed together, are called minimax (minmax) problems. Frequently, problems ofthis type arise in many areas like game theory, Chebychev approximation, economics,financial planning and facility location [1].

The optimization problems in which the objective function is a ratio of two functionsare commonly known as fractional programming problems. In the past few years, manyauthors have shown interest in the field of minimax fractional programming problems.Schmittendorf [2] first developed necessary and sufficient optimality conditions for aminimax programming problem. Tanimoto [3] applied the necessary conditions in [2] to formulate a dual problem and discussed the duality results, which wereextended to a fractional analogue of the problem considered in [2, 3] by several authors [4–10]. Liu [11] proposed the second-order duality theorems for a minimax programmingproblem under generalized second-order B-invex functions. Husain etal.[12] formulated two types of second-order dual models for minimax fractionalprogramming and derived weak, strong and converse duality theorems underη-convexity assumptions.

Ahmad et al.[13] and Husain et al.[14] discussed the second-order duality results for the followingnon-differentiable minimax programming problem:

$$\begin{array}{r}\text{Minimize }\underset{y\in Y}{sup}f(x,y)+{\left(x^{T}Bx\right)}^{1/2},\\ \text{subject to }h(x)\le 0,x\in R^n,\end{array}$$

(P)

where Y is a compact subset of $R^l$, $f(\cdot ,\cdot ):R^n\times R^l\to R$ and $h(\cdot ):R^n\to R^m$ are twice differentiable functions. B is an$n\times n$ positive semidefinite symmetric matrices. Ahmadet al.[15] formulated a unified higher-order dual of (P) and established appropriateduality theorems under higher-order $(F,\alpha ,\rho ,d)$-type I assumptions. Recently, Jayswal andStancu-Minasian [16] obtained higher-order duality results for (P).

In this paper, we formulate a higher-order dual for a non-differentiable minimaxfractional programming problem and establish weak, strong and strict converseduality theorems under generalized higher-order $(\mathcal{F},\alpha ,\rho ,d)$-type I assumptions. This paper generalizes severalresults that have appeared in the literature [11, 12, 14–22] and references therein.

2 Preliminaries

The problem to be considered in the present analysis is the followingnon-differentiable minimax fractional problem:

$$\begin{array}{r}\text{Minimize }\underset{y\in Y}{sup}\frac{f(x,y)+{(x^{T}Bx)}^{1/2}}{g(x,y)-{(x^{T}Cx)}^{1/2}},\\ \text{subject to }h(x)\le 0,x\in R^n,\end{array}$$

(NP)

where Y is a compact subset of $R^l$, $f(\cdot ,\cdot ),g(\cdot ,\cdot ):R^n\times R^l\to R$ and $h(\cdot ):R^n\to R^m$ are differentiable functions. B andC are $n\times n$ positive semidefinite symmetric matrices. It isassumed that for each $(x,y)$ in $R^n\times R^l$, $f(x,y)+{(x^{T}Bx)}^{\frac{1}{2}}\ge 0$ and $g(x,y)-{(x^{T}Cx)}^{\frac{1}{2}}>0$.

Let $\mathcal{X}=\{x\in R^n:h(x)\le 0\}$ denote the set of all feasible solutions of (NP). Anypoint $x\in \mathcal{X}$ is called the feasible point of (NP). For each$(x,y)\in \mathcal{X}\times Y$, we define

$$\psi (x,y)=\frac{f(x,y)+{(x^{T}Bx)}^{1/2}}{g(x,y)-{(x^{T}Cx)}^{1/2}}$$

such that for each $(x,y)\in \mathcal{X}\times Y$,

$$f(x,y)+{\left(x^{T}Bx\right)}^{1/2}\ge 0\text{and}g(x,y)-{\left(x^{T}Cx\right)}^{1/2}>0.$$

For each $(x,y)\in \mathcal{X}\times Y$, we define

$$J(x)=\{j\in J:h_j(x)=0\},$$

where

$$\begin{array}{r}J=\{1,2,\dots ,m\},\\ Y(x)=\{y\in Y:\frac{f(x,y)+{(x^{T}Bx)}^{1/2}}{g(x,y)-{(x^{T}Cx)}^{1/2}}=\underset{z\in Y}{sup}\frac{f(x,z)+{(x^{T}Bx)}^{1/2}}{g(x,z)-{(x^{T}Cx)}^{1/2}}\},\\ S(x)=\{(s,t,\tilde{y})\in N\times R_{+}^s\times R^{ls}:1\le s\le n+1,t=(t_1,t_2,\dots ,t_s)\in R_{+}^s\\ \phantom{S(x)=}\text{with }\sum _{i=1}^s{t}_i=1,\tilde{y}=({\overline{y}}_1,{\overline{y}}_2,\dots ,{\overline{y}}_s)\text{ with }{\overline{y}}_i\in Y(x)(i=1,2,\dots ,s)\}.\end{array}$$

Since f and g are continuously differentiable and Y iscompact in $R^l$, it follows that for each $x^{\ast }\in \mathcal{X}$, $Y(x^{\ast })\ne \mathrm{\varnothing }$, and for any ${\overline{y}}_i\in Y(x^{\ast })$, we have

$${\lambda }_{\circ }=\varphi (x^{\ast },{\overline{y}}_i)=\frac{f(x^{\ast },{\overline{y}}_i)+{({x^{\ast }}^{T}B{x}^{\ast })}^{1/2}}{g(x^{\ast },{\overline{y}}_i)-{({x^{\ast }}^{T}C{x}^{\ast })}^{1/2}}.$$

Lemma 2.1 (Generalized Schwarz inequality)

Let A be a positive-semidefinite matrix of order n. Then, for all$x,w\in R^n$,

$$x^{T}Aw\le {\left(x^{T}Ax\right)}^{\frac{1}{2}}{\left(w^{T}Aw\right)}^{\frac{1}{2}}.$$

(2.1)

The equality$Ax=\xi Aw$holds for some$\xi \ge 0$. Clearly, if${(w^{T}Aw)}^{\frac{1}{2}}\le 1$, we have

$$x^{T}Aw\le {\left(x^{T}Ax\right)}^{\frac{1}{2}}.$$

Let ℱ be a sublinear functional, and let $d(\cdot ,\cdot ):R^n\times R^n\to R$. Let $\rho =({\rho }^1,{\rho }^2)$, where ${\rho }^1=({\rho }_1^1,{\rho }_2^1,\dots ,{\rho }_s^1)\in R^s$ and ${\rho }^2=({\rho }_1^2,{\rho }_2^2,\dots ,{\rho }_m^2)\in R^m$, and let $\alpha =({\alpha }^1,{\alpha }^2):R^n\times R^n\to R_{+}\setminus \{0\}$. Let $\psi (\cdot ,\cdot ):R^n\times Y\to R$, $h(\cdot ):R^n\to R^m$, $K:R^n\times Y\times R^n\to R$ and $H_j:R^n\times Y\times R^n\to R$, $j=1,2,\dots ,m$, be differentiable functions at$\overline{x}\in R^n$.

Definition 2.1 A functional $\mathcal{F}:R^n\times R^n\times R^n\mapsto R$ is said to be sublinear in its third argument if forall $x,\overline{x}\in R^n$,

• (i) $\mathcal{F}(x,\overline{x};a+b)\le \mathcal{F}(x,\overline{x};a)+\mathcal{F}(x,\overline{x};b)$, $\mathrm{\forall }a,b\in R^n$;

• (ii) $\mathcal{F}(x,\overline{x};\beta a)=\beta \mathcal{F}(x,\overline{x};a)$, $\mathrm{\forall }\beta \in R$, $\beta \ge 0$, and $\mathrm{\forall }a\in R^n$.

From (ii), it is clear that $\mathcal{F}(x,\overline{x};0)=0$.

Definition 2.2[15]

For each $j\in J$, $(\psi ,h_j)$ is said to be higher-order $(\mathcal{F},\alpha ,\rho ,d)$-pseudoquasi-type I at $\overline{x}\in R^n$ if for all $x\in \mathcal{X}$, $p\in R^n$ and ${\overline{y}}_i\in Y(x)$,

$$\begin{array}{r}\psi (x,{\overline{y}}_i)<\psi (\overline{x},{\overline{y}}_i)+K(\overline{x},{\overline{y}}_i,p)-p^T{\mathrm{\nabla }}_{p}K(\overline{x},{\overline{y}}_i,p)\\ \Rightarrow \mathcal{F}(x,\overline{x};{\alpha }^1(x,\overline{x})({\mathrm{\nabla }}_{p}K(\overline{x},{\overline{y}}_i,p)))<-{\rho }_i^1{d}^2(x,\overline{x}),i=1,2,\dots ,s,\\ -[h_j(\overline{x})+H_j(\overline{x},p)-p^T{\mathrm{\nabla }}_p{H}_j(\overline{x},p)]\le 0\\ \Rightarrow \mathcal{F}(x,\overline{x};{\alpha }^2(x,\overline{x})({\mathrm{\nabla }}_p{H}_j(\overline{x},p)))\le -{\rho }_j^2{d}^2(x,\overline{x}),j=1,2,\dots ,m.\end{array}$$

In the above definition, if

$$\begin{array}{r}\mathcal{F}(x,\overline{x};{\alpha }^1(x,\overline{x})({\mathrm{\nabla }}_{p}K(\overline{x},{\overline{y}}_i,p)))\ge -{\rho }_i^1{d}^2(x,\overline{x})\\ \Rightarrow \psi (x,{\overline{y}}_i)>\psi (\overline{x},{\overline{y}}_i)+K(\overline{x},{\overline{y}}_i,p)-p^T{\mathrm{\nabla }}_{p}K(\overline{x},{\overline{y}}_i,p),i=1,2,\dots ,s,\end{array}$$

then we say that $(\psi ,g_j)$ is higher-order $(\mathcal{F},\alpha ,\rho ,d)$-strictly pseudoquasi-type I at$\overline{x}$.

If the functions f, g and h in problem (NP) arecontinuously differentiable with respect to $x\in R^n$, then Liu [11] derived the following necessary conditions for optimalityof (NP).

Theorem 2.1 (Necessary conditions)

If$x^{\ast }$is a solution of (NP) satisfying${x^{\ast }}^{T}B{x}^{\ast }>0$, ${x^{\ast }}^{T}C{x}^{\ast }>0$, and$\mathrm{\nabla }{h}_j(x^{\ast })$, $j\in J(x^{\ast })$are linearly independent, then there exist$(s,t^{\ast },\tilde{y})\in S(x^{\ast })$, ${\lambda }_0\in R_{+}$, $w,v\in R^n$, and${\mu }^{\ast }\in R_{+}^m$such that

$$\begin{array}{c}\sum _{i=1}^s{t}_i^{\ast }\{\mathrm{\nabla }f(x^{\ast },{\overline{y}}_i)+Bw-{\lambda }_0(\mathrm{\nabla }g(x^{\ast },{\overline{y}}_i)-Cv)\}+\mathrm{\nabla }\sum _{j=1}^m{\mu }_j^{\ast }{h}_j\left(x^{\ast }\right)=0,\hfill \\ f(x^{\ast },{\overline{y}}_i)+{\left({x^{\ast }}^{T}B{x}^{\ast }\right)}^{\frac{1}{2}}-{\lambda }_0(g(x^{\ast },{\overline{y}}_i)-{\left({x^{\ast }}^{T}C{x}^{\ast }\right)}^{\frac{1}{2}})=0,i=1,2,\dots ,s,\hfill \\ \sum _{j=1}^m{\mu }_j^{\ast }{h}_j\left(x^{\ast }\right)=0,\hfill \\ t_i^{\ast }\ge 0(i=1,2,\dots ,s),\sum _{i=1}^s{t}_i^{\ast }=1,\hfill \\ w^{T}Bw\le 1,v^{T}Cv\le 1,\hfill \\ {\left({x^{\ast }}^{T}B{x}^{\ast }\right)}^{1/2}={x^{\ast }}^{T}Bw,{\left({x^{\ast }}^{T}C{x}^{\ast }\right)}^{1/2}={x^{\ast }}^{T}Cv.\hfill \end{array}$$

In the above theorem, both matrices B and C are positivesemidefinite. If either ${x^{\ast }}^{T}B{x}^{\ast }$ or ${x^{\ast }}^{T}C{x}^{\ast }$ is zero, then the functions involved in the objectivefunction of problem (NP) are not differentiable. To derive these necessaryconditions under this situation, for $(s,t^{\ast },\tilde{y})\in S(x^{\ast })$, we define

$$U_{\tilde{y}}\left(x^{\ast }\right)=\{u\in R^n:u^t\mathrm{\nabla }{h}_j\left(x^{\ast }\right)\le 0,j\in J\left(x^{\ast }\right)\text{ satisfying one of the following conditions:}$$

• (i) ${x^{\ast }}^{T}B{x}^{\ast }>0$, ${x^{\ast }}^{T}C{x}^{\ast }=0$

  $$\Rightarrow u^T\left(\sum _{i=1}^s{t}_i\{\mathrm{\nabla }f(x^{\ast },{\overline{y}}_i)+\frac{B{x}^{\ast }}{{({x^{\ast }}^{t}B{x}^{\ast })}^{\frac{1}{2}}}-{\lambda }_{\circ }\mathrm{\nabla }g(x^{\ast },{\overline{y}}_i)\}\right)+{\left(u^T\left({\lambda }_{\circ }^{2}C\right)u\right)}^{\frac{1}{2}}<0,$$

• (ii) ${x^{\ast }}^{T}B{x}^{\ast }=0$, ${x^{\ast }}^{T}C{x}^{\ast }>0$

  $$\Rightarrow u^T\left(\sum _{i=1}^s{t}_i\{\mathrm{\nabla }f(x^{\ast },{\overline{y}}_i)-{\lambda }_{\circ }(\mathrm{\nabla }g(x^{\ast },{\overline{y}}_i)-\frac{C{x}^{\ast }}{{({x^{\ast }}^{T}C{x}^{\ast })}^{\frac{1}{2}}})\}\right)+{\left(u^{T}Bu\right)}^{\frac{1}{2}}<0,$$

• (iii) ${x^{\ast }}^{T}B{x}^{\ast }=0$, ${x^{\ast }}^{T}C{x}^{\ast }=0$

  $$\Rightarrow u^T\left(\sum _{i=1}^s{t}_i\{\mathrm{\nabla }f(x^{\ast },{\overline{y}}_i)-{\lambda }_{\circ }\mathrm{\nabla }g(x^{\ast },{\overline{y}}_i)\}\right)+{\left(u^T\left({\lambda }_{\circ }^{2}C\right)u\right)}^{\frac{1}{2}}+{\left(u^{T}Bu\right)}^{\frac{1}{2}}<0,$$

• (iv) ${x^{\ast }}^{T}B{x}^{\ast }>0$, ${x^{\ast }}^{T}C{x}^{\ast }>0$

  $$\Rightarrow u^T\left(\sum _{i=1}^s{t}_i\{\mathrm{\nabla }f(x^{\ast },{\overline{y}}_i)-{\lambda }_{\circ }\mathrm{\nabla }g(x^{\ast },{\overline{y}}_i)\}\right)+{\left(u^T\left({\lambda }_{\circ }^{2}C\right)u\right)}^{\frac{1}{2}}+{\left(u^{T}Bu\right)}^{\frac{1}{2}}<0\}.$$

If in addition, we insert $U_{\tilde{y}}(x^{\ast })=\mathrm{\varnothing }$, then the results of Theorem 2.1 still hold.

3 Higher-order non-differentiable fractional duality

In this section, we consider the following dual problem to (NP):

$$\underset{(s,t,\tilde{y})\in S(z)}{max}\underset{(z,\mu ,\lambda ,v,w,p)\in L(s,t,\tilde{y})}{sup}\lambda ,$$

(ND)

where $L(s,t,\tilde{y})$ denotes the set of all $(z,\mu ,\lambda ,v,w,p)\in R^n\times R_{+}^m\times R_{+}\times R^n\times R^n\times R^n$ satisfying

$$\sum _{i=1}^s{t}_i\left[{\mathrm{\nabla }}_p(F(z,{\overline{y}}_i,p)-\lambda G(z,{\overline{y}}_i,p))\right]+Bw+\lambda Cv+\sum _{j=1}^m{\mu }_j{\mathrm{\nabla }}_p{H}_j(z,p)=0,$$

(3.1)

$$\begin{array}{r}\sum _{i=1}^s{t}_i[f(z,{\overline{y}}_i)+z^{T}Bw-\lambda (g(z,{\overline{y}}_i)-z^{T}Cv)+F(z,{\overline{y}}_i,p)-\lambda G(z,{\overline{y}}_i,p)+\sum _{j\in J_0}{\mu }_j{h}_j(z)\\ -p^T{\mathrm{\nabla }}_p\{F(z,{\overline{y}}_i,p)-\lambda G(z,{\overline{y}}_i,p)\}]\\ +\sum _{j\in J_0}{\mu }_j{H}_j(z,p)-p^T\sum _{j\in J_0}{\mu }_j{\mathrm{\nabla }}_p{H}_j(z,p)\ge 0,\end{array}$$

(3.2)

$$\sum _{j\in J_{\beta }}{\mu }_j[h_j(z)+H_j(z,p)-p^T{\mathrm{\nabla }}_p{H}_j(z,p)]\ge 0,\beta =1,2,\dots ,r,$$

(3.3)

$$w^{T}Bw\le 1,v^{T}Cv\le 1,$$

(3.4)

where $F:R^n\times Y\times R^n\to R$, $G:R^n\times Y\times R^n\to R$, $J_{\beta }\subseteq M=\{1,2,\dots ,m\}$, $\beta =0,1,2,\dots ,r$ with ${\bigcup }_{\beta =0}^r{J}_{\beta }=M$ and $J_{\beta }\cap J\alpha =\mathrm{\varnothing }$ if $\beta \ne \alpha$. If for a triplet $(s,t,\tilde{y})\in S(z)$, the set $L(s,t,\tilde{y})=\mathrm{\varnothing }$, then we define the supremum over it to be∞.

Theorem 3.1 (Weak duality)

Let x and$(z,\mu ,\lambda ,s,t,v,w,\tilde{y},p)$be feasible solutions of (NP) and (ND), respectively.Suppose that

$$[\sum _{i=1}^s{t}_i\{f(\cdot ,{\overline{y}}_i)+{(\cdot )}^{T}Bw-\lambda (g(\cdot ,{\overline{y}}_i)-z^{T}Cv)\}+\sum _{j\in J_0}{\mu }_j{h}_j(\cdot ),\sum _{j\in j_{\beta }}{\mu }_j{h}_j(\cdot ),\beta =1,2,\dots ,r]$$

is higher-order$(F,\alpha ,\rho ,d)$-pseudoquasi-type I at z and

$$\frac{{\rho }_1^1}{{\alpha }^1(x,z)}+\sum _{\beta =1}^r\frac{{\rho }_{\beta }^2}{{\alpha }^2(x,z)}\ge 0.$$

Then

$$\underset{y\in Y}{sup}\frac{f(x,y)+{(x^{t}Bx)}^{1/2}}{g(x,y)-{(x^{t}Cx)}^{1/2}}\ge \lambda .$$

Proof Suppose to the contrary that

$$\underset{y\in Y}{sup}\frac{f(x,y)+{(x^{T}Bx)}^{1/2}}{g(x,y)-{(x^{T}Cx)}^{1/2}}<\lambda .$$

Then we have

$$f(x,{\overline{y}}_i)+{\left(x^{T}Bx\right)}^{1/2}-\lambda (g(x,{\overline{y}}_i)-{\left(x^{T}Cx\right)}^{1/2})<0,\text{for all }{\overline{y}}_i\in Y,i=1,2,\dots ,s.$$

It follows from $t_i\ge 0$, $i=1,2,\dots ,s$, that

$$t_i[f(x,{\overline{y}}_i)+{\left(x^{T}Bx\right)}^{1/2}-\lambda (g(x,{\overline{y}}_i)-{\left(x^{T}Cx\right)}^{1/2})]\le 0,i=1,2,\dots ,s,$$

with at least one strict inequality, since $t=(t_1,t_2,\dots ,t_s)\ne 0$. Taking summation over i and using${\sum }_{i=1}^s{t}_i=1$, we have

$$\sum _{i=1}^s{t}_i[f(x,{\overline{y}}_i)+{\left(x^{T}Bx\right)}^{1/2}-\lambda (g(x,{\overline{y}}_i)-{\left(x^{T}Cx\right)}^{1/2})]<0.$$

It follows from the generalized Schwarz inequality and (3.4) that

$$\sum _{i=1}^s{t}_i[f(x,{\overline{y}}_i)+x^{T}Bw-\lambda (g(x,{\overline{y}}_i)-x^{T}Cv)]<0.$$

(3.5)

By the feasibility of x for (NP) and $\mu \ge 0$, we obtain

$$\sum _{j\in J_0}{\mu }_j{h}_j(x)\le 0.$$

(3.6)

The above inequality with (3.5) gives

$$\sum _{i=1}^s{t}_i[f(x,{\overline{y}}_i)+x^{T}Bw-\lambda (g(x,{\overline{y}}_i)-x^{T}Cv)]+\sum _{j\in J_0}{\mu }_j{h}_j(x)<0.$$

(3.7)

From (3.2) and (3.7), we have

$$\begin{array}{r}\sum _{i=1}^s{t}_i[f(x,{\overline{y}}_i)+x^{T}Bw-\lambda (g(x,{\overline{y}}_i)-x^{T}Cv)]+\sum _{j\in J_0}{\mu }_j{h}_j(x)\\ <\sum _{i=1}^s{t}_i[f(z,{\overline{y}}_i)+z^{T}Bw-\lambda (g(z,{\overline{y}}_i)-z^{T}Cv)+F(z,{\overline{y}}_i,p)-\lambda G(z,{\overline{y}}_i,p)+\sum _{j\in J_0}{\mu }_j{h}_j(z)\\ -p^T{\mathrm{\nabla }}_p\{F(z,{\overline{y}}_i,p)-\lambda G(z,{\overline{y}}_i,p)\}]+\sum _{j\in J_0}{\mu }_j{H}_j(z,p)-p^T\sum _{j\in J_0}{\mu }_j{\mathrm{\nabla }}_p{H}_j(z,p).\end{array}$$

(3.8)

Also, from (3.3), we have

$$\sum _{j\in j_{\beta }}{\mu }_j[h_j(z)+H_j(z,p)-p^T{\mathrm{\nabla }}_p{H}_j(z,p)]\ge 0,\beta =1,2,\dots ,r.$$

(3.9)

The higher second-order $(\mathcal{F},\alpha ,\rho ,d)$-pseudoquasi-type I assumption on

$$[\sum _{i=1}^s{t}_i\{f(\cdot ,{\overline{y}}_i)+{(\cdot )}^{T}Bw-\lambda (g(\cdot ,{\overline{y}}_i)-{(\cdot )}^{T}Cv)\}+\sum _{j\in J_0}{\mu }_j{h}_j(\cdot ),\sum _{j\in j_{\beta }}{\mu }_j{h}_j(\cdot ),\beta =1,2,\dots ,r]$$

at z, with (3.8) and (3.9), implies

$$\begin{array}{c}\mathcal{F}(x,z;{\alpha }^1(x,z)\sum _{i=1}^s{t}_i\left\{{\mathrm{\nabla }}_p(F(z,{\overline{y}}_i,p)-\lambda G(z,{\overline{y}}_i,p))\right\}+Bw+\lambda Cv)<-{\rho }_1^1{d}^2(x,z),\hfill \\ \mathcal{F}(x,z;{\alpha }^2(x,z)\sum _{j\in j_{\beta }}{\mu }_j{\mathrm{\nabla }}_p{H}_j(z,p))\le -{\rho }_{\beta }^2{d}^2(x,z),\beta =1,2,\dots ,r.\hfill \end{array}$$

By using ${\alpha }^1(x,z)>0$, ${\alpha }^2(x,z)>0$, and the sublinearity of ℱ in the aboveinequalities, we summarize to get

$$\begin{array}{r}\mathcal{F}(x,z;\sum _{i=1}^s{t}_i\left\{{\mathrm{\nabla }}_p(F(z,{\overline{y}}_i,p)-\lambda G(z,{\overline{y}}_i,p))\right\}+Bw+\lambda Cv+\sum _{\beta =1}^r\sum _{j\in j_{\beta }}{\mu }_j{\mathrm{\nabla }}_p{H}_j(z,p))\\ <-(\frac{{\rho }_1^1}{{\alpha }^1(x,z)}+\sum _{\beta =1}^r\frac{{\rho }_{\beta }^2}{{\alpha }^2(x,z)})d^2(x,z).\end{array}$$

Since $(\frac{{\rho }_1^1}{{\alpha }^1(x,z)}+{\sum }_{\beta =1}^r\frac{{\rho }_{\beta }^2}{{\alpha }^2(x,z)})\ge 0$, therefore

$$\mathcal{F}(x,z;\sum _{i=1}^s{t}_i\left\{{\mathrm{\nabla }}_p(F(z,{\overline{y}}_i,p)-\lambda G(z,{\overline{y}}_i,p))\right\}+Bw+\lambda Cv+\sum _{j=1}^m{\mu }_j{\mathrm{\nabla }}_p{H}_j(z,p))<0,$$

which contradicts (3.1), as $\mathcal{F}(x,z;0)=0$. □

Theorem 3.2 (Strong duality)

Let$x^{\ast }$be an optimal solution of (NP) and let$\mathrm{\nabla }{h}_j(x^{\ast })$, $j\in J(x^{\ast })$be linearly independent. Assume that

$$\begin{array}{c}F(x^{\ast },{\overline{y}}_i^{\ast },0)=0;{\mathrm{\nabla }}_{p}F(x^{\ast },{\overline{y}}_i^{\ast },0)=\mathrm{\nabla }f(x^{\ast },{\overline{y}}_i^{\ast }),i=1,2,\dots ,s,\hfill \\ G(x^{\ast },{\overline{y}}_i^{\ast },0)=0;{\mathrm{\nabla }}_{p}G(x^{\ast },{\overline{y}}_i^{\ast },0)=\mathrm{\nabla }g(x^{\ast },{\overline{y}}_i^{\ast }),i=1,2,\dots ,s,\hfill \\ H_j(x^{\ast },0)=0;{\mathrm{\nabla }}_p{H}_j(x^{\ast },0)=\mathrm{\nabla }{h}_j\left(x^{\ast }\right),j\in J.\hfill \end{array}$$

Then there exist$(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })\in S$and$(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },p^{\ast })\in L(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })$such that$(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$is a feasible solution of (ND) and the two objectives have the samevalues. Furthermore, if the assumptions of weak duality(Theorem  3.1) hold for all feasible solutions of (NP)and (ND), then$(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$is an optimal solution of (ND).

Proof Since $x^{\ast }$ is an optimal solution of (NP) and$\mathrm{\nabla }{h}_j(x^{\ast })$, $j\in J(x^{\ast })$ are linearly independent, by Theorem 2.1, thereexist $(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })\in S$ and $(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },p^{\ast })\in L(s^{\ast },t^{\ast },{\tilde{y}}^{\ast })$ such that $(x^{\ast },{\mu }^{\ast },{\lambda }^{\ast },v^{\ast },w^{\ast },s^{\ast },t^{\ast },{\tilde{y}}^{\ast },p^{\ast }=0)$ is a feasible solution of (ND) and problems (NP) and(ND) have the same objectives values and

$${\lambda }^{\ast }=\frac{f(x^{\ast },{\overline{y}}_i^{\ast })+{({x^{\ast }}^{T}B{x}^{\ast })}^{1/2}}{g(x^{\ast },{\overline{y}}_i^{\ast })-{({x^{\ast }}^{T}C{x}^{\ast })}^{1/2}}.$$

 □

Theorem 3.3 (Strict converse duality)

Let$x^{\ast }$and$(z^{\ast },{\mu }^{\ast },{\lambda }^{\ast },s^{\ast },t^{\ast },v^{\ast },w^{\ast },{\tilde{y}}^{\ast },p^{\ast })$be the optimal solutions of (NP) and (ND), respectively.Suppose that

$$\begin{array}{r}[\sum _{i=1}^s{t}_i^{\ast }\{f(\cdot ,{\overline{y}}_i^{\ast })+{(\cdot )}^{T}B{w}^{\ast }-\lambda (g(\cdot ,{\overline{y}}_i^{\ast })-{(\cdot )}^{T}C{v}^{\ast })\}+\sum _{j\in J_0}{\mu }_j^{\ast }{h}_j(\cdot ),\\ \sum _{j\in j_{\beta }}{\mu }_j^{\ast }{h}_j(\cdot ),\beta =1,2,\dots ,r]\end{array}$$

is higher-order$(\mathcal{F},\alpha ,\rho ,d)$-strictly pseudoquasi-type I at$z^{\ast }$with

$$\frac{{\rho }_1^1}{{\alpha }^1(x^{\ast },z^{\ast })}+\sum _{\beta =1}^r\frac{{\rho }_{\beta }^2}{{\alpha }^2(x^{\ast },z^{\ast })}\ge 0,$$

and that$\mathrm{\nabla }{h}_j(x^{\ast })$, $j\in J(x^{\ast })$are linearly independent. Then$z^{\ast }=x^{\ast }$; that is, $z^{\ast }$is an optimal solution of (NP).

Proof We assume that $z^{\ast }\ne x^{\ast }$ and reach a contradiction. From the strong dualitytheorem (Theorem 3.2), it follows that

$$\underset{y\in Y}{sup}\frac{f(x^{\ast },{\tilde{y}}^{\ast })+{(x^{\ast T}B{x}^{\ast })}^{1/2}}{g(x^{\ast },{\tilde{y}}^{\ast })-{(x^{\ast T}C{x}^{\ast })}^{1/2}}={\lambda }^{\ast }.$$

(3.10)

Now, proceeding as in Theorem 3.1, we get

$$\sum _{i=1}^s{t}_i^{\ast }[f(x^{\ast },{\overline{y}}_i^{\ast })+x^{\ast T}B{w}^{\ast }-{\lambda }^{\ast }(g(x^{\ast },{\overline{y}}_i^{\ast })-x^{\ast T}C{v}^{\ast })]+\sum _{j\in J_0}{\mu }_j^{\ast }{h}_j\left(x^{\ast }\right)<0.$$

(3.11)

The feasibility of $x^{\ast }$ for (NP), ${\mu }^{\ast }\ge 0$ and (3.3) imply

$$\sum _{j\in J_{\beta }}{\mu }_j^{\ast }{h}_j\left(x^{\ast }\right)\le 0\le \sum _{j\in J_{\beta }}{\mu }_j^{\ast }[h_j\left(z^{\ast }\right)+H_j(z^{\ast },p^{\ast })-p^{\ast T}{\mathrm{\nabla }}_p{H}_j(z^{\ast },p^{\ast })],$$

which along with the second part of higher-order $(\mathcal{F},\alpha ,\rho ,d)$-strictly pseudoquasi-type I assumption on

$$\begin{array}{r}[\sum _{i=1}^s{t}_i^{\ast }\{f(\cdot ,{\overline{y}}_i^{\ast })+{(\cdot )}^{T}B{w}^{\ast }-\lambda (g(\cdot ,{\overline{y}}_i^{\ast })-{(\cdot )}^{T}C{v}^{\ast })\}+\sum _{j\in J_0}{\mu }_j^{\ast }{h}_j(\cdot ),\\ \sum _{j\in j_{\beta }}{\mu }_j^{\ast }{h}_j(\cdot ),\beta =1,2,\dots ,r]\end{array}$$

at $z^{\ast }$ gives

$$\mathcal{F}(x^{\ast },z^{\ast };{\alpha }^2(x^{\ast },z^{\ast })\sum _{j\in j_{\beta }}{\mu }_j^{\ast }{\mathrm{\nabla }}_p{H}_j(z^{\ast },p^{\ast }))<-{\rho }_{\beta }^2{d}^2(x^{\ast },z^{\ast }),\beta =1,2,\dots ,r.$$

As ${\alpha }^2(x^{\ast },z^{\ast })>0$ and as ℱ is sublinear, it follows that

$$\mathcal{F}(x^{\ast },z^{\ast };\sum _{j\in j_{\beta }}{\mu }_j^{\ast }{\mathrm{\nabla }}_p{H}_j(z^{\ast },p^{\ast }))<-\frac{{\rho }_{\beta }^2}{{\alpha }^2(x^{\ast },z^{\ast })}{d}^2(x^{\ast },z^{\ast }),\beta =1,2,\dots ,r.$$

(3.12)

From (3.1), (3.12) and the sublinearity of ℱ, we have

$$\begin{array}{r}\mathcal{F}(x^{\ast },z^{\ast };\sum _{i=1}^s{t}_i^{\ast }\left[{\mathrm{\nabla }}_p(F(z^{\ast },{\overline{y}}_i^{\ast },p^{\ast })-\lambda G(z^{\ast },{\overline{y}}_i^{\ast },p^{\ast }))\right]+B{w}^{\ast }+{\lambda }^{\ast }C{v}^{\ast }\\ +\sum _{j\in j_0}{\mu }_j^{\ast }{\mathrm{\nabla }}_p{H}_j(z^{\ast },p^{\ast }))\ge \frac{{\sum }_{\beta =1}^r{\rho }_{\beta }^2}{{\alpha }^2(x^{\ast },z^{\ast })}{d}^2(x^{\ast },z^{\ast }).\end{array}$$

In view of $(\frac{{\rho }_1^1}{{\alpha }^1(x^{\ast },z^{\ast })}+\frac{{\sum }_{\beta =1}^r{\rho }_{\beta }^2}{{\alpha }^2(x^{\ast },z^{\ast })})\ge 0$, ${\alpha }^1(x^{\ast },z^{\ast })>0$ and the sublinearity of ℱ, the above inequalitybecomes

$$\begin{array}{r}\mathcal{F}(x^{\ast },z^{\ast };{\alpha }^1(x^{\ast },z^{\ast })\sum _{i=1}^s{t}_i^{\ast }\left[{\mathrm{\nabla }}_p(F(z^{\ast },{\overline{y}}_i^{\ast },p^{\ast })-\lambda G(z^{\ast },{\overline{y}}_i^{\ast },p^{\ast }))\right]+B{w}^{\ast }+{\lambda }^{\ast }C{v}^{\ast }\\ +\sum _{j\in j_0}{\mu }_j^{\ast }{\mathrm{\nabla }}_p{H}_j(z^{\ast },p^{\ast }))\ge -{\rho }_1^1{d}^2(x^{\ast },z^{\ast }).\end{array}$$

By using the first part of the said assumption imposed on

$$\begin{array}{r}[\sum _{i=1}^s{t}_i^{\ast }\{f(\cdot ,{\overline{y}}_i^{\ast })+{(\cdot )}^{T}B{w}^{\ast }-{\lambda }^{\ast }(g(\cdot ,{\overline{y}}_i^{\ast })-{(\cdot )}^{T}C{v}^{\ast })\}+\sum _{j\in J_0}{\mu }_j^{\ast }{h}_j(\cdot ),\\ \sum _{j\in j_{\beta }}{\mu }_j^{\ast }{h}_j(\cdot ),\beta =1,2,\dots ,r]\end{array}$$

at $z^{\ast }$, it follows that

$$\begin{array}{r}\sum _{i=1}^s{t}_i^{\ast }[f(x^{\ast },{\overline{y}}_i^{\ast })+x^{\ast T}B{w}^{\ast }-{\lambda }^{\ast }(g(x^{\ast },{\overline{y}}_i^{\ast })-x^{\ast T}C{v}^{\ast })]+\sum _{j\in J_0}{\mu }_j^{\ast }{h}_j\left(x^{\ast }\right)\\ >\sum _{i=1}^s{t}_i^{\ast }[f(z^{\ast },{\overline{y}}_i^{\ast })+z^{\ast T}B{w}^{\ast }-{\lambda }^{\ast }(g(z^{\ast },{\overline{y}}_i^{\ast })-z^{\ast T}C{v}^{\ast })+F(z^{\ast },{\overline{y}}_i^{\ast },p^{\ast })\\ -\lambda G(z^{\ast },{\overline{y}}_i^{\ast },p^{\ast })+\sum _{j\in J_0}{\mu }_j^{\ast }{h}_j\left(z^{\ast }\right)-p^{\ast T}{\mathrm{\nabla }}_p\{F(z^{\ast },{\overline{y}}_i^{\ast },p^{\ast })-{\lambda }^{\ast }G(z^{\ast },{\overline{y}}_i^{\ast },p^{\ast })\}]\\ +\sum _{j\in J_0}{\mu }_j^{\ast }{H}_j(z^{\ast },p^{\ast })-p^{\ast T}\sum _{j\in J_0}{\mu }_j^{\ast }{\mathrm{\nabla }}_p{H}_j(z^{\ast },p^{\ast })\\ \ge 0\text{(by (3.2))},\end{array}$$

which contradicts (3.11). Hence the result. □

4 Special cases

Let $J_0=\mathrm{\varnothing }$, $F(z,{\overline{y}}_i,p)=p^T\mathrm{\nabla }f(z,{\overline{y}}_i)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^{2}f(z,{\overline{y}}_i)$, $G(z,{\overline{y}}_i,p)=p^T\mathrm{\nabla }g(z,{\overline{y}}_i)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^{2}g(z,{\overline{y}}_i)$, $i=1,2,\dots ,s$ and $H_j(z,p)=p^T\mathrm{\nabla }{h}_j(z)+\frac{1}{2}{p}^T{\mathrm{\nabla }}^2{h}_j(z)p$, $j=1,2,\dots ,m$. Then (ND) becomes the second-order dual studied in [17, 22]. If, in addition, $p=0$, then we obtain the dual formulated by Ahmad etal.[20].

5 Conclusion

The notion of higher-order $(\mathcal{F},\alpha ,\rho ,d)$-pseudoquasi-type I is adopted, which includes manyother generalized convexity concepts in mathematical programming as special cases.This concept is appropriate to discuss the weak, strong and strict converse dualitytheorems for a higher-order dual (ND) of a non-differentiable minimax fractionalprogramming problem (NP). The results of this paper can be discussed by formulatinga unified higher-order dual involving support functions on the lines of Ahmad [23].