Optimality and Duality for Multi-objective Fractional Programming in Complex Spaces

Abstract

In this paper, we consider a complex multi-objective fractional programming problem (CMFP). We establish the necessary optimality conditions of problem (CMFP) in the sense of Pareto optimality and derive its sufficient optimality conditions using generalized convexity. Finally, we construct the parametric dual problem to the primal problem (CMFP) and their duality theorems.

1 Introduction

In complex programming problems, the earliest study seems to be the paper published by Levinson [14] in 1966 for the case of complex linear programming. Since then, nonlinear complex programming for fractional or non-fractional was treated by numerous authors (Refs. [2, 6, 18]). Recently, authors considered some kind of minimax programming problems in complex spaces (Refs. [8, 11,12,13, 16, 17]); they derived the optimality conditions and established various dual problems so as to obtain their duality theorems.

Multi-objective fractional programming in real spaces has been of much interest in the recent past (Refs. [1, 7, 9, 10, 15, 21]). The basic idea in these methods has used some kind of transformation so as to obtain an equivalent single-objective program (Ref. [19]). The multi-objective programming in complex spaces is useful in engineering, economics and physics. In order to solve the multi-objective programming problems, the efficient points were introduced and characterized. For instance, Duca [3] formulated the linear vector-valued optimization problem in complex spaces, and the interested reader can consult papers: Duca [4, 5] and Stancu-Minasian et al. [20].

In this paper, we are interesting in a complex multi-objective fractional programming as follows:

$$\begin{aligned} \text{(CMFP) }&\min&\left( \frac{\mathrm{Re~} f_1({\mathbf {z}})}{\mathrm{Re~} g_1({\mathbf {z}}) }, \dots , \frac{\mathrm{Re~} f_m({\mathbf {z}})}{\mathrm{Re~} g_m({\mathbf {z}}) } \right) \\&\text{ subject } \text{ to }&{\mathbf {z}}\in X= \left\{ {\mathbf {z}}=(z,{\overline{z}})\in {\mathbb {C}}^{2n} ~|~ -h({\mathbf {z}})\in S \right\} , \end{aligned}$$

where S is a polyhedral cone in ${\mathbb {C}}^p$; for $i=1,\dots ,m$, $f_i(\cdot ),g_i(\cdot ):{\mathbb {C}}^{2n} \rightarrow {\mathbb {C}}$ and $h(\cdot ):{\mathbb {C}}^{2n} \rightarrow {\mathbb {C}}^p$ are analytic functions defined on $Q\subset {\mathbb {C}}^{2n}$, and $Q=\{ {\mathbf {z}}=(z,{\overline{z}}) ~|~ z\in {\mathbb {C}}^n \}\subset {\mathbb {C}}^{2n}$ is a linear manifold over real field. Without loss of generality, we may assume that $\mathrm{Re~} f_i({\mathbf {z}})\ge 0$, $\mathrm{Re~} g_i({\mathbf {z}})>0$ for all ${\mathbf {z}}\in X$, $i= 1,2,\ldots ,m$. Throughout the paper, we need the analytic function $\phi ( {\mathbf {z}})$ that is defined on the set $Q\subset {\mathbb {C}}^{2n}$ since a nonlinear analytic function $\phi (z)$ that is defined on ${\mathbb {C}}^{n}$ cannot have a convex real part in our requirement (Ref. [6]).

The paper is organized as follows. We will recall some notations and preliminaries in Sect. 2. In Sect. 3 and Sect. 4, we will derive the necessary and sufficient optimality conditions of the primal problem (CMFP) under some generalized convexity. Finally, we will establish the parametric dual problem of the primal problem (CMFP) and obtain their duality theorems.

2 Notations and Preliminary

Let ${\mathbb {R}}^n$ be an Euclidean space, and ${\mathbb {R}}^n_+$ be its nonnegative orthant. The following convention for vectors ${\mathbf {u}}=(u_1, \dots , u_n), {\mathbf {v}}=(v_1, \dots , v_n)\in {\mathbb {R}}^n$ will be adopted:

• (1) ${\mathbf {u}} = {\mathbf {v}}$ if and only if $u_i=v_i$, for all $i=1,\dots , n$.

• (2) ${\mathbf {u}} < {\mathbf {v}}$ if and only if $u_i< v_i$, for all $i=1,\dots , n$.

• (3) ${\mathbf {u}} \le {\mathbf {v}}$ if and only if $u_i \le v_i$, for all $i=1,\dots , n$.

• (4) ${\mathbf {u}} \leqq {\mathbf {v}}$ if and only if ${\mathbf {u}} \le {\mathbf {v}}$ and $u_i \ne v_i$ for some $i\in \{ 1,\dots , n\}$.

Given a general multi-objective programming (MP) as the following:

$$\begin{aligned} \begin{array}{rl} \mathrm {(MP)~ } \min &{} \phi ({\mathbf {x}})= (\phi _1({\mathbf {x}}),\phi _{2}({\mathbf {x}}), \dots , \phi _m({\mathbf {x}})) \\ s.t. &{} {\mathbf {x}}\in X, \end{array} \end{aligned}$$

where X is the feasible set and $\phi : X \rightarrow {\mathbb {R}}^m$ is the multi-function. The optimality of the multi-objective minimization problem (MP) is given in the sense of Pareto optimality, which is defined as follows.

Definition 1

The point ${\mathbf {x}}_0\in X$ is said to be an efficient solution (or Pareto point) of problem (MP) if there does not exist other ${\mathbf {x}}\in X$ such that

$$\begin{aligned} \phi ({\mathbf {x}}) \leqq \phi ({\mathbf {x}}_0). \end{aligned}$$

We recall some notations in complex spaces. Given $z\in {\mathbb {C}}^p$, the notations ${\overline{z}}$, $z^T$ and $z^H$ are the conjugate, transpose and conjugate transpose of z. Let $S=\{ z\in {\mathbb {C}}^p ~|~ \mathrm{Re}(K z) \ge 0 \}\subset {\mathbb {C}}^p$ be a polyhedral cone with matrix $K\in {\mathbb {C}}^{k\times p}$ where k is a positive integer. The dual cone $S^*$ of the convex cone S is defined by

$$\begin{aligned} S^*=\{ \eta \in {\mathbb {C}}^p ~|~ \mathrm{Re} \langle z,\eta \rangle \ge 0 \text{ for } z \in S\}, \end{aligned}$$

where $\langle z,\eta \rangle = \eta ^H z$ stands for the inner product in complex spaces. For $s_0\in S,$ the set $S(s_0)$ is the intersection of those closed half spaces that includes $s_0$ in their boundaries. Thus, if $s_0\in \text{ int }(S)$, $S(s_0)$ is the whole space ${\mathbb {C}}^p$.

Lat ${\mathbf {z}}=(z,{\overline{z}})\in {\mathbb {C}}^{2n}$ and analytic function $f:{\mathbb {C}}^{2n}\rightarrow {\mathbb {C}}$, the gradient expression $\nabla f({\mathbf {z}})$ is denoted by

$$\begin{aligned} \nabla f({\mathbf {z}})= \Big ( \nabla _z f({\mathbf {z}}), \nabla _{{\overline{z}}} f({\mathbf {z}}) \Big ) \in {\mathbb {C}}^{2n} \end{aligned}$$

with $\nabla _z f({\mathbf {z}}) = \Big ( f_{z_1}({\mathbf {z}}),\ldots ,f_{z_n}({\mathbf {z}}) \Big ) \in {\mathbb {C}}^{n}, \; \; \nabla _{{\overline{z}}} f({\mathbf {z}}) = \Big ( f_{\overline{z_1} }({\mathbf {z}}), \ldots , f_{\overline{z_n} }({\mathbf {z}}) \Big ) \in {\mathbb {C}}^{n}.$

We express the differential form of a complex function by using the gradient representations as the following lemma.

Lemma 1

Given ${\mathbf {z}}=(z,{\overline{z}}), {\mathbf {z}}_0=(z_0,\overline{z_0})\in Q \subset {\mathbb {C}}^{2n}$ and $v=z-z_0\in {\mathbb {C}}^n$.

Suppose that $f(\cdot ): {\mathbb {C}}^{2n} \rightarrow {\mathbb {C}}$, $h(\cdot ): {\mathbb {C}}^{2n} \rightarrow {\mathbb {C}}^p$, and $\tau \in {\mathbb {C}}^p$. If $\varPhi ({\mathbf {z}})= f({\mathbf {z}})+ \langle h({\mathbf {z}}),\tau \rangle$, then $\varPhi$ is differentiable at ${\mathbf {z}}_0=(z_0,\overline{z_0}) \in {\mathbb {C}}^{2n}$ and

$$\begin{aligned} \mathrm{Re} [ \varPhi '({\mathbf {z}}_0)({\mathbf {z}}-{\mathbf {z}}_0)]= \mathrm{Re} \Big \langle ~v,~ \overline{\nabla _z f({\mathbf {z}}_0)} + \nabla _{{\overline{z}}}f({\mathbf {z}}_0)+ \tau ^T \overline{\nabla _z h({\mathbf {z}}_0)} +\tau ^H \nabla _{{\overline{z}}}h({\mathbf {z}}_0)~ \Big \rangle . \end{aligned}$$

For the proof of Lemma 1, one can refer to Lai and Huang [11, Lemma 2].

3 Necessary Optimality Conditions

Observing that if ${\mathbf {z}}_0\in {\mathbb {C}}^n$ is an efficient solution of the primal problem (CMFP) with optimal value ${\mathbf {k}}=(k_1, \dots , k_m) \in {\mathbb {R}}^m$, there does not exist other ${\mathbf {z}}\in X$ such that

$$\begin{aligned} \frac{\mathrm{Re~} f_i({\mathbf {z}})}{\mathrm{Re~} g_i({\mathbf {z}}) } \le \frac{\mathrm{Re~} f_i({\mathbf {z}}_0)}{\mathrm{Re~} g_i({\mathbf {z}}_0) }=k_i \hbox { ~for all}\ i= 1 , \dots , m \end{aligned}$$

and

$$\begin{aligned} \frac{\mathrm{Re~} f_j({\mathbf {z}})}{\mathrm{Re~} g_j({\mathbf {z}}) } \ne \frac{\mathrm{Re~} f_j({\mathbf {z}}_0)}{\mathrm{Re~} g_j({\mathbf {z}}_0) }=k_j ~ \hbox { for some}\ j\in \{1, 2, \dots , m\}. \end{aligned}$$

By the above inequalities, we could give the single-objective problems (CFP$_i$) and lemma as follows; the proof is similar to Kanniappan [19, Lemma 3.1].

Lemma 2

${\mathbf {z}}_0\in {\mathbb {C}}^n$ is an efficient solution of the primal problem (CMFP) if and only if ${\mathbf {z}}_0$ solves (CFP$_i$), for all $i=1, \dots , m$. The problem (CFP$_i$) is defined by

$$\begin{aligned} \text{(CFP }_i)&\min&\frac{\mathrm{Re~} f_i({\mathbf {z}})}{\mathrm{Re~} g_i({\mathbf {z}}) } \\&\text{ s.t. }&{\mathbf {z}}\in M_i = \left\{ {\mathbf {z}}\in X ~\left| ~ \begin{array}{l} \frac{\mathrm{Re~} f_j({\mathbf {z}})}{\mathrm{Re~} g_j({\mathbf {z}}) } \le k_j, j=1, \dots , m, \text{ with } j \ne i \\ \text{ and } -h({\mathbf {z}})\in S \end{array} \right. \right\} \\&\quad = \left\{ {\mathbf {z}}\in X ~\left| ~ \begin{array}{l} \mathrm{Re~}[ f_j({\mathbf {z}}) - k_j g_j({\mathbf {z}}) ] \le 0, j=1, \dots , m,\\ \text{ with } j \ne i \text{ and } -h({\mathbf {z}})\in S \end{array} \right. \right\} , \end{aligned}$$

where ${\mathbf {k}}=(k_1, \dots , k_m)= \left( \frac{\mathrm{Re~} f_1({\mathbf {z}}_0)}{\mathrm{Re~} g_1({\mathbf {z}}_0) }, \dots , \frac{\mathrm{Re~} f_m({\mathbf {z}}_0)}{\mathrm{Re~} g_m({\mathbf {z}}_0) } \right) \in {\mathbb {R}}^m$, and S is a polyhedral cone in ${\mathbb {C}}^p$.

Remark that primal problem (CMFP) is said to have constraint qualification at ${\mathbf {z}}_0$, if for any nonzero $\mu \in S^*\subset {\mathbb {C}}^p$, such that

$$\begin{aligned} \mu ^T \overline{\nabla _z h({\mathbf {z}}_0)}+\mu ^H \nabla _{{\overline{z}}}h({\mathbf {z}}_0)\ne 0 ~ \text{ whenever } \mu \ne 0 \text{ in } S^*. \end{aligned}$$

Now, we drive the following necessary optimality conditions for the primal problem (CMFP).

Theorem 1

(Necessary optimality conditions) Suppose that ${\mathbf {z}}_0$ is an efficient solution of the primal problem (CMFP) with optimal value ${\mathbf {k}}=(k_1, \dots , k_m)\in {\mathbb {R}}^m$, and let the primal problem (CMFP) possess the constraint qualification at ${\mathbf {z}}_0$. Then, there are $\lambda _i \ge 0$ for $i=1, \dots , m$ and $\mu \in S^* \subset {\mathbb {C}}^p$ satisfied the following conditions.

$$\begin{aligned}&\sum _{i=1}^m \lambda _i [ (\overline{\nabla _z f_i({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}f_i({\mathbf {z}}_0))- k_i ( \overline{\nabla _z g_i({\mathbf {z}}_0)}+ \nabla _{{\overline{z}}}g_i({\mathbf {z}}_0))]\nonumber \\&\quad +\,\mu ^T \overline{\nabla _z h({\mathbf {z}}_0)}+\mu ^H \nabla _{{\overline{z}}}h({\mathbf {z}}_0)=0, \end{aligned}$$

(1)

$$\begin{aligned}&\mathrm{Re}~ \mu ^H h({\mathbf {z}}_0)=0. \end{aligned}$$

(2)

Proof

Let ${\mathbf {z}}_0$ be an efficient solution of the primal problem (CMFP) with optimal value ${\mathbf {k}}=(k_1, \dots , k_m)\in {\mathbb {R}}^m$. By Lemma 2, ${\mathbf {z}}_0$ is an optimal solution of problems (CFP$_i$), $i=1, \dots , m$.

Without loss of generality, the problem (CFP$_1$) is the form:

$$\begin{aligned}&\min&\frac{\mathrm{Re~} f_1({\mathbf {z}})}{\mathrm{Re~} g_1({\mathbf {z}}) } \\&\text{ s.t. }&{\mathbf {z}}\in M_1 = \left\{ {\mathbf {z}}\in X ~\left| ~ \begin{array}{l} \mathrm{Re~}[ f_j({\mathbf {z}}) - k_j g_j({\mathbf {z}}) ] \le 0, j=2, \dots , m \\ \text{ and } -h({\mathbf {z}})\in S \end{array} \right. \right\} . \end{aligned}$$

We could write out the Kuhn–Tucker-type optimality theorem as follows. If ${\mathbf {z}}_0$ is an optimal solution of problems (CFP$_1$), then there exist ${\overline{\lambda }} \ge 0$, $\lambda _i \ge 0$ for $i=2, \dots , m$, and $\mu \in S^* \subset {\mathbb {C}}^p$ such that the Lagrangian

$$\begin{aligned} \varPsi ({\mathbf {z}})= {\overline{\lambda }}\cdot \left( \frac{\mathrm{Re~} f_1({\mathbf {z}})}{\mathrm{Re~} g_1({\mathbf {z}})} \right) +\sum _{i=2}^m \lambda _i \cdot \mathrm{Re~}[f_i ({\mathbf {z}})- k_i g_i({\mathbf {z}}) ]+ \mathrm{Re}[\mu ^H h({\mathbf {z}})] \end{aligned}$$

satisfies the Kuhn–Tucker-type condition at ${\mathbf {z}}_0$. That is

$$\begin{aligned}&\mathrm{Re}~[\varPsi ' ({\mathbf {z}}_0) ({\mathbf {z}} - {\mathbf {z}}_0)] =0, \end{aligned}$$

(3)

$$\begin{aligned}&\lambda _i\mathrm{Re~}[f_i ({\mathbf {z}}_0))-k_i g_i({\mathbf {z}}_0)]=0,~~~ i= 2, 3, \dots ,m, \end{aligned}$$

(4)

$$\begin{aligned}&\mathrm{Re}~ \mu ^H h({\mathbf {z}}_0)=0. \end{aligned}$$

(5)

By Lemma 1, the condition (3) becomes

$$\begin{aligned}&\frac{{\overline{\lambda }}}{\mathrm{Re~}g_1({\mathbf {z}}_0)} [ (\overline{\nabla _z f_1({\mathbf {z}}_0)}+ \nabla _{{\overline{z}}}f_1({\mathbf {z}}_0))- k_1 ( \overline{\nabla _z g_1({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}g_i({\mathbf {z}}_0))] \nonumber \\&+\, \sum _{i=2}^m \lambda _i [ (\overline{\nabla _z f_i({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}f_i({\mathbf {z}}_0))- k_i ( \overline{\nabla _z g_i({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}g_i({\mathbf {z}}_0))] \nonumber \\&+\,\mu ^T \overline{\nabla _z h({\mathbf {z}}_0)}+\mu ^H \nabla _{{\overline{z}}}h({\mathbf {z}}_0)=0. \end{aligned}$$

Hence, setting $\lambda _1= \displaystyle \frac{{\overline{\lambda }}}{ \mathrm{Re~}g_1({\mathbf {z}}_0)}$, we obtain

$$\begin{aligned}&\displaystyle \sum _{i=1}^m \lambda _i [ (\overline{\nabla _z f_i({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}f_i({\mathbf {z}}_0))- k_i ( \overline{\nabla _z g_i({\mathbf {z}}_0)}+ \nabla _{{\overline{z}}}g_i({\mathbf {z}}_0))]\nonumber \\&\quad +\,\mu ^T \overline{\nabla _z h({\mathbf {z}}_0)}+\mu ^H \nabla _{{\overline{z}}}h({\mathbf {z}}_0)=0. \end{aligned}$$

Thus, the result is proved. $\square$

4 Sufficient Optimality Conditions

In order to get the sufficient optimality conditions and duality theorems, we need some generalizations of convexity as follows.

Definition 2

(Lai and Huang [12]) The real part of an analytic function $f(\cdot )$ from ${\mathbb {C}}^{2n}$ to ${\mathbb {R}}$ is called, respectively,

• (i) convex (strictly) at ${\mathbf {z}}_0\in Q\subset {\mathbb {C}}^{2n}$ if $\begin{array}{lcl} \mathrm{Re~} \big [ f({\mathbf {z}})-f({\mathbf {z}}_0) \big ]&\ge (>)&\mathrm{Re~} \big [ f'({\mathbf {z}}_0)({\mathbf {z}}-{\mathbf {z}}_0) \big ], \end{array}$

• (ii) pseudoconvex (strictly) at ${\mathbf {z}}_0\in Q$ if $\begin{array}{lc} \mathrm{Re~} \big [ f'({\mathbf {z}}_0)({\mathbf {z}}-{\mathbf {z}}_0) \big ] \ge 0 \Rightarrow \mathrm{Re~} \big [ f({\mathbf {z}})-f({\mathbf {z}}_0) \big ]&\ge (>) 0, \end{array}$

• (iii) quasiconvex at ${\mathbf {z}}_0\in Q$ if $\mathrm{Re~} \big [ f({\mathbf {z}})-f({\mathbf {z}}_0) \big ] \le 0 \Rightarrow \mathrm{Re~} \big [ f'({\mathbf {z}}_0)({\mathbf {z}}-{\mathbf {z}}_0) \big ] \le 0.$

Definition 3

(Lai and Huang [12]) An analytic mapping $h(\cdot ):{\mathbb {C}}^{2n}\rightarrow {\mathbb {C}}^p$ is called, respectively,

• (i) convex at ${\mathbf {z}}_0\in Q$ with respect to (w.r.t.) a polyhedral cone S in ${\mathbb {C}}^p$ if there is a nonzero $\mu \in S^* \big ( \subset {\mathbb {C}}^p\big ),$ the dual cone of S, such that $\mathrm{Re~} \langle h({\mathbf {z}})-h({\mathbf {z}}_0), \mu \rangle \ge \mathrm{Re~} \langle h'({\mathbf {z}}_0)({\mathbf {z}}-{\mathbf {z}}_0),\mu \rangle$.

• (ii) pseudoconvex (strictly) at ${\mathbf {z}}_0\in Q$ w.r.t. S if there is a nonzero $\mu \in S^* \big ( \subset {\mathbb {C}}^p\big )$ the dual cone of S, such that $\begin{array}{lc} \mathrm{Re~} \langle h'({\mathbf {z}}_0)({\mathbf {z}}-{\mathbf {z}}_0),\mu \rangle \ge 0 \Rightarrow \mathrm{Re~} \langle h({\mathbf {z}})-h({\mathbf {z}}_0), \mu \rangle&\ge (>) 0, \end{array}$

• (iii) quasiconvex at ${\mathbf {z}}_0\in Q$ w.r.t. S if there is a nonzero $\mu \in S^* \big ( \subset {\mathbb {C}}^p\big )$ such that $\mathrm{Re~} \langle h({\mathbf {z}})-h({\mathbf {z}}_0), \mu \rangle \le 0 \Rightarrow \mathrm{Re~} \langle h'({\mathbf {z}}_0)({\mathbf {z}}-{\mathbf {z}}_0),\mu \rangle \le 0$.

The following is the sufficient optimality conditions for the primal problem (CMFP) under generalized convexity.

Theorem 2

(Sufficient optimality conditions) Let S be a polyhedral cone in ${\mathbb {C}}^p$, $f_i(\cdot ),~g_i(\cdot ): {\mathbb {C}}^{2n} \rightarrow {\mathbb {C}}$, $i=1, \dots , m$, and $h(\cdot ): {\mathbb {C}}^{2n} \rightarrow {\mathbb {C}}^p$ be analytic on $X \subseteq Q$. Suppose that ${\mathbf {z}}_0$ is a feasible solution of the primal problem (CMFP), and there exist $\lambda _i \ge 0$, $k_i\ge 0$, for $i=1, \dots , m$, $\mu \in S^* \subset {\mathbb {C}}^p$ satisfying conditions (1) and (2) in Theorem 1. Assume that any one of the following conditions (i)–(iv) holds:

• (i) one of $\mathrm{Re}~ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]$ and $\mathrm{Re} [\mu ^H h(\cdot ) ]$ is strictly convex and another is convex at ${\mathbf {z}}_0\in Q$, or both are strictly convex at ${\mathbf {z}}_0\in Q$,

• (ii) $\mathrm{Re}~ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]$ is quasiconvex at ${\mathbf {z}}_0\in Q$ and $\mathrm{Re}~ [\mu ^H h(\cdot )]$ is strictly pseudoconvex at ${\mathbf {z}}_0\in Q$,

• (iii) $\mathrm{Re}~ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]$ is strictly pseudoconvex at ${\mathbf {z}}_0\in Q$ and $\mathrm{Re}~ [\mu ^H h(\cdot )]$ is quasiconvex at ${\mathbf {z}}_0\in Q$,

• (iv) $\mathrm{Re} \left\{ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]+ \mu ^H h(\cdot ) \right\}$ is strictly pseudoconvex at ${\mathbf {z}}_0\in Q$.

Then, ${\mathbf {z}}_0$ is an efficient solution of the primal problem (CMFP).

Proof

Suppose that ${\mathbf {z}}_0=(z_0, \overline{z_0})$ is not an efficient solution of the primal problem (CMFP). Then, there is a feasible solution ${\mathbf {z}}=(z, {\overline{z}})$ with ${\mathbf {z}}\ne {\mathbf {z}}_0$ such that

$$\begin{aligned} \frac{\mathrm{Re~} f_i({\mathbf {z}})}{\mathrm{Re~} g_i({\mathbf {z}}) } \le \frac{\mathrm{Re~} f_i({\mathbf {z}}_0)}{\mathrm{Re~} g_i({\mathbf {z}}_0) }=k_i \hbox { ~for all}\ i= 1 , \dots , m \end{aligned}$$

and

$$\begin{aligned} \frac{\mathrm{Re~} f_j({\mathbf {z}})}{\mathrm{Re~} g_j({\mathbf {z}}) } \ne \frac{\mathrm{Re~} f_j({\mathbf {z}}_0)}{\mathrm{Re~} g_j({\mathbf {z}}_0) }=k_j ~ \hbox { for some}\ j\in \{1, 2, \dots , m\}. \end{aligned}$$

Without loss of generality, we could let

$$\begin{aligned} \mathrm{Re~ }[f_1({\mathbf {z}}) - k_1 g_1({\mathbf {z}})] <0, \text{ and } \mathrm{Re~ } [f_i({\mathbf {z}}) - k_i g_i({\mathbf {z}})] \le 0, ~ \text{ for } i=2, \dots ,m. \end{aligned}$$

From $f_i({\mathbf {z}}_0) - k_i g_i({\mathbf {z}}_0)=0$ and $\lambda _i \ge 0$ for $i=1, \dots , m$, we get

$$\begin{aligned} \mathrm{Re~ } \sum _{i=1}^m \lambda _i [f_i({\mathbf {z}}) - k_i g_i({\mathbf {z}})] \le 0= \mathrm{Re~ } \sum _{i=1}^m \lambda _i [f_i({\mathbf {z}}_0) - k_i g_i({\mathbf {z}}_0)]. \end{aligned}$$

That is

$$\begin{aligned} \mathrm{Re~ } \displaystyle \sum _{i=1}^m \lambda _i \left[ \big (f_i({\mathbf {z}})-f_i({\mathbf {z}}_0)\big ) - k_i \big (g_i({\mathbf {z}})-g_i({\mathbf {z}}_0)\big )\right] \le 0. \end{aligned}$$

(6)

With the feasibility of ${\mathbf {z}}$ for the primal problem (CMFP), that is $\mathrm{Re ~}\mu ^H h({\mathbf {z}}) \le 0$ for $\mu \in S^*$. Since condition (2) holds, $\mathrm{Re ~}\mu ^H h({\mathbf {z}}_0) =0$. We obtain

$$\begin{aligned} \mathrm{Re ~}\mu ^H[ h({\mathbf {z}})- h({\mathbf {z}}_0)] \le 0. \end{aligned}$$

(7)

• (a) If hypothesis (i) holds, without loss of generality, assume that $\mathrm{Re}\sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]$ is strictly convex and $\mathrm{Re} [\mu ^H h(\cdot ) ]$ is convex at ${\mathbf {z}}_0\in Q$. From conditions (6), (7) and definition of general convexity, we get

  $$\begin{aligned} \mathrm{Re}\sum _{i=1}^m \lambda _i[ f'_i({\mathbf {z}}_0)- k_i g'_i({\mathbf {z}}_0)]({\mathbf {z}}-{\mathbf {z}}_0) <0 \text{ and } \mathrm{Re~} \mu ^H[ h'({\mathbf {z}}_0)({\mathbf {z}}-{\mathbf {z}}_0)] \le 0. \end{aligned}$$

  Using Lemma 2, the above inequalities will be

  $$\begin{aligned}&\displaystyle \sum _{i=1}^m \lambda _i [ (\overline{\nabla _z f_i({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}f_i({\mathbf {z}}_0))- k_i ( \overline{\nabla _z g_i({\mathbf {z}}_0)} +\nabla _{{\overline{z}}}g_i({\mathbf {z}}_0))]\nonumber \\&\quad +\,\mu ^T \overline{\nabla _z h({\mathbf {z}}_0)}+\mu ^H \nabla _{{\overline{z}}}h({\mathbf {z}}_0)<0. \end{aligned}$$

  (8)

  This contradicts the equality of (3).

• (b) If hypothesis (ii) holds, $\mathrm{Re}~ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]$ is quasiconvex at ${\mathbf {z}}_0$, and from inequality (6), it implies that

  $$\begin{aligned} \mathrm{Re} \langle v, \sum _{i=1}^m \lambda _i [ (\overline{\nabla _z f_i({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}f_i({\mathbf {z}}_0))- k_i ( \overline{\nabla _z g_i({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}g_i({\mathbf {z}}_0))] \rangle \le 0. \end{aligned}$$

  (9)

  Since $\mathrm{Re}[ \mu ^H g(\cdot )]$ is strictly pseudoconvex at ${\mathbf {z}}_0$ with inequality (7), we obtain

  $$\begin{aligned} \mathrm{Re} \langle v, \mu ^T \overline{\nabla _z h({\mathbf {z}}_0)}+\mu ^H \nabla _{{\overline{z}}}h({\mathbf {z}}_0) \rangle <0. \end{aligned}$$

  (10)

  Adding inequalities (9) and (10), then the inequality (8) still holds, this contradicts the equality of (3).

• (c) The proof is similar to case (b).

• (d) If we combine inequalities (6) and (7), then

  $$\begin{aligned} \mathrm{Re} \left\{ \sum _{i=1}^m \lambda _i \left[ \big (f_i({\mathbf {z}})-f_i({\mathbf {z}}_0)\big ) - k_i \big (g_i({\mathbf {z}})-g_i({\mathbf {z}}_0)\big )\right] + \mu ^H[ h({\mathbf {z}})- h({\mathbf {z}}_0)] \right\} \le 0. \end{aligned}$$

  If hypothesis (iii) holds, $\mathrm{Re} \left\{ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]+ \mu ^H h(\cdot )\right\}$ is strictly pseudoconvex at ${\mathbf {z}}_0$, and from above inequality, we will the same get inequality (8), which contradicts the equality of (3).

Thus, ${\mathbf {z}}_0=(z_0, \overline{z_0})$ is an efficient solution of the primal problem (CMFP). $\square$

5 Parametric Duality Model

In this section, we introduce the following parametric dual problem (D) to the primal problem (CMFP) by using the necessary optimality conditions in Theorem 1 with some constraints.

$$\begin{aligned} \text{(D) } \max _{{\mathcal {F}}_{D}}~ \gamma =(r_1,~ \ldots , ~r_m ), \end{aligned}$$

where ${\mathcal {F}}_{D}$ is the set of all feasible solutions $(\lambda ,{\mathbf {u}},\mu ,\gamma )$ subject to

$$\begin{aligned}&\sum _{i=1}^m \lambda _i [ (\overline{\nabla _z f_i({\mathbf {u}} )}+\nabla _{{\overline{z}}}f_i({\mathbf {u}}))- r_i ( \overline{\nabla _z g_i({\mathbf {u}})}+ \nabla _{{\overline{z}}}g_i({\mathbf {u}}))] \nonumber \\&\quad +\,\mu ^T \overline{\nabla _z h({\mathbf {u}})}+\mu ^H \nabla _{{\overline{z}}} h({\mathbf {u}})=0, \end{aligned}$$

(11)

$$\begin{aligned}&\mathrm{Re} [f_i({\mathbf {u}}) -r_i g_i({\mathbf {u}})] \ge 0, \quad i=1, \dots , m, \end{aligned}$$

(12)

$$\begin{aligned}&\mathrm{Re} \langle h({\mathbf {u}}),\mu \rangle \ge 0, \quad \mu \ne 0 \text{ in } S^*, \end{aligned}$$

(13)

for ${\mathbf {u}}=(u,{\overline{u}})\in Q\subset {\mathbb {C}}^{2n}$, $\gamma =(r_1, \dots , r_m) \text{ and } \lambda =(\lambda _1, \dots , \lambda _m)\ge 0 \hbox { in}\ {\mathbb {R}}^m$.

The duality theorems of dual problem (D) to the primal problem (CMFP) are established as follows.

Theorem 3

(Weak Duality) Let ${\mathbf {z}}=(z,{\overline{z}})$ be (CMFP)-feasible solution, and $(\lambda ,{\mathbf {u}},\mu ,\gamma )$ be (D)-feasible solution. If any one of the conditions (i)–(iv) of Theorem 2 holds, i.e.,

• (i) one of $\mathrm{Re}~ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]$ and $\mathrm{Re} [\mu ^H h(\cdot ) ]$ is strictly convex and another is convex at ${\mathbf {u}}\in Q$, or both are strictly convex at ${\mathbf {u}}\in Q$,

• (ii) $\mathrm{Re}~ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]$ is quasiconvex at ${\mathbf {u}}\in Q$ and $\mathrm{Re}~ [\mu ^H h(\cdot )]$ is strictly pseudoconvex at ${\mathbf {u}}\in Q$,

• (iii) $\mathrm{Re}~ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]$ is strictly pseudoconvex at ${\mathbf {u}}\in Q$ and $\mathrm{Re}~ [\mu ^H h(\cdot )]$ is quasiconvex at ${\mathbf {u}}\in Q$,

• (iv) $\mathrm{Re} \left\{ \sum _{i=1}^m \lambda _i[ f_i(\cdot )- k_i g_i(\cdot )]+ \mu ^H h(\cdot ) \right\}$ is strictly pseudoconvex at ${\mathbf {u}}\in Q$,

then there are not exist feasible solution ${\mathbf {z}}\in X$ of the primal problem (CMFP) such that

$$\begin{aligned} \left( \frac{\mathrm{Re~} f_1({\mathbf {z}})}{\mathrm{Re~} g_1({\mathbf {z}}) }, \dots , \frac{\mathrm{Re~} f_m({\mathbf {z}})}{\mathrm{Re~} g_m({\mathbf {z}}) } \right) \leqq \gamma . \end{aligned}$$

Proof

Suppose on the contrary that

$$\begin{aligned} \left( \frac{\mathrm{Re~} f_1({\mathbf {z}})}{\mathrm{Re~} g_1({\mathbf {z}}) }, \dots , \frac{\mathrm{Re~} f_m({\mathbf {z}})}{\mathrm{Re~} g_m({\mathbf {z}}) } \right) \leqq \gamma =(r_1, \dots , r_m). \end{aligned}$$

(14)

That is $\mathrm{Re~} [f_i({\mathbf {z}}) - r_i g_i({\mathbf {z}})] \le 0$ for all $i= 1 , \dots , m$ and $\mathrm{Re~} [f_j({\mathbf {z}}) - r_j g_j({\mathbf {z}})] \ne 0$ for some $j\in \{1, 2, \dots , m\}$. Since condition (12),

$$\begin{aligned} \mathrm{Re} [f_i({\mathbf {z}}) - r_i g_i({\mathbf {z}})] \le 0 \le \mathrm{Re} [f_i({\mathbf {u}}) - r_i g_i({\mathbf {u}})], \text{ for } i=1, \dots , m\text{. } \end{aligned}$$

From $(\lambda _1, \dots , \lambda _m) \ge 0$, we get

$$\begin{aligned} \mathrm{Re} \sum _{i=1}^m \lambda _i \left\{ [f_i({\mathbf {z}}) - f_i({\mathbf {u}})]+ r_i[ g_i({\mathbf {z}})- g_i({\mathbf {u}})] \right\} \le 0. \end{aligned}$$

(15)

On the other hand, since ${\mathbf {z}}=(z,{\overline{z}})$ is a feasible solution of the primal problem (CMFP), for a nonzero vector $\mu \in S^*,$ such that $\mathrm{Re} \langle h({\mathbf {z}}),\mu \rangle \le 0$. For any ${\mathbf {u}}\in Q$ in dual problem (D), and from inequality (13), thus

$$\begin{aligned} \mathrm{Re} [\mu ^H h({\mathbf {z}})-\mu ^H h({\mathbf {u}})] \le 0. \end{aligned}$$

(16)

From inequalities (15), (16) and by similar proofs of case (a)–(d) in Theorem 2, then we will obtain inequality

$$\begin{aligned} \begin{array}{l} \displaystyle \sum _{i=1}^m \lambda _i [ (\overline{\nabla _z f_i({\mathbf {u}} )}+\nabla _{{\overline{z}}}f_i({\mathbf {u}}))- r_i ( \overline{\nabla _z g_i({\mathbf {u}})}+\nabla _{{\overline{z}}}g_i({\mathbf {u}}))]\\ +\mu ^T \overline{\nabla _z h({\mathbf {u}})}+\mu ^H \nabla _{{\overline{z}}} h({\mathbf {u}})< 0, \end{array} \end{aligned}$$

which contradicts the equality of (11). Thus, there does not exist feasible solution ${\mathbf {z}}\in X$ of the primal problem (CMFP) such that

$$\begin{aligned} \left( \frac{\mathrm{Re~} f_1({\mathbf {z}})}{\mathrm{Re~} g_1({\mathbf {z}}) }, \dots , \frac{\mathrm{Re~} f_m({\mathbf {z}})}{\mathrm{Re~} g_m({\mathbf {z}}) } \right) \leqq \gamma , \end{aligned}$$

the proof is complete. $\square$

Theorem 4

(Strong Duality) Suppose that ${\mathbf {z}}_0$ is an efficient solution of the primal problem (CMFP) with optimal value $\gamma =(r_1, \dots ,r_m)$. Then, there exists $(\lambda ,{\mathbf {z}}_0,\mu ,\gamma )$ is a feasible solution of the dual problem (D). If the hypotheses of Theorem 3 are fulfilled, then $(\lambda ,{\mathbf {z}}_0,\mu ,\gamma )$ is an efficient solution of (D), and the two problems (CMFP) and (D) have the same optimal value.

Proof

Let ${\mathbf {z}}_0=(z_0,\overline{z_0})\in Q$ be an efficient solution of the primal problem (CMFP) with optimal value $\gamma$. By Theorem 1, there exist $\lambda \in {\mathbb {R}}^m_+$ and $\mu \in S^* \subset {\mathbb {C}}^p$ such that

$$\begin{aligned} \begin{array}{l} \sum _{i=1}^m \lambda _i [ (\overline{\nabla _z f_i({\mathbf {z}}_0)}+\nabla _{{\overline{z}}}f_i({\mathbf {z}}_0))- r_i ( \overline{\nabla _z g_i({\mathbf {z}}_0)}+ \nabla _{{\overline{z}}}g_i({\mathbf {z}}_0))]\\ +\mu ^T \overline{\nabla _z h({\mathbf {z}}_0)}+\mu ^H \nabla _{{\overline{z}}}h({\mathbf {z}}_0)=0, \\ \mathrm{Re}[ \mu ^H h({\mathbf {z}}_0)]=0. \end{array} \end{aligned}$$

We obtain conditions (11) and (13). Because $\gamma$ is the optimal value of problem (CMFP), that is $\gamma =(r_1, \dots , r_m) = \left( \frac{\mathrm{Re~}f_1({\mathbf {z}}_0)}{\mathrm{Re~}g_1({\mathbf {z}}_0)}, \dots , \frac{\mathrm{Re~}f_m({\mathbf {z}}_0)}{\mathrm{Re~}g_m({\mathbf {z}}_0)} \right) = \min \left( \frac{\mathrm{Re~}f_1({\mathbf {z}})}{\mathrm{Re~}g_1({\mathbf {z}})}, \dots , \frac{\mathrm{Re~}f_m({\mathbf {z}})}{\mathrm{Re~}g_m({\mathbf {z}})} \right)$. It implies that

$$\begin{aligned} \mathrm{Re} [f_i({\mathbf {z}}_0) -r_i g_i({\mathbf {z}}_0)] = 0, \text{ for } i=1, \dots , m\text{. } \end{aligned}$$

The condition (12) holds. Hence, $(\lambda ,{\mathbf {z}}_0,\mu ,\gamma )$ is a feasible solution of the dual problem (D). From Theorem 3, the optimality of the feasible solution $(\lambda ,{\mathbf {z}}_0,\mu ,\gamma )$ for (D) reduces to be the optimal value of (D). $\square$

Theorem 5

(Strictly Converse Duality) Suppose that $\widehat{{\mathbf {z}}}$ is an efficient solution of the primal problem (CMFP) with optimal value ${\widehat{\gamma }}=(\widehat{r_1}, \dots , \widehat{r_m})$, and $({\widehat{\lambda }}, \widehat{{\mathbf {u}}},{\widehat{\mu }},{\widehat{\gamma }})$ is an efficient solution of dual problem (D). Assume that the assumptions of Theorem 4 are fulfilled, and any one of the following conditions (i)–(iv) holds:

• (i) one of $\mathrm{Re}~ \sum _{i=1}^m \widehat{\lambda _i}[ f_i(\cdot )- k_i g_i(\cdot )]$ and $\mathrm{Re} [{\widehat{\mu }}^H h(\cdot ) ]$ is strictly convex and another is convex at $\widehat{{\mathbf {u}}}\in Q$, or both are strictly convex at $\widehat{{\mathbf {u}}}\in Q$,

• (ii) $\mathrm{Re}~ \sum _{i=1}^m \widehat{\lambda _i}[ f_i(\cdot )- k_i g_i(\cdot )]$ is quasiconvex at $\widehat{{\mathbf {u}}}\in Q$ and $\mathrm{Re}~ [{\widehat{\mu }}^H h(\cdot )]$ is strictly pseudoconvex at $\widehat{{\mathbf {u}}}\in Q$,

• (iii) $\mathrm{Re}~ \sum _{i=1}^m \widehat{\lambda _i}[ f_i(\cdot )- k_i g_i(\cdot )]$ is strictly pseudoconvex at $\widehat{{\mathbf {u}}}\in Q$ and $\mathrm{Re}~ [{\widehat{\mu }}^H h(\cdot )]$ is quasiconvex at $\widehat{{\mathbf {u}}}\in Q$,

• (iv) $\mathrm{Re} \left\{ \sum _{i=1}^m \widehat{\lambda _i}[ f_i(\cdot )- k_i g_i(\cdot )]+ {\widehat{\mu }}^H h(\cdot ) \right\}$ is strictly pseudoconvex at $\widehat{{\mathbf {u}}}\in Q$.

Then, $\widehat{{\mathbf {z}}}=\widehat{{\mathbf {u}}}$, and the optimal values of (CMFP) and (D) are equal.

Proof

We assume that $\widehat{{\mathbf {z}}}\ne \widehat{{\mathbf {u}}}$; then, there will be a contradiction. Since $\widehat{{\mathbf {z}}}$ is an efficient solution of the primal problem (CMFP) with optimal value ${\widehat{\gamma }}$ ,

$$\begin{aligned} {\widehat{\gamma }} = (\widehat{r_1}, \dots , \widehat{r_m}) = \left( \frac{\mathrm{Re~}f_1(\widehat{{\mathbf {z}}})}{\mathrm{Re~}g_1(\widehat{{\mathbf {z}}})}, \dots , \frac{\mathrm{Re~}f_m(\widehat{{\mathbf {z}}})}{\mathrm{Re~}g_m(\widehat{{\mathbf {z}}})} \right) = \min \left( \frac{\mathrm{Re~}f_1({\mathbf {z}})}{\mathrm{Re~}g_1({\mathbf {z}})}, \dots , \frac{\mathrm{Re~}f_m({\mathbf {z}})}{\mathrm{Re~}g_m({\mathbf {z}})} \right) . \end{aligned}$$

Hence, $\mathrm{Re} [ f_i(\widehat{{\mathbf {z}}})-\widehat{r_i}g_i(\widehat{{\mathbf {z}}})] = 0$, for $i=1, \dots , m$. We pick $\widehat{\lambda _i} \ge 0$, $i=1, \dots , m$, then

$$\begin{aligned} \mathrm{Re~}\sum _{i=1}^m \widehat{\lambda _i} [ f_i(\widehat{{\mathbf {z}}})-\widehat{r_i}g_i(\widehat{{\mathbf {z}}})] = 0. \end{aligned}$$

Using condition (12) and assume that $\widehat{{\mathbf {z}}}\ne \widehat{{\mathbf {u}}}$, we obtain

$$\begin{aligned} \mathrm{Re~}\sum _{i=1}^m \widehat{\lambda _i} [ f_i(\widehat{{\mathbf {z}}})-\widehat{r_i}g_i(\widehat{{\mathbf {z}}})] = 0 \le \mathrm{Re~}\sum _{i=1}^m \widehat{\lambda _i} [ f_i(\widehat{{\mathbf {u}}})-\widehat{r_i}g_i(\widehat{{\mathbf {u}}})] . \end{aligned}$$

(17)

On the other hand, using the feasibility of $\widehat{{\mathbf {z}}}$ for the primal problem (CMFP) with ${\widehat{\mu }}\in S^*$, and by inequality (13), we have

$$\begin{aligned} \mathrm{Re} [{\widehat{\mu }}^H h(\widehat{{\mathbf {z}}})] \le 0 \le \mathrm{Re} [{\widehat{\mu }}^H h(\widehat{{\mathbf {u}}})]. \end{aligned}$$

(18)

From inequalities (17) and (18), the result is proved by similar methods of case (a)–(d) in Theorem 2. $\square$

6 Further Plausible Works

In this paper, we derived the necessary and sufficient conditions of complex multi-objective fractional problem (CMFP) and established the parametric dual (D) as well as their duality theorems. In the further plausible work, we will establish parametric free dual problems of the primal problem (CMFD), and these parametric free dual problems are called Mond–Weir-type dual problem and Wolfe-type dual problem.