On optimality conditions and duality for multiobjective fractional optimization problem with vanishing constraints

1.   Introduction

Recently, there has been a lot of attention on mathematical programming problems with vanishing constraints, which serve as a unified framework for several applications in topological optimization and optimal contral. The optimality conditions and duality theorems of these problems have been extensively researched since their introduction by Achtziger and Kanzow^[1]. Mishra et al.^[2] developed and analyzed dual models and obtained some duality results under differentiable assumptions. Hu and his co-authors in ^[3] provided some new dual models based on the dual models proposed by ^[2], which do not require computing the index sets. Tung ^[4] extended the single objective programming to multiobjective semi-infinite cases with vanishing constraints and investigated the KKT optimality conditions and duality results of the Wolfe and Mond–Weir-type dual models for this problem. Furthermore, Tung ^[5] established the KKT optimality conditions and the duality theorems for nonsmooth multiobjective semi-infinite optimization problems with vanishing constraints in terms of Clarke subdifferentials. By proposing new constraints for ACQ and VC-ACQ, Antczak ^[6] derived optimality conditions and duality results for differentiable semi-infinite multiobjective optimization problems with vanishing constraints. Additionally, Antczak ^[7] addressed the KKT optimality conditions for a class of nondifferentiable multiobjective programming problems with vanishing constraints under the VC-Cottle constraint qualification. However, duality results are not taken into account in ^[7]. Meanwhile, for directionally differentiable vector optimization problems, Antczak ^[8] also discussed the KKT necessary optimality conditions under both ACQ and m-ACQ; the sufficient optimality conditions and Wolfe-type duality theorems were also established under appropriate convexity hypotheses. Huang and Zhu ^[9] studied optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints in terms of Clark subdifferential. Guu et al.^[10] provided strong KKT sufficient optimality conditions for multiobjective semi-infinite programming problems with vanishing constraints under generalized convexity assumptions. Wang and Wang ^[11,12] established optimality conditions for a class of nonsmooth interval-valued optimization problems with vanishing constraints, along with duality theorems for the corresponding dual models. The principal challenge inherent in optimization problems with vanishing constraints stems from the inclusion of a product of two functions within the constraint conditions. This situation gives rise to two notable issues: firstly, the feasible set is generally non-convex; secondly, when one of the functions in the product equals zero, the constraint properties of the other function become ineffective.

A fundamental question here is why we should study optimality conditions and duality in the framework of multiobjective fractional programming problems with vanishing constraints, as well as their corresponding Mond–Weir and Wolfe-type dual problems. We try to address this question succinctly. While many studies have been published over the past decade concerning optimization problems with vanishing constraints, there remains a scarcity of research specifically focused on multiobjective fractional programming problems with vanishing constraints (see ^{[1,2,3,4,5,6,7,8,9,10,11,12]}). Notably, the Mond–Weir and Wolfe types of dual problems have garnered significant attention in this field due to their practical applicability.

Due to the fact that in numerous optimization problems, the objective functions are expressed as quotients of two functions. There are many authors who established optimality conditions and employed the conditions to search for optimal solutions as well as duality theorems for such vector optimization problems (see ^{[13,14,15,16,17]}). Kim et al.^[13] derived optimality conditions and duality results for nondifferentiable multiobjective fractional programming. Long^[14] discussed similar results for this type of problem using $(C, \alpha, \rho, d)$-convexity. Later, under higher-order $(C, \alpha, \gamma, \rho, d)$-assumptions, Dubey et al.^[15] established higher-order optimality conditions and duality results for such a problem. In addition, for nonsmooth fractional multiobjective optimization problems with equality or inequality constraints, several optimality conditions and duality theorems are studied in ^[16,17,18]. We note that there is relatively little literature on optimality conditions and duality theorems for nonsmooth multiobjective fractional programming problems with vanishing constraints.

Motivated by the above works, this paper aims to investigate nonsmooth multiobjective fractional optimization problems with vanishing constraints (abbreviated as, (FPVC)), and establish necessary and sufficient optimality conditions for (FPVC). Subsequently, duality theorems of Wolfe type and Mond–Weir-type for (FPVC) will be formulated. The organization of this paper is outlined as follows: In Section 2, essential notions and definitions are reviewed for subsequent discussion. Section 3 focuses on the optimality conditions for the (weak) Pareto minimum of (FPVC) subject to VC-Cottle constraints. Section 4 establishes Wolfe-type and Mond–Weir type dual models for (FPVC) and studies the weak, strong and converse duality theorems between (FPVC) and its dual problems.

2.   Preliminaries

Let $\mathbb{R}^{n}$ be the n-dimensional Euclidean space. For any $a, b \in \mathbb{R}^{n}$, we define:

(ⅰ) $a < b \Leftrightarrow a_{i} < b_{i}$ for all $i = 1, 2, \ldots, n$;

(ⅱ) $a\leqq b\Leftrightarrow a_{i}\leqq b_{i}$ for all $i = 1, 2, \ldots, n$;

(ⅲ) $a\leq b\Leftrightarrow a_{i}\leqq b_{i}$ for all $i = 1, 2, \ldots, n$ and $a\neq b$;

(ⅳ) $a\nleq b$ is the negation of $a\leq b$.

Row and column vectors will be treated with the same notation in this paper when the interpretation is obvious.

Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a locally Lipschitz function. The Clarke subdifferential of $f$ at $\bar{x}$ is defined as follows:

where

Lemma 2.1. ^[19] Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be locally Lipschitz at $\bar{x}\in\mathbb{R}^{n}$ and attain its minimum at $\bar{x}$. Then $0\in\partial_{c} f(\bar{x}).$

Lemma 2.2. ^[19] Let $f_{k}:\mathbb{R}^{n}\rightarrow \mathbb{R}, \ k\in K: = \{1, \ldots, l\}$ be a locally Lipschitz function at a point $\bar{x}\in\mathbb{R}^{n}$. Then

where $\lambda_{k}\in \mathbb{R}$. If $f(x): = \max\limits_{k\in K}f_{k}(x)$, then the function $f(x)$ is also locally Lipschitz at $\bar{x}$. In addition,

where $K(\bar{x}): = \{k\in K:f(\bar{x}) = f_{k}(\bar{x})\}$, and $conv$ is an abbreviation for convex hull.

Lemma 2.3. ^[19] Let $f, g: \mathbb{R}^{n}\rightarrow \mathbb{R}$ be locally Lipschitz functions at $\bar{x}\in\mathbb{R}^{n}$. Then $f g$ is a locally Lipschitz function at $\bar{x}$, and

If $g(\bar{x})\neq0$, $\frac{f}{g}$ is also a locally Lipschitz function at $\bar{x}$, and

Accordingly, we consider multiobjective fractional optimization with vanishing constraints (FPVC) as follows:

where $f_{i}, g_{i}, h_{j}, U_{s}, V_{s}:\mathbb{R}^{n}\rightarrow \mathbb{R}$, $i\in I: = \{1, \ldots, p\}$, $j\in J$, $s\in S$, are locally Lipschitz functions. For all $i\in I$, we set $f_{i}(x)\geqq0$, $g_{i}(x) > 0$. The set $D$ stands for the feasible set of problems (FPVC).

Definition 2.1. Let $\bar{x}\in D$,

(i) $\bar{x}$ is said to be a weak Pareto solution for (FPVC) if there is no other $x\in D$ such that $F(x) < F(\bar{x})$.

(ii) $\bar{x}$ is said to be a Pareto solution for (FPVC) if there is no other $x\in D$ such that $F(x)\leq F(\bar{x})$.

Now, for any feasible point $\bar{x}\in D$, we denote the following index sets:

Obviously, $S_{0}(\bar{x}) = S_{0+}(\bar{x})\cup S_{00}(\bar{x})\cup S_{0-}(\bar{x})$, $S_{+}(\bar{x}) = S_{+0}(\bar{x})\cup S_{+-}(\bar{x})$, $S_{UV}(\bar{x}) = S_{0}(\bar{x})\cup S_{+0}(\bar{x}).$

3.   Optimality conditions

In the sequel, the KKT-necessary optimality conditions of the (weak) Pareto solution for (FPVC) are presented. Firstly, we introduce the following VC-Cottle constraint qualification given by Antczak ^[7].

Definition 3.1. ^[7] The VC-Cottle constraint qualification is fulfilled at $\bar{x}\in D$ for (FPVC) if either $h_{j}(\bar{x}) < 0$, $\forall j\in J$, $U_{s}(\bar{x}) > 0$ and $V_{s}(\bar{x}) < 0$, $\forall s\in S$ or

Theorem 3.1. Suppose that $\bar{x}\in D$ is a weak Pareto solution in (FPVC) and that the VC-Cottle constraint qualification is satisfied at $\bar{x}$. Then there exist $\alpha\in\mathbb{R}^{p}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$ and $\gamma^{V}\in\mathbb{R}^{q}$ such that certain conditions hold:

where $r_{i} = \frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}\ (\forall i\in I)$.

Proof. We define an auxiliary function $\Psi(x):\mathbb{R}^{n}\rightarrow \mathbb{R}$, where

Since $\bar{x}$ is a weak Pareto solution of (FPVC), it can be deduced that $\Psi(x)\geqq0$ for all $x\in\mathbb{R}^{n}$, and that $\Psi(\bar{x}) = 0$. Which implies that $\Psi$ attains its global minimum at $\bar{x}$. It follows from Lemma 2.1 one has

Furthermore, since

From Lemma 2.2, one has

Case 1. We suppose that $h_{j}(\bar{x}) < 0$, $\forall j\in J$, $U_{s}(\bar{x}) > 0$ and $V_{s}(\bar{x}) < 0$, $\forall s\in S$. Then, by (3.9) and (3.11), one has $0\in conv\left\{\partial_{c}\left(\frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}\right) :\ i\in I\right\}$. Then there exist $\mu\in\mathbb{R}^{p}, \ \mu\geq0, \ \sum\limits_{i = 1}^{p}\mu_{i} = 1$ such that $0\in\sum\limits_{i = 1}^{p}\mu_{i}\partial_{c}\left(\frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}\right).$

From Lemma 2.3, one has

Thus,

Setting $r_{i} = \frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}$ and $\alpha_{i} = \mu_{i}\frac{1}{g_{i}(\bar{x})}$, $\forall i\in I$, we obtain $\alpha\geq0$ and

Therefore, we have (3.1)–(3.7) by setting $\beta_{j} = 0, \ j\in J$, $\gamma^{U}_{S} = 0, \ s\in S_{+}(\bar{x})$, $\gamma^{V}_{S} = 0, \ s\in S_{+-}(\bar{x})$.

Case 2. If there exists $j\in J$ such that $h_{j}(\bar{x}) = 0$ or $s\in S$ such that $U_{s}(\bar{x}) = 0$ or $V_{s}(\bar{x}) = 0$, then there exist $\mu\in\mathbb{R}^{p}$, $\mu\geqq0$, $\beta\in \mathbb{R}^{m}$, $\beta\geqq0$, $\omega\in\mathbb{R}^{S_{0}(\bar{x})}$, $\omega\geqq0$ and $\upsilon\in\mathbb{R}^{S_{UV}(\bar{x})}$, $\upsilon\geqq0$ with $\sum_{i = 1}^p u_i + \sum_{j\in J(\bar{x})} \beta_j + \sum_{s\in S_{UV}(\bar{x}) } v_s = 1$ such that

Therefore, we obtain

where $\beta_{j} = 0, \ j\notin J(\bar{x})$, $\omega_{s} = 0, \ s\notin S_{0}(\bar{x})$ and $\upsilon_{s} = 0, \ s\notin S_{UV}(\bar{x})$. From 2.3.13 in ^[18], one has

Let $r_{i} = \frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}$ and $\alpha_{i} = \mu_{i}\frac{1}{g_{i}(\bar{x})}$ for all $i\in I$. Combining (3.12), (3.14), and (3.15), we have

Now, setting $\gamma^{U}_{s} = \omega_{s}-\upsilon_{s}V_{s}(\bar{x})$ and $\gamma^{V}_{s} = \upsilon_{s}U_{s}(\bar{x})$ for all $s\in S$, we have

The proofs of (3.6) and (3.7) are coupled with Theorem 3.1 in ^[7]. Then, (3.4) and (3.5) hold. By the VC-Cottle constraint qualification, we have Lagrange multiplier $\alpha$ is not equal to 0 (i.e., $\alpha\geq0$). In this case, the conditions (3.1)–(3.7) hold. □

Remark 1. When $\bar{x}$ is a Pareto solution of (FPVC), the conditions (3.1)–(3.7) hold as well. The proof of this statement is similar to that of Theorem 1 and is thus omitted in this paper. Further, note that the conditions (3.1)–(3.7) are KKT necessary optimality conditions due to the fact that $\alpha\neq0$.

Remark 2. It is noted that when $g_{i}(x)\equiv1$ $(\forall i\in I)$, the nonsmooth multiobjective fractional optimization problems with vanishing constraints (FPVC) transforms into the nonsmooth multiobjective optimization problems with vanishing constraints (MPVC) in ^[7]. Consequently, Theorem 1 in our study enhances the corresponding conclusions in ^[7].

Definition 3.2. The point $\bar{x}\in D$ is called an S-stationary point for (FPVC) if there exist $\alpha\in\mathbb{R}^{p}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$ and $\gamma^{V}\in\mathbb{R}^{q}$ not equal to 0, such that the conditions

hold, where $r_{i} = \frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}\ (\forall i\in I)$.

An example is provided to demonstrate the application of Theorem 3.1.

Example 3.1. Consider the problem (FPVC) with the following parameters: $I = \{1, 2\}$, $J = \{1\}$. For all $x = (x_{1}, x_{2})\in \mathbb{R}^{2}$,

where $f_{1}(x) = x_{1}+x_{2}^{2}$, $f_{2}(x) = \left|x_{1}\right|+\left|x_{2}\right|$, $g_{1}(x) = 1-x_{1}^{2}$, $g_{2}(x) = -3x_{1}^{2}+x_{2}+2$, $V_{1}(x) = x_{1}+\left|x_{2}\right|-1$. We have that $D = \{(x_{1}, x_{2})\in\mathbb{R}^{2}:-x_{1}-x_2\leqq0, \ x_{2}\geqq0, \ x_{2}(x_{1}+\left|x_{2}\right|-1)\leqq 0 \}$ and $\bar{x} = (0, 0)\in D$. The sets $J(\bar{x}) = \{1\}$, $S_{0-}(\bar{x}) = \{1\}$, $S_{+0}(\bar{x}) = S_{+-}(\bar{x}) = S_{0+}(\bar{x}) = S_{00}(\bar{x}) = \emptyset$, and the parameter $(r_{1}, r_{2}) = (0, 0)$. Thus, we have

Since $0\notin conv\{\partial_{c}h_{1}(\bar{x}), \ -\partial_{c}U_{1}(\bar{x}), \ \partial_{c}(U_{1}V_{1})(\bar{x})\}$ when $S_{00}(\bar{x}) = \emptyset$, the VC-Cottle constraint qualification is fulfilled at $\bar{x}$. Further, there exist $\alpha_{1} = \frac{1}{2}$, $\alpha_{2} = \frac{1}{2}$, $\beta_{1} = \frac{1}{2}$, $\gamma_{1}^{U} = 0$, $\gamma_{1}^{V} = 0$, and $\xi_{1} = (1, 0)\in\partial_{c}(f_{1}-r_{1}g_{1})(\bar{x})$, $\xi_{2} = (0, 1)\in\partial_{c}(f_{2}-r_{2}g_{2})(\bar{x})$, $\rho_{1} = (-1, -1)\in\partial_{c}h_{1}(\bar{x})$, $\delta_{1} = (0, 1)\in\partial_{c}U_{1}(\bar{x})$, $\nu_{1} = (1, -1)\in\partial_{c}V_{1}(\bar{x})$ satisfying $\alpha_1 \xi_1+ \alpha_2 \xi_2+\beta_1 \rho_1- \gamma_{1}^{U} v_1 + \gamma_{1}^{V} v_2 = 0$, that is

Hence, the conditions of Theorem 1 are met.

Definition 3.3. Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ be a locally Lipschitz function.

(i) $f$ is said to be generalized quasiconvex at $\bar{x}$ if, for each $x\in\mathbb{R}^{n}$,

(ii) $f$ is said to be strictly generalized quasiconvex at $\bar{x}$ if, for each $x\in\mathbb{R}^{n}$ with $x \neq \bar{x}$,

Lemma 3.1. ^[8] Let $f_{0}$ be strictly generalized quasiconvex and $f_{1}, f_{2}, \ldots, f_{s}$ be generalized quasiconvex at $\bar{x}$. If $\lambda_{0} > 0$ and $\lambda_{l}\geqq0, \ l = 1, \ldots, s$, then $\sum_{l = 0}^{s}\lambda_{l}f_{l}$ is strictly generalized quasiconvex at $\bar{x}$.

Let $\bar{x}\in D$ be an S-stationary point for (FPVC). According to Definition 3.2, if there exist $\alpha\in\mathbb{R}^{p}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$ and $\gamma^{V}\in\mathbb{R}^{q}$ not equal to 0, such that (3.18)–(3.21) are fulfilled at $\bar{x}$, then we introduce the following denotations:

Theorem 3.2. Let $\bar{x}\in D$ be an S-stationary point for (FPVC). Suppose that the conditions (3.18)–(3.21) are fulfilled at $\bar{x}$ and the following assumptions are satisfied:

$(a)$ $D_{V+}: = \bigcup_{t\in S_{+0}(\bar{x})}\{x\in D\backslash\{\bar{x}\}\mid V_{s} > 0\} = \emptyset$ or $S_{+0}^{V+}(\bar{x}) = \emptyset$,

$(b)$ $S_{0+}^{U-}(\bar{x}) = \emptyset$.

Additionally, it is assumed that the functions $f_{i}, \ i\in I$, $h_{j}, \ j\in J(\bar{x})$, $-g_{i}, \ i\in I$, $-U_{s}, \ s\in S_{00}(\bar{x})\cup S_{0-}(\bar{x})\cup S_{0+}^{U+}(\bar{x})$ and $V_{s}, \ s\in S_{+0}(\bar{x})$ are generalized quasiconvex at $\bar{x}$. Among the functions $f_{i}-r_{i}g_{i}, \ i\in I$, $h_{j}, \ j\in J(\bar{x})$, $-U_{s}$ and $V_{s}, \ s\in S$, at least one is strictly generalized quasiconvex at $\bar{x}$. Then, $\bar{x}$ is a weak Pareto solution of (FPVC).

Proof. Given that $\bar{x}\in S$ is an S-stationary point for (FPVC), it follows from Definition 3.2 that there exist $\alpha\in\mathbb{R}^{p}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$ and $\gamma^{V}\in\mathbb{R}^{q}$ such that

and (3.19)–(3.21) hold. Then, there are $\xi_{i}\in\partial_{c} f_{i}(\bar{x})-r_{i}\partial_{c} g_{i}(\bar{x}), \ i\in I$, $\rho_{j}\in\partial_{c} h_{j}(\bar{x}), \ j\in J$, $\delta_{s}\in\partial_{c} U_{s}(\bar{x})$ and $\nu_{s}\in\partial_{c} V_{s}(\bar{x}), \ s\in S$, such that

Assuming the contrary, if $\bar{x}$ is not a weak Pareto solution of (FPVC), then there exists $\tilde{x}\in S$ that satisfies

Therefore, one has

where $r_{i} = \frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}\ (\forall i\in I)$. Thus, there exists $\alpha\in\mathbb{R}^{p}, \ \alpha\geq0$, such that

By $\tilde{x}\in S$ and Definition 3.2, we have

According to the conditions $(a)$ and $(b)$, one has

Thus, combining (3.23)–(3.26), we have

By the generalized quasiconvex hypotheses of the functions $f_{i}$ and $-g_{i}$, $\forall i\in I$, it can be deduced that the function $f_{i}-r_{i}g_{i}(\forall i\in I)$ is generalized quasiconvex at $\bar{x}$, where $r_{i} = \frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}\geqq0$ for all $i\in I$. By applying Lemma 2.2, it follows that

is strictly generalized quasiconvex at $\bar{x}$, and

Therefore,

which contradicts (3.22). □

Example 3.2. In Example 3.1, the functions $f_{1}, \ f_{2}$, $h_{1}$, $-g_{1}, \ -g_{2}$, $-U_{1}$ and $V_{1}$ are generalized quasiconvex at $\bar{x}$ on $D$, $g_{i}(x) > 0$ and $r_{i} = \frac{f_{i}(\bar{x})}{g_{i}(\bar{x})}\geqq0$, $\forall i\in I$. One can see that $\bar{x} = (0, 0)$ is an S-stationary point of the problem in Example 3.1. Since $S_{+0}(\bar{x}) = S_{0+}(\bar{x}) = \emptyset$, the conditions $(a)$ and $(b)$ are satisfied. Furthermore, we can verify that $h_1$ is strictly generalized quasiconvex at $\bar{x}$. In fact, for any $x = (x_1, x_2) \neq \bar{x}$ satisfying $h_1(x) \leqq h_1(\bar{x}) = 0$, then $x_1+x_2 > 0$, and so $\langle\eta, \ x-\bar{x}\rangle = -x_1-x_2 < 0$, where $\eta \in \partial_{c} h_1(\bar{x}) = \{(-1, -1)\}\}$. Therefore, $\bar{x} = (0, 0)$ is a weak Pareto solution.

4.   Duality theorems

The aim of this section is to consider the Wolfe and Mond–Weir-type dual problems for (FPVC). We prove the duality results between (FPVC) and its dual problems under the generalized quasiconvexity and strictly generalized quasiconvexity assumptions imposed on the functions involved.

Let $y\in \mathbb{R}^{n}$, $\alpha\in \mathbb{R}^{p}_{+}\backslash\{0\}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$ and $\gamma^{V}\in\mathbb{R}^{q}$. The vector Lagrange function $\Phi$ is defined as follows:

where $e = (1, \cdots, 1)\in \mathbb{R}^{p}$.

For any $\bar{x}\in D$, the Wolfe-type dual model $(D_{W}(\bar{x}))$ associated with the problem (FPVC) is defined as:

Let

denote the feasible set of $(D_{W}(\bar{x}))$.

The other Wolfe-type dual model, which does not rely on $\bar{x}$, is

where the sets $\Omega_{W} = \bigcap_{\bar{x}\in S}\Omega_{W}(\bar{x})$.

Definition 4.1. The point $(\tilde{y}, \tilde{\alpha}, \tilde{\beta}, \tilde{\gamma}^{U}, \tilde{\gamma}^{V}, \tilde{\upsilon}, \tilde{\omega})\in \Omega_{W}$ is said to be a Pareto solution of $(D_{W})$, if there is no $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)\in \Omega_{W}$ satisfying

In what follows, weak, strong, and converse duality theorems between (FPVC) and the Wolfe type duality problem $(D_{W})$ are given.

Theorem 4.1. (Weak duality) Let $\bar{x}\in D$ and $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)\in \Omega_{W}$ be any feasible solutions for (FPVC) and $(D_{W})$, respectively. If $\Phi(\cdot, \alpha, \beta, \gamma^{U}, \gamma^{V})$ is strictly generalized quasiconvex at $y\in \mathbb{R}^{n}$, then

Proof. Suppose, contrary to the result, that

That is

By $\bar{x}\in D$, it holds that

Thus,

In (4.1) and (4.2), we have

By utilizing the strictly generalized quasiconvex at $y\in \mathbb{R}^{n}$ of $\Phi(\cdot, \alpha, \beta, \gamma^{U}, \gamma^{V})$, it can be deduced that there exist $\bar{\xi_{i}}\in\partial_{c} f_{i}(y)-r_{i}\partial_{c} g_{i}(y), \ i\in I$, $\bar{\rho_{j}}\in\partial_{c} h_{j}(y), \ j\in J$, $\bar{\delta_{s}}\in\partial_{c} U_{s}(y), \ \bar{\nu_{s}}\in\partial_{c} V_{s}(y), \ s\in S$, such that

This contradicts the constraint of $(D_{W})$. □

Theorem 4.2. (Strong duality) Let $\bar{x}\in D$ be a weak Pareto solution of problem (FPVC), and suppose the VC-Cottle constraint qualification is fulfilled at $\bar{x}$., then there exist Lagrange multipliers $\alpha\in\mathbb{R}^{p}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$, $\gamma^{V}\in\mathbb{R}^{q}$, $\upsilon\in\mathbb{R}^{q}$ and $\omega\in\mathbb{R}^{q}$ such that $(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is feasible in $(D_{W})$ and $F(\bar{x}) = \Phi(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V})$. If $\Phi(\cdot, \alpha, \beta, \gamma^{U}, \gamma^{V})$ is strictly generalized quasiconvex at $y\in \mathbb{R}^{n}$, then $(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is a Pareto solution of $(D_{W})$.

Proof. From the assumptions that $\bar{x}\in D$ and the VC-Cottle constraint qualification holds, there exist $\alpha\in\mathbb{R}^{p}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$ and $\gamma^{V}\in\mathbb{R}^{q}$ such that the necessary optimality conditions (Theorem 3.1) are fulfilled. Then, by the definition of $\Omega_{W}$ and (3.1)–(3.7), we conclude that $(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is feasible in $(D_{W})$ and

Thus, $F(\bar{x}) = \Phi(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V})$.

Suppose, on the contrary, that $(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is not a Pareto solution of $(D_{W})$, then we have $(\tilde{y}, \tilde{\alpha}, \tilde{\beta}, \tilde{\gamma^{U}}, \tilde{\gamma^{V}}, \tilde{\upsilon}, \tilde{\omega})$ such that

Then, $F(\bar{x})\leq\Phi(\tilde{y}, \tilde{\alpha}, \tilde{\beta}, \tilde{\gamma^{U}}, \tilde{\gamma^{V}}),$ which contradicts Theorem 4.1. □

Theorem 4.3. (Converse duality) Suppose that $\bar{x}\in D$ is a feasible solution of (FPVC), $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is a weak Pareto solution of $(D_{W})$, and the inequalities

hold, such that $y\in D$. If one of the following assumptions is fulfilled:

$(i)$ $\Phi(\cdot, \alpha, \beta, \gamma^{U}, \gamma^{V})$ is strictly generalized quasiconvex at $y$;

$(ii)$ $f_{i}\geqq0$, $-g_{i} < 0 (i\in I)$ are strictly generalized quasiconvex at $y$, $h_{j}, \ j\in J(\bar{x})$, $-U_{s}, \ s\in S_{00}(\bar{x})\cup S_{0-}(\bar{x})\cup S_{0+}^{U+}(\bar{x})$ and $V_{s}, \ s\in S_{+0}(\bar{x})$ are generalized quasiconvex at $\bar{x}$,

then $y$ is a Pareto solution in $(FPVC)$.

Proof. Suppose, on the contrary, that $y\in D$ is not a Pareto solution in $(FPVC)$. Then, there exists $\tilde{y}\in D$ such that

For the assumption $(i)$, since $\tilde{y}$ and $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ are feasible points for (FPVC) and $(D_{W})$, respectively, combined with (4.2) and (4.3), one gets

Hence,

Due to the fact that $\Phi(\cdot, \alpha, \beta, \gamma^{U}, \gamma^{V})$ is strictly generalized quasiconvex at $y$, there exist $\bar{\xi_{i}}\in\partial_{c} f_{i}(y)-r_{i}\partial_{c} g_{i}(y), \ i\in I$, $\bar{\rho_{j}}\in\partial_{c} h_{j}(y), \ j\in J$, $\bar{\delta_{s}}\in\partial_{c} U_{s}(y)$ and $\bar{\nu_{s}}\in\partial_{c} V_{s}(y), \ s\in S$, such that

This contradicts the constraint of $(D_{W})$.

For the assumption $(ii)$, since $\tilde{y}$ and $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ are feasible points for (FPVC) and $(D_{W})$ respectively, by (4.3), we have

Thus

Using the generalized quasiconvex of the functions in assumption $(ii)$ and (4.5), the inequalities

hold, that is

Since $0\in\sum\limits_{i = 1}^{p}\alpha_{i}(\partial_{c}f_{i}(y)-r_{i}\partial_{c}g_{i}(y))+\sum\limits_{j = 1}^{m}\beta_{j}\partial_{c} h_{j}(y)-\sum\limits_{s = 1}^{q}\gamma_{s}^{U}\partial_{c} U_{s}(y)+\sum\limits_{s = 1}^{q}\gamma_{s}^{V}\partial_{c} V_{s}(y)$, there exists $\bar{\xi_{i}}\in\partial_{c} f_{i}(y)-r_{i}\partial_{c} g_{i}(y), \ i\in I$, such that

By $F(\tilde{y})-F(y)\leq0\Longleftrightarrow f_{i}(\tilde{y})-\hat{r}_{i}g_{i}(\tilde{y})\leq0$, where $\hat{r}_{i} = \frac{f_{i}(y)}{g_{i}(y)}, \ \forall i\in I$. Hence, there exists $\alpha\in\mathbb{R}^{p}, \ (\alpha\geq0)$, such that

For all $i\in I$, the functions $f_{i}\geqq0$ and $-g_{i} < 0$ are strictly generalized quasiconvex at $y$ and $\hat{r}_{i} = \frac{f_{i}(y)}{g_{i}(y)}\geqq0$, it follows that $f_{i}-\hat{r}_{i}*g_{i}$ $(\forall i\in I)$ is strictly generalized quasiconvex at $y$. Then, there exists $\bar{\xi_{i}}\in\partial_{c} f_{i}(y)-r_{i}\partial_{c} g_{i}(y)$ $(\forall i\in I)$ such that

This contradicts (4.6). □

For $\bar{x}\in D$, the Mond–Weir type dual model $(D_{MW}(\bar{x}))$ is follows:

Let

denote the feasible set of $(D_{MW}(\bar{x}))$.

The other Mond–Weir type dual model, which does not rely on $\bar{x}$, is

where $\Omega_{MW} = \bigcap_{\bar{x}\in S}\Omega_{MW}(\bar{x})$.

Definition 4.2. The point $(\tilde{y}, \tilde{\alpha}, \tilde{\beta}, \tilde{\gamma}^{U}, \tilde{\gamma}^{V}, \tilde{\upsilon}, \tilde{\omega})\in \Omega_{MW}$ is said to be a Pareto solution of $(D_{MW})$, if there is no $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)\in \Omega_{MW}$ satisfying

Next, we present the duality theorems between (FPVC) and the Mond–Weir type dual problem $(D_{MW})$.

Theorem 4.4. (Weak duality) Let $\bar{x} \in D$ and $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)\in \Omega_{MW}$ be any feasible solutions for (FPVC) and $(D_{MW})$, respectively. If one of the following assumptions is fulfilled:

$(i)$ $f_{i}\geqq0$, $-g_{i} < 0(i\in I)$ are strictly generalized quasiconvex at $y$, $\sum\limits_{j = 1}^{m}\beta_{j} h_{j}(\cdot)-\sum\limits_{s = 1}^{q}\gamma_{s}^{U} U_{s}(\cdot)+\sum\limits_{s = 1}^{q}\gamma_{s}^{V} V_{s}(\cdot)$ is generalized quasiconvex at $y$;

$(ii)$ $f_{i}\geqq0$, $-g_{i} < 0(i\in I)$ are strictly generalized quasiconvex at $y$, $h_{j}, \ j\in J(\bar{x})$, $-U_{s}, \ s\in S_{00}(\bar{x})\cup S_{0-}(\bar{x})\cup S_{0+}^{U+}(\bar{x})$ and $V_{s}, \ s\in S_{+0}(\bar{x})$ are generalized quasiconvex at $\bar{x}$,

then

Proof. Suppose, contrary to the result, that

By the strictly generalized quasiconvex at $y$ of $f_{i}$ and $-g_{i}$ $(\forall i\in I)$, we have that $f_{i}-\hat{r_{i}} g_{i}(\forall i\in I)$ is strictly generalized quasiconvex at $y$, where $\hat{r_{i}} = \frac{f_{i}(y)}{g_{i}(y)}\geqq0$. Then, there exists $\bar{\xi_{i}}\in\partial_{c} f_{i}(y)-r_{i}\partial_{c} g_{i}(y)$ for all $i\in I$ such that

For the assumption $(i)$, since $\bar{x}\in D$ and $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)\in \Omega_{MW}$, it follows that

Since $\sum\limits_{j = 1}^{m}\beta_{j} h_{j}(\cdot)-\sum\limits_{s = 1}^{q}\gamma_{s}^{U} U_{s}(\cdot)+\sum\limits_{s = 1}^{q}\gamma_{s}^{V} V_{s}(\cdot)$ is generalized quasiconvex at $y$, there are $\bar{\rho_{j}}\in\partial_{c} h_{j}(y), \ j\in J$, $\bar{\delta_{s}}\in\partial_{c} U_{s}(y), \ \bar{\nu_{s}}\in\partial_{c} V_{s}(y), \ s\in S$, such that

According to the condition

there exists $\bar{\xi_{i}}\in\partial_{c} f_{i}(y)-r_{i}\partial_{c} g_{i}(y), \ i\in I$ satisfying

which contradicts (4.8).

For the assumption $(ii)$, since $\bar{x}\in D$ and $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)\in \Omega_{MW}$, it holds that

By the generalized quasiconvex of the functions in conditions $(ii)$ and (4.9), the inequalities

hold, that is

The rest of the proof is omitted because it is consistent with assumption $(i)$. □

Theorem 4.5. Let $\bar{x}\in D$ be a weak Pareto solution of the problem (FPVC). The VC-Cottle constraint qualification holds at $\bar{x}$. Then, there exist $\alpha\in\mathbb{R}^{p}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$, $\gamma^{V}\in\mathbb{R}^{q}$, $\bar{\upsilon}\in\mathbb{R}^{q}$ and $\bar{\beta}\in\mathbb{R}^{q}$ such that $(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is feasible in $(D_{MW})$. If the assumptions of Theorem 4.4 are satisfied, then $(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is a Pareto solution of $(D_{MW})$.

Proof. From the assumption that $\bar{x}\in D$ and the VC-Cottle constraint qualification holds at $\bar{x}$, there exist $\alpha\in\mathbb{R}^{p}$, $\beta\in \mathbb{R}^{m}$, $\gamma^{U}\in\mathbb{R}^{q}$, and $\gamma^{V}\in\mathbb{R}^{q}$, such that necessary optimality conditions (Theorem 3.1) are fulfilled. Then, by the definitions of $\Omega_{MW}$ and (3.1)–(3.7), we conclude that $(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is feasible in $(D_{MW})$.

Suppose, on the contrary, that $(\bar{x}, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ is not a Pareto solution of $(D_{W})$, then we get $(\tilde{y}, \tilde{\alpha}, \tilde{\beta}, \tilde{\gamma^{U}}, \tilde{\gamma^{V}}, \tilde{\upsilon}, \tilde{\omega})\in \Omega_{MW}$ such that

which contradicts Theorem 4.4. □

Theorem 4.6. (Converse duality) Let $\bar{x}\in D$ be feasible in (FPVC) and $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ be a weak Pareto solution in $(D_{MW})$ such that $y\in D$. If the hypotheses of Theorem 4.4 hold, then $y$ is a Pareto solution in $(FPVC)$.

Proof. Suppose on the contrary that $y$ is not a Pareto solution in $(FPVC)$. Then there exists $\tilde{y}\in D$ such that

Since $\tilde{y}\in D$ and $(y, \alpha, \beta, \gamma^{U}, \gamma^{V}, \upsilon, \omega)$ are feasible points for (FPVC) and $(D_{MW})$, respectively, it holds that $F(\tilde{y})\nleq F(y)$ by Theorem 4.4, which contradicts to (4.10). □

5.   Concluding remarks

The optimality conditions and duality results for the problem (FPVC) with both inequality and vanishing constraints are presented. Utilizing the Clarke subdifferential, the necessary KKT optimality conditions are derived under the VC-Cottle constraint. By assuming generalized quasiconvexity and strictly generalized quasiconvexity, sufficient optimality conditions and duality theorems are established. The results in this paper improve the existing ones in ^[7]. In further research, it will be interesting to consider the second-order optimality conditions for (FPVC).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the special fund for Science and Technology Innovation Teams of Shanxi Province (NO. 202204051002018).

Conflict of interest

The authors declare there is no conflicts of interest.