Quasi Efficient Solutions and Duality Results in a Multiobjective Optimization Problem with Mixed Constraints via Tangential Subdifferentials

Abstract

We take up a nonsmooth multiobjective optimization problem with tangentially convex objective and constraint functions. In employing a suitable constraint qualification, we formulate both necessary and sufficient optimality conditions for (local) quasi efficient solutions in terms of tangential subdifferentials. Furthermore, under generalized convexity assumptions, we state strong, weak and converse duality relations of Wolfe and Mond–Weir types. We give a number of examples to illustrate the new concepts and main results of this paper.

1. Introduction

Nonsmooth multiobjective optimization (NMO) is still an area of active research thanks to the large number of applications in engineering [1], mechanics [2], economics [3], etc. Among the new aspects in this context, we can cite the recent works on quasi efficiency, which is a flexible concept that allows us to make the error dependent on the decision variables, taking into account that most of the optimization problems only yield approximate solutions due to data imprecision. For instance, to characterize quasi efficient solutions of an NMO problem, optimality conditions were investigated in the literature through the use of Clarke’s subdifferential as in [4,5,6], or the limiting subdifferential as in [7]. Moreover, the Mond–Weir duality type was examined in [8], where weak, strong and converse duality theorems were proven.

On the other hand, many papers [9,10,11,12] recently studied NMO problems by utilizing the notion of tangential subdifferentials, which was introduced by Pschenichnyi [13]. The results in these papers depict the importance of this subdifferential as it includes many types of subdifferentials such as Gâteaux derivatives, convex subdifferentials or those of Clarke and Michel–Penot. Accordingly, the optimality conditions, given in terms of this general tool, provide sharper results and cover a wide spectrum of optimization problems, including those with nonconvex and nondifferentiable functions. For instance, Martínez-Legaz [9] derived optimality conditions for pseudoconvex functions by assuming the Slater constraint qualification and convexity of the feasible set. This last condition was dropped in [10] to obtain an extended result for NMO problems. More recently, under generalized convexity assumptions, Tung [11,12] established Karush–Kuhn–Tucker (KKT) optimality conditions and duality for NMO problems subject to an infinite number of inequalities.

Motivation and Contribution. As far as the authors are aware, to date, none of the previous research has tried to simultaneously employ both the notions of local quasi efficiency and tangential subdifferentials to establish duality relations and optimality conditions of NMO problems, which we aim to undertake in the present work. In particular, we take up an NMO problem subject to inequality, equality and set constraints

$$\begin{array}{c}\hfill \left(\mathrm{NMOP}\right)\left\{\begin{array}{cc}min\hfill & f\left(x\right)=(f_1\left(x\right),\dots ,f_m\left(x\right)),\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & x\in \Omega ,\hfill \\ \hfill & g\left(x\right)=(g_1\left(x\right),\dots ,g_p\left(x\right))\le 0,\hfill \\ \hfill & h\left(x\right)=(h_1\left(x\right),\dots ,h_q\left(x\right))=0,\hfill \end{array}\right.\end{array}$$

where $\Omega \subseteq {\mathbb{R}}^n$ is a closed convex set, the scalar functions $f_i$, $i\in I=\{1,\dots ,m\}$, $g_j$, $j\in J=\{1,\dots ,p\}$, $h_k$, $k\in K=\{1,\dots ,q\}$ are defined on ${\mathbb{R}}^n$ and are all tangentially convex at a given point in the feasible region

$$\Pi =\{x\in \Omega :g(x)\le 0,h(x)=0\}.$$

It can be shown that neither f, g nor h are generally convex. However, we will ensure that, under specific constraint qualifications, KKT conditions can be proven to be necessary and sufficient for optimality. We will also consider dual programs of Wolfe and Mond–Weir types that correspond to (NMOP), and prove different duality theorems. Finally, we will give a number of examples to illustrate the new concepts and main results of our work.

We can summarize our motivation and contributions as follows:

• We investigate an NMO problem involving functions which are neither convex nor differentiable nor even locally Lipschitz. The studied problem contains different types of constraints such as geometric constraints as well as equality and inequality constraints.
• We focus on the concept of quasi efficient solutions due to their extreme importance in real-life applications in which an efficient solution fails to exist.
• We obtain new general results in terms of the tangential subdifferential, which is known to generalize both Clarke regular and Michel–Penot regular subdifferentials.
• We use a suitable constraint qualification that is weaker than many standard constraint qualifications such as Cottle, Mangasarian–Fromovitz, Kuhn–Tucker, and Zangwill.
• We establish new duality results of Wolfe and Mond–Weir types.

The paper consists of four additional sections. In the next section, we present the notations used in this paper, and recall and introduce some definitions. In Section 3, we propose KKT necessary conditions for local quasi efficient solutions of (NMOP) involving appropriate constraint qualifications. In Section 4, by employing a generalized convexity concept, we formulate sufficient optimality conditions. Finally, in Section 5, we consider dual programs of Wolfe and Mond–Weir types and prove weak, strong and converse duality theorems.

2. Preliminaries

Hereafter, for $x,y\in {\mathbb{R}}^n$, $x<y$ means $x_i<y_i$ for all $i=1,2,\dots ,n$, and $x\le y$ means $x_i\le y_i$ for all $i=1,2,\dots ,n$ and $x_k<y_k$ for some k. The set $\mathbb{B}=\{x\in \mathbb{R}:\parallel x\parallel \le 1\}$ is the closed unit ball of ${\mathbb{R}}^n$. Given a nonempty subset $\mathcal{S}$ of ${\mathbb{R}}^n$, co $\mathcal{S}$, int $\mathcal{S}$, cl $\mathcal{S}$ and cl co $\mathcal{S}$ denote the convex hull, interior, closure and closed convex hull of $\mathcal{S}$, respectively. In addition, the polar cone, the strictly negative polar cone and the orthogonal complement of $\mathcal{S}$ are respectively defined by

$$\begin{array}{ccc}{\mathcal{S}}^{\circ }& =& \left\{x\in {\mathbb{R}}^n:\langle x,d\rangle \le 0,\forall d\in \mathcal{S}\right\},\hfill \\ {\mathcal{S}}^s& =& \left\{x\in {\mathbb{R}}^n:\langle x,d\rangle <0,\forall d\in \mathcal{S}\backslash \left\{0\right\}\right\},\hfill \\ {\mathcal{S}}^{\perp }& =& \left\{x\in {\mathbb{R}}^n:\langle x,d\rangle =0,\forall d\in \mathcal{S}\right\}.\hfill \end{array}$$

It can easily be shown that, if ${\mathcal{S}}^s\ne \varnothing$, then $\mathrm{cl}\left({\mathcal{S}}^s\right)={\mathcal{S}}^{\circ }$. Moreover, ${\mathcal{S}}^{\perp }={\mathcal{S}}^{\circ }\cap {(-\mathcal{S})}^{\circ }$.

Furthermore, for $\overline{x}\in \mathrm{cl}\mathcal{S}$, the cone of feasible directions, the cone of weak feasible directions, the tangent cone and the normal cone are respectively given by

$$\begin{array}{ccc}D(\mathcal{S},\overline{x})& =& \left\{\upsilon \in {\mathbb{R}}^n:\exists r>0,\forall \lambda \in (0,r),\overline{x}+\lambda \upsilon \in \mathcal{S}\right\},\hfill \\ WD(\mathcal{S},\overline{x})& =& \left\{\upsilon \in {\mathbb{R}}^n:\exists t_m\downarrow 0,\overline{x}+t_m\upsilon \in \mathcal{S},\forall m\in \mathbb{N}\right\},\hfill \\ T(\mathcal{S},\overline{x})& =& \left\{\upsilon \in {\mathbb{R}}^n:\exists t_m\downarrow 0,\exists {\upsilon }_m\to \upsilon ,\overline{x}+t_m{\upsilon }_m\in \mathcal{S},\forall m\in \mathbb{N}\right\},\hfill \\ N(\mathcal{S},\overline{x})& =& \left\{\xi \in {\mathbb{R}}^n:\langle \xi ,\upsilon \rangle \le 0,\forall \upsilon \in T(\mathcal{S},\overline{x})\right\}={\left(T(\mathcal{S},\overline{x})\right)}^{\circ }.\hfill \end{array}$$

Observe that $D(\mathcal{S},\overline{x})$ is neither closed nor convex, and that $T(\mathcal{S},\overline{x})$ is closed but not necessarily convex. We also have $D(\mathcal{S},\overline{x})\subseteq WD(\mathcal{S},\overline{x})\subseteq T(\mathcal{S},\overline{x})$, and when $\mathcal{S}$ is convex, $D(\mathcal{S},\overline{x})=WD(\mathcal{S},\overline{x})$ and $\mathrm{cl}\left(D(\mathcal{S},\overline{x})\right)=T(\mathcal{S},\overline{x})$.

The convex cone generated by $\mathcal{S}$ and the linear hull of $\mathcal{S}$ are

$$\begin{array}{ccc}\mathrm{cone}\left(\mathcal{S}\right)& =& {\displaystyle \left\{y\in {\mathbb{R}}^n:\exists k\in \mathbb{N}\hspace{0.17em}\mathrm{s}.\mathrm{t}.y=\sum _{i=1}^k{\lambda }_i{y}_i,{\lambda }_i\ge 0,y_i\in \mathcal{S},i=1,2,\dots ,k\right\},}\hfill \\ \mathrm{lin}\left(\mathcal{S}\right)& =& {\displaystyle \left\{y\in {\mathbb{R}}^n:\exists l\in \mathbb{N}\hspace{0.17em}\mathrm{s}.\mathrm{t}.y=\sum _{i=1}^l{\lambda }_i{y}_i,{\lambda }_i\in \mathbb{R},y_i\in \mathcal{S},i=1,2,\dots ,l\right\}.}\hfill \end{array}$$

Recall that for any two sets, ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ in ${\mathbb{R}}^n$, one has

$$\begin{array}{c}\hfill \mathrm{cone}({\mathcal{S}}_1\cup {\mathcal{S}}_2)=\mathrm{cone}\left({\mathcal{S}}_1\right)+\mathrm{cone}\left({\mathcal{S}}_2\right),\mathrm{lin}({\mathcal{S}}_1\cup {\mathcal{S}}_2)=\mathrm{lin}\left({\mathcal{S}}_1\right)+\mathrm{lin}\left({\mathcal{S}}_2\right),\end{array}$$

(1)

and that ${\left({\mathcal{S}}^{\circ }\right)}^{\circ }=\mathrm{cl}\left(\mathrm{cone}\left(\mathcal{S}\right)\right)$.

Definition 1 

([13]). A function $\phi :{\mathbb{R}}^n\to \mathbb{R}$ is called tangentially convex at $\overline{x}\in {\mathbb{R}}^n$ if its directional derivative (also known as Dini derivative) at $\overline{x}$,

$${\phi }^{\prime }(\overline{x},d)=\underset{t\downarrow 0}{lim}\frac{\phi (\overline{x}+td)-\phi \left(\overline{x}\right)}{t},$$

is finite for any direction $d\in {\mathbb{R}}^n$ and convex with respect to the second argument.

Observe that the directional derivative of a tangentially convex function is sublinear because it is positive homogeneous. The following definition concerns a subdifferential concept that is closely connected to this class of functions.

Definition 2 

([13,14]). The tangential subdifferential of $\phi :{\mathbb{R}}^n\to \mathbb{R}$ at $\overline{x}\in {\mathbb{R}}^n$ is given by

$${\partial }_T\phi \left(\overline{x}\right)=\left\{y^{*}\in {\mathbb{R}}^n:\langle y^{*},d\rangle \le {\phi }^{\prime }(\overline{x},d),\forall d\in {\mathbb{R}}^n\right\}.$$

Realize that, for a tangentially convex function, the tangential subdifferential is nonempty and its support functional coincides with the directional derivative because, as we said above, the latter is sublinear [9]; that is ${\displaystyle {\phi }^{\prime }(\overline{x},d)=\underset{y^{*}\in {\partial }_T\phi \left(\overline{x}\right)}{max}\langle y^{*},d\rangle }$. Note also that the tangential subdifferential is compact and convex.

An appealing property of tangentially convex functions is that their class is properly large. In particular, convex functions on open domains are tangentially convex everywhere and their tangential subdifferential falls into the classical Fenchel subdifferential. Moreover, Gâteaux differentiable functions on open domains are also tangentially convex and their tangential subdifferential reduces to the singleton set of the gradient. This class also contains locally Lipschitz functions that are either Clarke regular [15] or Michel–Penot regular [16], and their tangential subdifferential coincides with the Clarke subdifferential in the first case and the Michel–Penot subdifferential in the second one.

We conclude this section by introducing the concept of quasi efficiency that will be used throughout this paper. To this end, assume we are given $\rho \in \mathrm{int}\left({\mathbb{R}}_{+}^m\right)$ and $\overline{x}\in \Pi$.

Definition 3. 

A point $\overline{x}\in \Pi$ will be called a local (weak) efficient solution of (NMOP) if there exists a neighborhood V of $\overline{x}$ such that $f\left(x\right)\le (<)f\left(\overline{x}\right)$ does not hold for any $x\in V\cap \Pi$. It will be called local (weak) ρ-quasi efficient if, instead of $f\left(x\right)\le (<)f\left(\overline{x}\right)$, we consider $f\left(x\right)-f\left(\overline{x}\right)\le (<)-\rho \parallel x-\overline{x}\parallel$. Finally, it will be called (weak) ρ-quasi efficient if the last statement holds for $V={\mathbb{R}}^n$.

It is easy to see that local (weak) efficient solution is local (weak) $\rho$-quasi efficient solution for some $\rho \in \mathrm{int}\left({\mathbb{R}}_{+}^m\right)$, but we do not have the converse in general as demonstrated below.

Example 1 

([4,17]). We can easily check that $\overline{x}=0$ is a local $(1,1)$-quasi efficient solution to $minf\left(x\right)=\left(ln(x+1)-x,x^3-x\right)$ subject to $x\ge 0$ but not a local efficient solution. Moreover, if we change the multiobjective function to $f\left(x\right)=\left(x^2-x,x^3-x^2\right)$, the feasible point $\overline{x}=0$ becomes local weak $(1,1)$-quasi efficient solution but not a local weak efficient solution.

3. KKT Necessary Conditions for Local Quasi Efficient Solutions

We derive KKT necessary conditions for local quasi efficient solutions of (NMOP) by means of the tangential subdifferential. To proceed, we introduce the following notations:

• ${\displaystyle F_T=\bigcup _{i\in I}({\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B})}$.
• ${\displaystyle G_T=\bigcup _{j\in J\left(\overline{x}\right)}{\partial }_T{g}_j\left(\overline{x}\right)}$, where $J\left(\overline{x}\right)=\{j\in J:g_j\left(\overline{x}\right)=0\}$ is the index set of active constraints at $\overline{x}$.
• ${\displaystyle H_T=\bigcup _{k\in K}{\partial }_T{h}_k\left(\overline{x}\right)}$.
• $A\left(\overline{x}\right)=\{\mu \in {\mathbb{R}}_{+}^n:{\mu }_j{g}_j\left(\overline{x}\right)=0,\forall j\in J\}$, which is called the set of active constraint multipliers of (NMOP) at $\overline{x}$.

These sets are employed to define three constraint qualifications:

• Abadie’s type constraint qualification (ACQ):

  $${\left(G_T\right)}^{\circ }\cap {\left(H_T\right)}^{\perp }\cap T(\Omega ,\overline{x})\subseteq T(\Pi ,\overline{x}).$$
• Zangwill’s type constraint qualification (ZCQ):

  $${\left(G_T\right)}^{\circ }\cap {\left(H_T\right)}^{\perp }\cap T(\Omega ,\overline{x})\subseteq \mathrm{cl}\left(D(\Pi ,\overline{x})\right).$$
• Weak Zangwill’s type constraint qualification (WZCQ):

  $${\left(G_T\right)}^{\circ }\cap {\left(H_T\right)}^{\perp }\cap T(\Omega ,\overline{x})\subseteq \mathrm{cl}\left(WD(\Pi ,\overline{x})\right).$$

Contrary to [8], we choose to use ${\left(H_T\right)}^{\perp }$ instead of

$${\left(\bigcup _{k\in K}{\partial }_T{h}_k\left(\overline{x}\right)\cup \bigcup _{k\in K}{\partial }_T(-h_k\left(\overline{x}\right))\right)}^{\circ }$$

because $-h_k$ can not be tangentially convex at $\overline{x}$ even when $h_k$ is tangentially convex at $\overline{x}$. Realize that, if we remove the equality or set constraints in (NMOP), we could simply ignore the corresponding term in the expression of constraint qualifications. Since $\mathrm{cl}\left(D(\Pi ,\overline{x})\right)\subset \mathrm{cl}\left(WD(\Pi ,\overline{x})\right)\subset T(\Pi ,\overline{x})$, the following implications hold:

$$\left(\mathrm{ZCQ}\right)\Rightarrow \left(\mathrm{WZCQ}\right)\Rightarrow \left(\mathrm{ACQ}\right).$$

The converse implications do not hold by default. In particular, the following example demonstrates that (ACQ) does not generally imply (WZCQ).

Example 2. 

Let $\Pi =\left\{(x,y)\in {\mathbb{R}}^2:g_i(x,y)\le 0;i=1,2;h(x,y)=0;(x,y)\in \Omega \right\},$ where

$$g_1(x,y)={\left|x\right|}^3-y,g_2(x,y)=-x^2+y,h(x,y)=\left\{\begin{array}{ccc}0,\hfill & & y\ge 0,\hfill \\ -y,\hfill & & y<0,\hfill \end{array}\right.$$

and $\Omega =\mathbb{R}\times [0,1].$ It is straightforward to check that $g_1$, $g_2$ and h are all tangentially convex at $\overline{x}=(0,0)\in \Pi$. We also have ${\partial }_T{g}_1\left(\overline{x}\right)=\left\{(0,-1)\right\}$, ${\partial }_T{g}_2\left(\overline{x}\right)=\left\{(0,1)\right\}$, ${\partial }_{T}h\left(\overline{x}\right)=\left\{0\right\}\times [-1,0]$ and $T(\Pi ,\overline{x})={\left(G_T\right)}^{\circ }\cap {\left(H_T\right)}^{\perp }\cap T(\Omega ,\overline{x})=\mathbb{R}\times \left\{0\right\}$. This yields that (ACQ) holds at $\overline{x}$. However, as $WD(\Pi ,\overline{x})=\left\{(0,0)\right\}$, (WZCQ) does not hold at $\overline{x}$.

The formulation of the KKT necessary conditions will rely on the two following lemmas.

Lemma 1 

([18]). Let $\left\{{\mathcal{S}}_j\right|j\in J\}$ be a family of nonempty convex sets in ${\mathbb{R}}^n$. Then, every nonzero vector of $\mathcal{C}=\mathrm{cone}({\displaystyle \bigcup _{j\in J}}{\mathcal{S}}_j)$ can be written as a non-negative linear combination of at most n linear independent vectors, each belonging to a different ${\mathcal{S}}_j$.

Lemma 2. 

Let $S_1$ and $S_2$ two subsets of ${\mathbb{R}}^n$. If $S_1\cap S_2=\varnothing$, then one has

$$\mathrm{int}\left(S_1\right)\cap cl\left(S_2\right)=\varnothing ,$$

Proof. 

Suppose that there exists $s\in \mathrm{int}\left(S_1\right)\cap \mathrm{cl}\left(S_2\right)$. Then, there exists $\epsilon >0$ such that $B(s,\epsilon )\subset S_1$ and for all $r>0$ one has $B(s,r)\cap S_2\ne \varnothing$. By choosing $r=\epsilon$, one obtains $S_1\cap S_2\ne \varnothing$, which contradicts the fact that $S_1\cap S_2=\varnothing$. □

The next lemma is the semi-infinite version of Motzkin’s theorem of the alternative discussed in [19] and given in [20].

Lemma 3. 

Let $S,T$ and P be three arbitrary index sets (possibly infinite). Consider the maps $\phi :S\to {\mathbb{R}}^n$, $\varphi :T\to {\mathbb{R}}^n$ and $\psi :P\to {\mathbb{R}}^n$. If the set

$$\mathrm{co}\left\{\phi \right(s),s\in S\}+\mathrm{cone}\left\{\varphi \right(t),t\in T\}+\mathrm{lin}\left\{\psi \right(p),p\in P\}$$

is closed, then the following two assertions are equivalent:

(i) 

There is no $d\in {\mathbb{R}}^n$ solution to the system

$$\left\{\begin{array}{c}\langle \phi \left(s\right),d\rangle <0,s\in S,S\ne \varnothing ,\hfill \\ \langle \varphi \left(t\right),d\rangle \le 0,t\in T,\hfill \\ \langle \psi \left(p\right),d\rangle =0,p\in P;\hfill \end{array}\right.$$

(ii) 

$0\in \mathrm{co}\left\{\phi \right(s),s\in S\}+\mathrm{cone}\left\{\varphi \right(t),t\in T\}+\mathrm{lin}\left\{\psi \right(p),p\in P\}.$

We are now ready to give necessary optimality conditions of KKT-type for local $\rho$-quasi efficient solutions of (NMOP).

Theorem 1. 

Let $\overline{x}$ be a local ρ-quasi efficient solution of (NMOP). Suppose that (WZCQ) is satisfied at $\overline{x}$. Moreover, assume that the set

$$\begin{array}{c}\hfill {\displaystyle D=\mathrm{cone}\left((\bigcup _{j=1}^p{\partial }_T{g}_j\left(\overline{x}\right))\cup N(\Omega ,\overline{x})\right)+\mathrm{lin}\left(\bigcup _{k=1}^q{\partial }_T{h}_k\left(\overline{x}\right)\right)}\end{array}$$

(2)

is closed. Then, there exists $\lambda =({\lambda }_1,\dots ,{\lambda }_m)\in {\mathbb{R}}_{+}^m$ with ${\displaystyle \sum _{i=1}^m{\lambda }_i=1}$, $\mu =({\mu }_1,\dots ,{\mu }_p)\in A\left(\overline{x}\right)$ and $\nu =({\nu }_1,\dots ,{\nu }_q)\in {\mathbb{R}}^q$ such that

$$\begin{array}{c}\hfill 0\in \sum _{i=1}^m{\lambda }_i\left({\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B}\right)+\sum _{j=1}^p{\mu }_j{\partial }_T{g}_j\left(\overline{x}\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(\overline{x}\right)+N(\Omega ,\overline{x}).\end{array}$$

(3)

Proof. 

We claim that

$$\begin{array}{c}\hfill {\left(F_T\right)}^s\cap WD(\Pi ,\overline{x})=\varnothing .\end{array}$$

(4)

Indeed, contrary to our claim, suppose that there exists $y^{*}\in {\left(F_T\right)}^s\cap WD(\Pi ,\overline{x})$, then we obtain $y^{*}\in {\left(F_T\right)}^s$ and $y^{*}\in WD(\Pi ,\overline{x})$.

Note first that, from ${\displaystyle y^{*}\in {\left(F_T\right)}^s={\left(\bigcup _{i=1}^m({\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B})\right)}^s}$, it follows that

$$\begin{array}{c}\hfill \langle x^{*},y^{*}\rangle +{\rho }_i\parallel y^{*}\parallel <0,\forall x^{*}\in {\partial }_T{f}_i\left(\overline{x}\right),\forall i\in I.\end{array}$$

(5)

For each $i\in I$, define ${\phi }_i:{\partial }_T{f}_i\left(\overline{x}\right)\subset {\mathbb{R}}^n\to \mathbb{R}$ as ${\phi }_i\left(x^{*}\right)=\langle x^{*},y^{*}\rangle$ for all $x^{*}\in {\partial }_T{f}_i\left(\overline{x}\right).$

The continuity of ${\phi }_i$ on the compact set ${\partial }_T{f}_i\left(\overline{x}\right)$ implies the existence of ${\overline{x}}_i^{*}\in {\partial }_T{f}_i\left(\overline{x}\right)$ with

$${\displaystyle {\phi }_i\left({\overline{x}}_i^{*}\right)=\underset{x^{*}\in {\partial }_T{f}_i\left(\overline{x}\right)}{max}\langle x^{*},y^{*}\rangle .}$$

Thus,

$$\langle {\overline{x}}_i^{*},y^{*}\rangle =f_i^{\prime }(\overline{x},y^{*}).$$

Hence, according to (5), we obtain for every $i\in I$

$$f_i^{\prime }(\overline{x},y^{*})+{\rho }_i\parallel y^{*}\parallel =\langle {\overline{x}}_i^{*},y^{*}\rangle +{\rho }_i\parallel y^{*}\parallel <0.$$

(6)

Now, since $y^{*}\in WD(\Pi ,\overline{x})$, there is $t_k\downarrow 0$ satisfying $\overline{x}+t_k{y}^{*}\in \Pi$ for all k. Because $\overline{x}$ is a local $\rho$-quasi efficient solution of (NMOP), there is $\overline{x}+t_k{y}^{*}\in B(\overline{x},r)$, for some $r>0$ and for large k such that there is $i_0\in I$ verifying

$$f_{i_0}(\overline{x}+t_k{y}^{*})\ge f_{i_0}\left(\overline{x}\right)-{\rho }_{i_0}\parallel t_k{y}^{*}\parallel .$$

In combining this with the fact that

$$\begin{array}{c}\hfill f_{i_0}^{\prime }(\overline{x},y^{*})+{\rho }_{i_0}\parallel y^{*}\parallel =\underset{k\to \infty }{lim}\frac{f_{i_0}(\overline{x}+t_k{y}^{*})-f_{i_0}\left(\overline{x}\right)}{t_k}+{\rho }_{i_0}\parallel y^{*}\parallel ,\end{array}$$

we obtain $f_{i_0}^{\prime }(\overline{x},y^{*})+{\rho }_{i_0}\parallel y^{*}\parallel \ge 0$ for $i_0\in I$, which contradicts (6), and consequently, (4) is fulfilled.

Taking into account (4) and applying Lemma 2 to $S_1={\left(F_T\right)}^s$ and $S_2=WD(\Pi ,\overline{x})$, we obtain

$$\mathrm{int}{\left(F_T\right)}^s\cap \mathrm{cl}\left(WD(\Pi ,\overline{x})\right)=\varnothing .$$

Thus,

$${\left(F_T\right)}^s\cap \mathrm{cl}\left(WD(\Pi ,\overline{x})\right)=\varnothing .$$

On the basis of (WZCQ), we have

$$\begin{array}{c}\hfill {\left(\bigcup _{i\in I}({\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B})\right)}^s\cap {\left(\bigcup _{j\in J\left(\overline{x}\right)}{\partial }_T{g}_j\left(\overline{x}\right)\right)}^{\circ }\cap {\left(\bigcup _{k\in K}{\partial }_T{h}_k\left(\overline{x}\right)\right)}^{\perp }\cap T(\Omega ,\overline{x})=\varnothing .\end{array}$$

(7)

The combination of the closedness and convexity of $\Omega$ with ([21] Corollary 5.2.5]) tells us that $T(\Omega ,\overline{x})={\left(N(\Omega ,\overline{x})\right)}^{\circ }$. Then, from (7), we see that the system

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\langle {\zeta }_i,y^{*}\rangle <0,\forall {\zeta }_i\in {\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B},\forall i\in I\hfill \\ \langle {\vartheta }_j,y^{*}\rangle \le 0,\forall {\vartheta }_j\in {\partial }_T{g}_j\left(\overline{x}\right),\forall j\in J\left(\overline{x}\right)\hfill \\ \langle {\eta }_k,y^{*}\rangle =0,\forall {\eta }_k\in {\partial }_T{h}_k\left(\overline{x}\right),\forall k\in K,\hfill \\ \langle \theta ,y^{*}\rangle \le 0,\forall \theta \in N(\Omega ,\overline{x}),\hfill \end{array}\right.\end{array}$$

has no solution $y^{*}\in {\mathbb{R}}^n$. On the other hand, since ${\partial }_T{f}_i\left(\overline{x}\right)$ is compact for all $i\in I$, the set ${\displaystyle \bigcup _{i=1}^m({\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B})}$ is also compact, and hence ${\displaystyle \bigcup _{i=1}^m({\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B})+D}$ is closed because so is D. Thus, by virtue of Lemma 3, we are led to

$$0\in \mathrm{co}\left(\bigcup _{i=1}^m({\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B})\right)+\mathrm{cone}\left(\bigcup _{j\in J\left(\overline{x}\right)}{\partial }_T{g}_j\left(\overline{x}\right)\cup N(\Omega ,\overline{x})\right)+\mathrm{lin}\left(\bigcup _{k=1}^q{\partial }_T{h}_k\left(\overline{x}\right)\right),$$

as well as

$$0\in \mathrm{co}\left(\bigcup _{i=1}^m({\partial }_T{f}_i\left(\overline{x}\right)+{\rho }_i\mathbb{B})\right)+\mathrm{cone}\left(\bigcup _{j\in J\left(\overline{x}\right)}{\partial }_T{g}_j\left(\overline{x}\right)\right)+\mathrm{lin}\left(\bigcup _{k=1}^q{\partial }_T{h}_k\left(\overline{x}\right)\right)+N(\Omega ,\overline{x}).$$

This yields (3) on the basis of Lemma 1. □

To illustrate Theorem 1, we present in what follows two examples of (NMOP).

Example 3. 

Let $\rho =(1,1)$, $\Omega =[0,1]$ and $f=(f_1,f_2):\mathbb{R}\to {\mathbb{R}}^2$, $g:\mathbb{R}\to \mathbb{R}$, $h:\mathbb{R}\to \mathbb{R}$ be given by

$$\begin{array}{cccc}\hfill f_1\left(x\right)& =\left\{\begin{array}{cc}{x}^{2}sin(2/x),\hfill & x\ne 0,\hfill \\ 0,\hfill & x=0,\hfill \end{array}\right.\hfill & \hfill f_2\left(x\right)& =x^2-x,\hfill \\ \hfill g\left(x\right)& =\left\{\begin{array}{cc}\hfill x-1,& x\ge 0,\hfill \\ \hfill -x-1,& x<0,\hfill \end{array}\right.\hfill & \hfill h\left(x\right)& =\left\{\begin{array}{cc}0,\hfill & x\ge 0,\hfill \\ -x,\hfill & x<0.\hfill \end{array}\right.\hfill \end{array}$$

Let $\overline{x}=0\in \Pi =[0,1]$. Since for $V=(-1,1)$, which is a neighborhood of $\overline{x}$, one has $f_1\left(x\right)\le -\left|x\right|$ and $f_2\left(x\right)\le -\left|x\right|$ does not hold for all $x\in V\cap \Pi$, then $\overline{x}$ is a local ρ-quasi efficient solution of (NMOP) (and not a local efficient solution since, for all neighborhoods V of 0, one has $f\left(x\right)\le f\left(0\right)$ for some $x\in V\cap \Pi$). An easy calculation shows that

$$\begin{array}{c}{\partial }_T{f}_1\left(\overline{x}\right)=\left\{0\right\},{\partial }_T{f}_2\left(\overline{x}\right)=\{-1\},{\partial }_{T}g\left(\overline{x}\right)=[-1,1],{\partial }_{T}h\left(\overline{x}\right)=[-1,0],\\ N(\Omega ,\overline{x})=(-\infty ,0],clWD(\Pi ,\overline{x})=[0,+\infty ),{\left({\partial }_{T}g\left(\overline{x}\right)\right)}^{\circ }=\left\{0\right\},{\left({\partial }_{T}h\left(\overline{x}\right)\right)}^{\perp }=\left\{0\right\}.\end{array}$$

Hence, $D=\mathrm{cone}\left({\partial }_{T}g\left(\overline{x}\right)\cup N(\Omega ,\overline{x})\right)+\mathrm{lin}\left({\partial }_{T}h\left(\overline{x}\right)\right)=\mathbb{R}$ is closed and

$${\left({\partial }_{T}g\left(\overline{x}\right)\right)}^{\circ }\cap {\left({\partial }_{T}h\left(\overline{x}\right)\right)}^{\perp }\cap T(\Omega ,\overline{x})\subseteq \mathrm{cl}\left(WD(\Pi ,\overline{x})\right).$$

Thus, (WZCQ) is fulfilled at $\overline{x}$. Consequently, $\overline{x}$ satisfies the assumptions of Theorem 1. In taking ${\lambda }_1={\lambda }_2=\mu =\nu =\frac{1}{2}$, the optimality condition (3) is verified.

Remark 1. 

The above example confirms that most known subdifferentials (Clarke, Michel Penot, Mordukhovich, etc.) may often contain the tangential subdifferential, and hence optimality conditions in terms of the latter provide a more general framework. Indeed, according to ([12], Example 2.2), the function $f_1$ is MP regular ([16]) at $\overline{x}$ and ${\partial }_T{f}_1\left(\overline{x}\right)=\left\{0\right\}⊊{\partial }_C{f}_1\left(\overline{x}\right)=[-2,2]$.

The next example demonstrates that ([8], Theorem 3.10) can not be applied to obtain a necessary optimality condition, while we can do so by employing Theorem 1.

Example 4. 

Consider the functions $f=(f_1,f_2):{\mathbb{R}}^2\to {\mathbb{R}}^2$, $g:{\mathbb{R}}^2\to \mathbb{R}$, $h:{\mathbb{R}}^2\to \mathbb{R}$ defined by

$$\begin{array}{cccc}\hfill f_1(x,y)& =\left\{\begin{array}{cc}{x}^3/y-x,\hfill & y\ne 0,\hfill \\ 0,\hfill & y=0,\hfill \end{array}\right.\hfill & \hfill f_2(x,y)& =max\{x,y\},\hfill \\ \hfill g(x,y)& =-x^2+y,\hfill & \hfill h(x,y)& =-x^3+y.\hfill \end{array}$$

In taking $\Omega =[0,1]\times [0,1]$, we have $\Pi =\{(x,y)\in [0,1]\times [0,1]|y=x^3\}$. Let $\rho =(1,1)$ and $\overline{x}=(0,0)\in \Pi$. We can see that $\overline{x}$ is a local ρ-quasi efficient solution of (NMOP), and, again, we can easily show that

$$\begin{array}{c}{\partial }_T{f}_1\left(\overline{x}\right)=\left\{(-1,0)\right\},{\partial }_T{f}_2\left(\overline{x}\right)=\mathrm{co}\{(1,0),(0,1)\},\\ {\partial }_{T}g\left(\overline{x}\right)=\left\{(0,1)\right\},{\partial }_{T}h\left(\overline{x}\right)=\left\{(0,1)\right\},\\ T(\Omega ,\overline{x})={\mathbb{R}}_{+}^2,N(\Omega ,\overline{x})=-{\mathbb{R}}_{+}^2,clWD(\Pi ,\overline{x})=WD(\Pi ,\overline{x})={\mathbb{R}}_{+}\times \left\{0\right\},\\ {\left({\partial }_{T}g\left(\overline{x}\right)\right)}^{\circ }=\mathbb{R}\times {\mathbb{R}}_{-},{\left({\partial }_{T}h\left(\overline{x}\right)\right)}^{\perp }=\mathbb{R}\times \left\{0\right\}.\end{array}$$

Hence, $D=\mathrm{cone}\left({\partial }_{T}g\left(\overline{x}\right)\cup N(\Omega ,\overline{x})\right)+\mathrm{lin}\left({\partial }_{T}h\left(\overline{x}\right)\right)={\mathbb{R}}_{-}\times \mathbb{R}$ is closed and

$${\left({\partial }_{T}g\left(\overline{x}\right)\right)}^{\circ }\cap {\left({\partial }_{T}h\left(\overline{x}\right)\right)}^{\perp }\cap T(\Omega ,\overline{x})={\mathbb{R}}_{+}\times \left\{0\right\}\subseteq cl\left(WD(\Pi ,\overline{x})\right).$$

Thus, (WZCQ) is fulfilled at $\overline{x}$. Consequently, $\overline{x}$ satisfies the assumptions of Theorem 1. In taking ${\lambda }_1={\lambda }_2=\frac{1}{2},\mu =1$ and $\nu =-1$, the optimality condition (3) is verified.

Remark 2. 

It is worth noting that ([8], Theorem 3.10) can not be employed for this example because $f_1$ is not locally Lipschitz at the local quasi efficient solution $\overline{x}=(0,0)$.

4. Sufficient Optimality Conditions for Quasi (Weak) Efficient Solutions

To provide sufficient optimality conditions in this section as well as the duality results in the next section, we introduce the definition of (strictly) generalized convexity in terms of tangentially subdifferentials. Since we shall employ ${\partial }_{T}h(.)$ and ${\partial }_T(-h)(.)$ evaluated at the reference point, we need the equality constraint function h together with its additive inverse $-h$ to both be tangentially convex. Note that this condition holds true for a large class of functions such as polynomials and rational functions, which are differentiable at the considered point.

Definition 4. 

Assume that $f_i$, $i\in I$, $g_j$, $j\in J$ and $\pm h_k$, $k\in K$, are all tangentially convex at a reference point $\overline{x}\in \Pi$. The triplet $(f,g,h)$ is called generalized convex on Ω at $\overline{x}\in \Omega$ if, for each $x\in \Omega \backslash \left\{\overline{x}\right\}$, $x_i^{*}\in {\partial }_T{f}_i\left(\overline{x}\right),i\in I$, $y_j^{*}\in {\partial }_T{g}_j\left(\overline{x}\right),j\in J$, $z_k^{*}\in {\partial }_T{h}_k\left(\overline{x}\right)$ and $w_k^{*}\in {\partial }_T(-h_k)\left(\overline{x}\right),k\in K$, there exists $\upsilon \in T(\Omega ,\overline{x})$ satisfying

$$\begin{array}{c}\hfill f_i\left(x\right)-f_i\left(\overline{x}\right)\ge \langle x_i^{*},\upsilon \rangle \forall i\in I\\ \hfill g_j\left(x\right)-g_j\left(\overline{x}\right)\ge \langle y_j^{*},\upsilon \rangle \forall j\in J\\ \hfill h_k\left(x\right)-h_k\left(\overline{x}\right)\ge \langle z_k^{*},\upsilon \rangle \forall k\in K\\ \hfill (-h_k)\left(x\right)-(-h_k)\left(\overline{x}\right)\ge \langle w_k^{*},\upsilon \rangle \forall k\in K\\ \hfill \langle b^{*},\upsilon \rangle \le \parallel x-\overline{x}\parallel \forall b^{*}\in \mathbb{B}.\end{array}$$

Moreover, if the first inequality is strict, then $(f,g,h)$ is called strictly generalized convex on Ω at $\overline{x}$.

Remark 3. 

Recall that a Dini-convex function at $\overline{x}\in \Omega$ [11] means a function $\phi :{\mathbb{R}}^n\to \mathbb{R}$ that satisfies for each $x\in \Omega$

$$\phi \left(x\right)-\phi \left(\overline{x}\right)\ge \langle x^{*},x-\overline{x}\rangle ,\forall x^{*}\in {\partial }_T\phi \left(\overline{x}\right).$$

Obviously, if the functions $f_i$, $g_j$ and $\pm h_k$ are all tangentially convex and Dini-convex at $\overline{x}\in \Omega$, then $(f,g,h)$ will be generalized convex on Ω at $\overline{x}$ by taking $\upsilon =x-\overline{x}$ for all $x\in \Omega$. That the converse is not always true can be seen from the following example:

Example 5. 

For $x\in \mathbb{R}$, let

$$f\left(x\right)=-x^3+x,g\left(x\right)=\left\{\begin{array}{cc}2x,\hfill & if\hspace{0.17em}x\ge 0\hfill \\ -x^2+x,\hfill & otherwise,\hfill \end{array}\right.h\left(x\right)=-2x^2+x,$$

Observe that $(f,g,h)$ is generalized convex on $\Omega =[-1,1]$ at $\overline{x}=0$. Indeed for all $x\in \Omega \backslash \left\{\overline{x}\right\}$, $x^{*}\in {\partial }_{T}f\left(\overline{x}\right)=\left\{1\right\}$, $y^{*}\in {\partial }_{T}g\left(\overline{x}\right)=[1,2]$, $z^{*}\in {\partial }_{T}h\left(\overline{x}\right)=\left\{1\right\}$ and $w^{*}\in {\partial }_T(-h)\left(\overline{x}\right)=\{-1\}$, there exists $\upsilon =-2x^2+x\in T(\Omega ,\overline{x})=\mathbb{R}$ satisfying all the inequalities of Definition 4. However, we see that f, g and h are all not Dini-convex at $\overline{x}=0$. Indeed, for $x=1$, there exists $x^{*}\in {\partial }_{T}f\left(0\right)={\partial }_{T}h\left(0\right)=\left\{1\right\}$ for which $f\left(x\right)-f\left(\overline{x}\right)\ge \langle x^{*},x-\overline{x}\rangle$ and $h\left(x\right)-h\left(\overline{x}\right)\ge \langle x^{*},x-\overline{x}\rangle$ are not satisfied, and, for $x=-1$, there exists $x^{*}\in {\partial }_{T}g\left(0\right)=[1,2]$ for which $g\left(x\right)-g\left(\overline{x}\right)\ge \langle x^{*},x-\overline{x}\rangle$ is not satisfied.

Remark 4. 

From the tangentially convexity of h and $-h$ at $\overline{x}$, it is easy to prove that ${\partial }_T(-h)\left(\overline{x}\right)=-{\partial }_{T}h\left(\overline{x}\right)$, and thereafter we can see that the following system of two inequalities

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{h}_k\left(x\right)-h_k\left(\overline{x}\right)\hfill & \ge \langle z_k^{*},\upsilon \rangle ,\forall k\in K,\hfill \\ (-h_k)\left(x\right)-(-h_k)\left(\overline{x}\right)\hfill & \ge \langle w_k^{*},\upsilon \rangle ,\forall k\in K,\hfill \end{array}\right.\end{array}$$

which appears in Definition 4, implies that

$$h_k\left(x\right)-h_k\left(\overline{x}\right)=\langle z_k^{*},\upsilon \rangle ,\forall k\in K.$$

Our next step is to investigate sufficient optimality conditions for quasi (weak) efficient solutions of (NMOP). We always suppose that $f_i$, $i\in I$, $g_j$, $j\in J$ and $\pm h_k$, $k\in K$ are all tangentially convex at a reference point $\overline{x}\in \Pi$.

Theorem 2. 

If $\overline{x}$ is feasible to (NMOP) such that the optimality condition (3) holds and $(f,g,h)$ is strictly generalized convex on Ω at $\overline{x}$; then, $\overline{x}$ is a ρ-quasi efficient solution of (NMOP).

Proof. 

Contrary to our claim, suppose $\overline{x}$ is not a $\rho$-quasi efficient solution. This means that we can find a vector $x\in \Pi$ satisfying

$$\begin{array}{c}\hfill f_i\left(x\right)+{\rho }_i\parallel x-\overline{x}\parallel \le f_i\left(\overline{x}\right),\mathrm{for}\hspace{0.17em}\mathrm{all}\hspace{0.17em}i\in I,\end{array}$$

(8)

with strict inequality for at least one i. In multiplying both sides of this inequality by ${\lambda }_i$ and summing over i, we are led to

$$\begin{array}{c}\hfill \sum _{i=1}^m{\lambda }_i{f}_i\left(x\right)+\sum _{i=1}^m{\lambda }_i{\rho }_i\parallel x-\overline{x}\parallel \le \sum _{i=1}^m{\lambda }_i{f}_i\left(\overline{x}\right).\end{array}$$

(9)

On the other side, on the basis of (3), there exist $x_i^{*}\in {\partial }_T{f}_i\left(\overline{x}\right)$, $i=1,\dots ,m$, $y_j^{*}\in {\partial }_T{g}_j\left(\overline{x}\right)$, $j=1,\dots ,p$, $z_k^{*}\in {\partial }_T{h}_k\left(\overline{x}\right),k=1,\dots ,q$, $b^{*}\in \mathbb{B}$ and ${\omega }^{*}\in N(\Omega ,\overline{x})$ satisfying

$$\sum _{i=1}^m{\lambda }_i{x}_i^{*}+\sum _{j=1}^p{\mu }_j{y}_j^{*}+\sum _{k=1}^q{\nu }_k{z}_k^{*}+\sum _{i=1}^m{\lambda }_i{\rho }_i{b}^{*}+{\omega }^{*}=0,$$

or, equivalently,

$$\sum _{i=1}^m{\lambda }_i{x}_i^{*}+\sum _{j=1}^p{\mu }_j{y}_j^{*}+\sum _{k=1}^q{\nu }_k{z}_k^{*}+\sum _{i=1}^m{\lambda }_i{\rho }_i{b}^{*}=-{\omega }^{*}.$$

Since $(f,g,h)$ is strictly generalized convex on $\Omega$ at $\overline{x}$, then, by Definition 4 and Remark 4, one has for x stated in (8) that there is $\upsilon \in T(\Omega ,\overline{x})$ such that

$$\begin{array}{cc}\hfill f_i\left(x\right)-f_i\left(\overline{x}\right)& >\langle x_i^{*},\upsilon \rangle ,\forall i\in I,\hfill \\ \hfill g_j\left(x\right)-g_j\left(\overline{x}\right)& \ge \langle y_j^{*},\upsilon \rangle ,\forall j\in J,\hfill \\ \hfill 0& =\langle z_k^{*},\upsilon \rangle ,\forall k\in K,\hfill \\ \hfill \langle b^{*},\upsilon \rangle & \le \hspace{0.17em}\parallel x-\overline{x}\parallel ,\forall b^{*}\in \mathbb{B}.\hfill \end{array}$$

Multiplying both sides of the above inequalities respectively by ${\lambda }_i$, ${\mu }_j$, ${\nu }_k$ and ${\lambda }_i{\rho }_i$ and summing over i, we obtain

$$\begin{array}{cc}\hfill 0\le -\langle {\omega }^{*},\upsilon \rangle & =\sum _{i=1}^m{\lambda }_i\langle x_i^{*},\upsilon \rangle +\sum _{j=1}^p{\mu }_j\langle y_j^{*},\upsilon \rangle +\sum _{k=1}^q{\nu }_k\langle z_k^{*},\upsilon \rangle +\sum _{i=1}^m{\lambda }_i{\rho }_i\langle b^{*},\upsilon \rangle \hfill \\ \hfill & <\sum _{i=1}^m{\lambda }_i\left(f_i\left(x\right)-f_i\left(\overline{x}\right)\right)+\sum _{j=1}^p{\mu }_j\left(g_j\left(x\right)-g_j\left(\overline{x}\right)\right)+\sum _{i=1}^m{\lambda }_i{\rho }_i\parallel x-\overline{x}\parallel .\hfill \end{array}$$

Therefore, from $x\in \Pi \backslash \left\{\overline{x}\right\}$ and $\mu \in A\left(\overline{x}\right)$,

$$\begin{array}{c}\hfill \sum _{i=1}^m{\lambda }_i{f}_i\left(\overline{x}\right)<\sum _{i=1}^m{\lambda }_i{f}_i\left(x\right)+\sum _{i=1}^m{\lambda }_i{\rho }_i\parallel x-\overline{x}\parallel ,\end{array}$$

which presents a contradiction to (9). □

Remark 5. 

If in the previous theorem, the triplet $(f,g,h)$ is only generalized convex on Ω at $\overline{x}$, then, following the same previous proof steps, we can show that $\overline{x}$ is a weak ρ-quasi efficient solution of (NMOP).

5. Duality

Related to (NMOP), we investigate in this section dual problems according to Wolfe [22] and Mond–Weir [23], and explore weak, strong and converse duality relations.

5.1. The Wolfe Duality

Let $\tau =(1,\dots ,1)\in {\mathbb{R}}^m$. We first formulate the Wolfe type dual problem for (NMOP)

$$\begin{array}{c}\hfill \left(\mathrm{WP}\right)\left\{\begin{array}{cc}max\hfill & \tilde{f}(u,\lambda ,\mu ,\nu ,\rho )=f\left(u\right)+\left({\displaystyle \sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)}\right)\tau ,\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & {\displaystyle 0\in \sum _{i=1}^m{\lambda }_i\left({\partial }_T{f}_i\left(u\right)+{\rho }_i\mathbb{B}\right)+\sum _{j=1}^p{\mu }_j{\partial }_T{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(u\right)+N(\Omega ,u),}\hfill \\ \hfill & {\displaystyle u\in \Omega ,\lambda \in {\mathbb{R}}_{+}^m,\sum _{i=1}^m{\lambda }_i=1,\mu \in {\mathbb{R}}_{+}^p,\nu \in {\mathbb{R}}^q,\rho \in \mathrm{int}\left({\mathbb{R}}_{+}^m\right).}\hfill \end{array}\right.\end{array}$$

Its feasible set is given by

$$\begin{array}{cc}\hfill {\Pi }_W=& \{(u,\lambda ,\mu ,\nu ,\rho )\in \Omega \times {\mathbb{R}}_{+}^m\times {\mathbb{R}}_{+}^p\times {\mathbb{R}}^q\times \mathrm{int}\left({\mathbb{R}}_{+}^m\right):\sum _{i=1}^m{\lambda }_i=1,\hfill \\ \hfill & 0\in \sum _{i=1}^m{\lambda }_i\left({\partial }_T{f}_i\left(u\right)+{\rho }_i\mathbb{B}\right)+\sum _{j=1}^p{\mu }_j{\partial }_T{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(u\right)+N(\Omega ,u)\}.\hfill \end{array}$$

Let us start with a weak Wolfe duality theorem.

Theorem 3.

(Weak duality) Assume that $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_W$. If $(f,g,h)$ is generalized convex on Ω at u, then, for each $\xi \ge \rho$, there exists $r>0$ such that, for each $x\in B(u,r)\cap \Pi$, we do not have $f\left(x\right)<\tilde{f}(u,\lambda ,\mu ,\nu ,\rho )-\xi \parallel x-u\parallel .$

Proof. 

Let $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_W$; then, there exist $x_i^{*}\in {\partial }_T{f}_i\left(u\right)$, $i=1,\dots ,m$, $y_j^{*}\in {\partial }_T{g}_j\left(u\right)$, $j=1,\dots ,p$, $z_k^{*}\in {\partial }_T{h}_k\left(u\right)$, $k=1,\dots ,q$, $b^{*}\in \mathbb{B}$ and $w^{*}\in N(\Omega ,u)$ satisfying

$$\begin{array}{c}\hfill \sum _{i=1}^m{\lambda }_i\left(x_i^{*}+{\rho }_i{b}^{*}\right)+\sum _{j=1}^p{\mu }_j{y}_j^{*}+\sum _{k=1}^q{\nu }_k{z}_k^{*}+w^{*}=0.\end{array}$$

(10)

By contradiction, assume that there is $\xi \ge \rho$ such that, for each $r>0$, there is $x\in B(u,r)\cap \Pi$ such that $f\left(x\right)<\tilde{f}(u,\lambda ,\mu ,\nu ,\rho )-\xi \parallel x-u\parallel$. Then,

$$\begin{array}{c}\hfill \sum _{i=1}^m{\lambda }_i\left(f_i\left(x\right)-{\tilde{f}}_i(u,\lambda ,\mu ,\nu ,\rho )+{\xi }_i\parallel x-u\parallel \right)<0.\end{array}$$

Hence,

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^m{\lambda }_i\left(f_i\left(x\right)-f_i\left(u\right)+{\xi }_i\parallel x-u\parallel -\sum _{j=1}^p{\mu }_j{g}_j\left(u\right)-\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)\right)<0.}\end{array}$$

Thus,

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^m{\lambda }_i\left(f_i\left(x\right)-f_i\left(u\right)+{\xi }_i\parallel x-u\parallel \right)-\sum _{i=1}^m{\lambda }_i\left(\sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)\right)<0.}\end{array}$$

From ${\displaystyle \sum _{i=1}^m{\lambda }_i=1}$, it follows that

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^m{\lambda }_i(f_i\left(x\right)-f_i\left(u\right)+{\xi }_i\parallel x-u\parallel )-\left(\sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)\right)<0.}\end{array}$$

(11)

However, by virtue of the generalized convexity of $(f,g,h)$ on $\Omega$ at u and Remark 4, there is $\upsilon \in T(\Omega ,u)$ such that

$$\begin{array}{ccc}{f}_i\left(x\right)-f_i\left(u\right)& \ge & \langle x_i^{*},\upsilon \rangle ,\forall i\in I,\\ g_j\left(x\right)-g_j\left(u\right)& \ge & \langle y_j^{*},\upsilon \rangle ,\forall j\in J,\\ h_k\left(x\right)-h_k\left(u\right)& =& \langle z_k^{*},\upsilon \rangle ,\forall k\in K,\\ \langle b^{*},\upsilon \rangle & \le & \parallel x-u\parallel ,\forall b^{*}\in \mathbb{B}.\end{array}$$

In combining the above inequalities with (10) while considering that $g_j\left(x\right)\le 0(j\in J)$ and $h_k\left(x\right)=0(k\in K)$, we obtain

$$\begin{array}{cc}\hfill 0\le & -\langle w^{*},\upsilon \rangle \hfill \\ \hfill =& \sum _{i=1}^m{\lambda }_i\langle x_i^{*},\upsilon \rangle +\sum _{j=1}^p{\mu }_j\langle y_j^{*},\upsilon \rangle +\sum _{k=1}^q{\nu }_k\langle z_k^{*},\upsilon \rangle +\sum _{i=1}^m{\lambda }_i{\rho }_i\langle b^{*},\upsilon \rangle \hfill \\ \hfill \le & \sum _{i=1}^m{\lambda }_i\left(f_i\left(x\right)-f_i\left(u\right)+{\xi }_i\parallel x-u\parallel \right)-\left(\sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)\right),\hfill \end{array}$$

which stands in contradiction to (11). □

Now, we establish the following strong Wolfe duality theorem.

Theorem 4.

(Strong duality) Let u be a local ρ-quasi efficient solution of (NMOP), D in (2) be closed and (WZCQ) holds at u. Then, there are $\lambda \in {\mathbb{R}}_{+}^m$, $\mu \in A\left(u\right)$ and $\nu \in {\mathbb{R}}^q$ such that $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_W$ and $f\left(u\right)=\tilde{f}(u,\lambda ,\mu ,\nu ,\rho )$. Moreover, if $(f,g,h)$ is generalized convex on Ω at u, then, for each $\xi \ge \rho$, one has $(u,\lambda ,\mu ,\nu ,\rho )$ is a local weak ξ-quasi efficient solution of (WP).

Proof. 

From Theorem 1, there exist $\lambda \in {\mathbb{R}}_{+}^m$, $\mu \in A\left(u\right)$ and $\nu \in {\mathbb{R}}^q$ satisfying

$$\begin{array}{c}\hfill 0\in \sum _{i=1}^m{\lambda }_i\left({\partial }_T{f}_i\left(u\right)+{\rho }_i\mathbb{B}\right)+\sum _{j=1}^p{\mu }_j{\partial }_T{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(u\right)+N(\Omega ,u).\end{array}$$

On the basis of ${\mu }_j{g}_j\left(u\right)=0$, for any $j\in J$ and $h_k\left(u\right)=0$ for any $k\in K$, we see that

$${\displaystyle \sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)=0.}$$

Consequently, $f\left(u\right)=\tilde{f}(u,\lambda ,\mu ,\nu ,\rho ),$ which means that $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_W$.

Furthermore, recalling the generalized convexity of $(f,g,h)$ on $\Omega$ at u, we see from Theorem 3 that, for each $\xi \ge \rho$, we can find $r>0$ such that, for each $x\in B(u,r)\cap \Pi$, we do not have $f\left(x\right)<\tilde{f}(u,\lambda ,\mu ,\nu ,\rho )-\xi \parallel x-u\parallel$. Hence, we conclude that $(u,\lambda ,\mu ,\nu ,\rho )$ is a local weak $\xi$-quasi efficient solution of (WP). □

Remark 6. 

We can easily verify that, if we assume that the triplet $(f,g,h)$ is strictly generalized convex, then the strict inequality in the conclusion of Theorem 3 will hold non-strict; i.e., we do not have $f\left(x\right)\le \tilde{f}(u,\lambda ,\mu ,\nu ,\rho )-\xi \parallel x-u\parallel$. This stronger assumption of strict generalized convexity when added in Theorem 4 will yield a local ξ-quasi efficient solution of (WP).

The following example presents an illustration of Theorem 4.

Example 6. 

Consider the functions $f=(f_1,f_2):{\mathbb{R}}^2\to {\mathbb{R}}^2$, $g:{\mathbb{R}}^2\to \mathbb{R}$ and $h:{\mathbb{R}}^2\to \mathbb{R}$ defined respectively by $f_1(x,y)=x+y^2,f_2(x,y)=\left|x\right|+\left|y\right|,g(x,y)=-x$ and $h(x,y)=y$. Let $\Omega =[0,1]\times [0,1]$ and $\rho =(1,1)$. We have $\Pi =[0,1]\times \left\{0\right\}$ and for any $u=(u_1,u_2)\in {\mathbb{R}}^2$, the Wolfe type dual problem of (NMOP) is

$$\begin{array}{c}\left\{\begin{array}{cc}max\hfill & \tilde{f}(u,\lambda ,\mu ,\nu ,\rho )=\hspace{0.17em}(u_1+u_2^2,|u_1|+|u_2\left|\right)+\left(-\mu u_1+\nu u_2\right)(1,1),\hfill \\ s.t.\hfill & 0\in {\lambda }_1{\partial }_T{f}_1\left(u\right)+{\lambda }_2{\partial }_T{f}_2\left(u\right)+\mu {\partial }_{T}g\left(u\right)+\nu {\partial }_{T}h\left(u\right)+\mathbb{B}+N(\Omega ,u),\hfill \\ \hfill & (u,\lambda ,\mu ,\nu ,\rho )\in \Omega \times {\mathbb{R}}_{+}^2\times {\mathbb{R}}_{+}\times \mathbb{R}with{\lambda }_1+{\lambda }_2=1.\hfill \end{array}\right.\end{array}$$

(12)

It is straightforward to verify that $\overline{u}=(0,0)\in \Pi$ is a local ρ-quasi efficient solution of (NMOP). An easy calculation reveals that

$$\begin{array}{ccc}{\partial }_T{f}_1\left(\overline{u}\right)=\left\{(1,0)\right\},& {\partial }_T{f}_2\left(\overline{u}\right)=[-1,1]\times [-1,1],& {\partial }_{T}g\left(\overline{u}\right)=\left\{(-1,0)\right\},\\ {\partial }_{T}h\left(\overline{u}\right)=\left\{(0,1)\right\},& T(\Omega ,\overline{u})={\mathbb{R}}_{+}^2,& N(\Omega ,\overline{u})=-{\mathbb{R}}_{+}^2,\\ {\left({\partial }_{T}g\left(\overline{u}\right)\right)}^{\circ }={\mathbb{R}}_{+}\times \mathbb{R},& {\left({\partial }_{T}h\left(\overline{u}\right)\right)}^{\perp }=\mathbb{R}\times \left\{0\right\},& clWD(\Pi ,\overline{u})={\mathbb{R}}_{+}\times \left\{0\right\}.\end{array}$$

Hence, ${\left({\partial }_{T}g\left(\overline{u}\right)\right)}^{\circ }\cap {\left({\partial }_{T}h\left(\overline{u}\right)\right)}^{\perp }\cap T(\Omega ,\overline{u})={\mathbb{R}}_{+}\times \left\{0\right\}\subseteq cl\left(WD(\Pi ,\overline{u})\right).$

Thus, (WZCQ) is fulfilled at $\overline{u}$. In taking ${\lambda }_1={\lambda }_2=\frac{1}{2},\mu =1$ and $\nu =-\frac{1}{2}$, we obtain $(\overline{u},\lambda ,\mu ,\nu ,\rho )\in {\Pi }_W$. Since D is closed and $(f,g,h)$ is generalized convex on Ω at $\overline{u}$, then, by Theorem 4, we deduce that, for each $\xi \ge \rho$, one has that $(\overline{u},\lambda ,\mu ,\nu ,\rho )$ is a local weak ξ-quasi efficient solution of (12).

Remark 7. 

• Since the notions of quasi efficiency and generalized convexity are respectively more general than those of efficiency and Dini-convexity, our results improve and extend the corresponding recent results in [11,12] when the number of constraints is finite.
• Taking into account that our objective and constraint functions are not necessarily assumed to be locally Lipschitz as is the case in [4,8], the results of this section are more general in this aspect.

The following results present two new converse duality theorems.

Theorem 5.

(Converse duality) Assume that $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_W$ such that $u\in \Pi$. If $(f,g,h)$ is strictly generalized convex on Ω at u, then u is a ρ-quasi efficient solution of (NMOP).

Proof. 

Suppose that $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_W$ with $u\in \Pi$. Then,

$$0\in \sum _{i=1}^m{\lambda }_i\left({\partial }_T{f}_i\left(u\right)+{\rho }_i\mathbb{B}\right)+\sum _{j=1}^p{\mu }_j{\partial }_T{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(u\right)+N(\Omega ,u).$$

Since $(f,g,h)$ is strictly generalized convex on $\Omega$ at u, then, by Theorem 2, we deduce that u is a $\rho$-quasi efficient solution of (NMOP). □

Remark 8. 

Notice that, if in the previous theorem, the triplet $(f,g,h)$ is only generalized convex on Π at u, then following the same previous proof steps, we can show that u is a weak ρ-quasi efficient solution of (NMOP).

Theorem 6.

(Strict converse duality) Assume that u is a ρ-quasi efficient solution of (NMOP) and $(\overline{u},\lambda ,\mu ,\nu ,\rho )$ is a ρ-quasi efficient solution of (WP) such that

$$\begin{array}{c}\hfill {\lambda }^{T}f\left(u\right)+{\lambda }^T\rho \parallel u-\overline{u}\parallel \le {\lambda }^T\tilde{f}(\overline{u},\lambda ,\mu ,\nu ,\rho ),\end{array}$$

(13)

where ${\lambda }^T$ is the transpose of the vector $({\lambda }_1,\dots ,{\lambda }_m)$.

If $(f,g,h)$ is strictly generalized convex on Ω at $\overline{u}$, then $u=\overline{u}$.

Proof. 

By contradiction, assuming that $u\ne \overline{u}$. Since $(\overline{u},\lambda ,\mu ,\nu ,\rho )\in {\Pi }_W$, then

there exist $x_i^{*}\in {\partial }_T{f}_i\left(\overline{u}\right)$, $i=1,\dots ,m$, $y_j^{*}\in {\partial }_T{g}_j\left(\overline{u}\right)$, $j=1,\dots ,p$, $z_k^{*}\in {\partial }_T{h}_k\left(\overline{u}\right)$, $k=1,\dots ,q$, $b^{*}\in \mathbb{B}$ and $w^{*}\in N(\Omega ,\overline{u})$ satisfying

$$\begin{array}{c}\hfill \sum _{i=1}^m{\lambda }_i\left(x_i^{*}+{\rho }_i{b}^{*}\right)+\sum _{j=1}^p{\mu }_j{y}_j^{*}+\sum _{k=1}^q{\nu }_k{z}_k^{*}+w^{*}=0.\end{array}$$

Hence, for all $\upsilon \in T(\Omega ,\overline{u})$, one has

$$\begin{array}{c}\hfill -\langle w^{*},\upsilon \rangle =\sum _{i=1}^m{\lambda }_i\langle x_i^{*},\upsilon \rangle +\sum _{j=1}^p{\mu }_j\langle y_j^{*},\upsilon \rangle +\sum _{k=1}^q{\nu }_k\langle z_k^{*},\upsilon \rangle +\sum _{i=1}^m{\lambda }_i{\rho }_i\langle b^{*},\upsilon \rangle \ge 0\end{array}$$

(14)

However, since $u\ne \overline{u}$ and $(f,g,h)$ is strictly generalized convex on $\Omega$ at $\overline{u}$, then, by Definition 4 and Remark 4, there is $\upsilon \in T(\Omega ,\overline{u})$ such that

$$\begin{array}{ccc}{f}_i\left(u\right)-f_i\left(\overline{u}\right)& >& \langle x_i^{*},\upsilon \rangle ,\forall i\in I,\\ g_j\left(u\right)-g_j\left(\overline{u}\right)& \ge & \langle y_j^{*},\upsilon \rangle ,\forall j\in J,\\ h_k\left(u\right)-h_k\left(\overline{u}\right)& =& \langle z_k^{*},\upsilon \rangle ,\forall k\in K,\\ \langle b^{*},\upsilon \rangle & \le & \parallel u-\overline{u}\parallel ,\forall b^{*}\in \mathbb{B}.\end{array}$$

In combining the above inequalities with (14) while considering that $g_j\left(u\right)\le 0(j\in J)$ and $h_k\left(u\right)=0(k\in K)$, we obtain

$$\sum _{i=1}^m{\lambda }_i\left(f_i\left(u\right)-f_i\left(\overline{u}\right)+{\rho }_i\parallel u-\overline{u}\parallel \right)-\left(\sum _{j=1}^p{\mu }_j{g}_j\left(\overline{u}\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(\overline{u}\right)\right)>0.$$

Since ${\displaystyle \sum _{i=1}^m{\lambda }_i=1}$, then

$$\sum _{i=1}^m{\lambda }_i\left(f_i\left(u\right)-f_i\left(\overline{u}\right)+{\rho }_i\parallel u-\overline{u}\parallel \right)-\sum _{i=1}^m{\lambda }_i\left(\sum _{j=1}^p{\mu }_j{g}_j\left(\overline{u}\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(\overline{u}\right)\right)>0.$$

Thus,

$$\sum _{i=1}^m{\lambda }_i\left(f_i\left(u\right)-f_i\left(\overline{u}\right)+{\rho }_i\parallel u-\overline{u}\parallel -\sum _{j=1}^p{\mu }_j{g}_j\left(\overline{u}\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(\overline{u}\right)\right)>0.$$

Therefore,

$${\lambda }^{T}f\left(u\right)+{\lambda }^T\rho \parallel u-\overline{u}\parallel >{\lambda }^T\tilde{f}(\overline{u},\lambda ,\mu ,\nu ,\rho ),$$

which stands in contradiction to (13). □

5.2. The Mond–Weir Duality

We are concerned here with the Mond–Weir duality type for (NMOP). To proceed, we consider

$$\begin{array}{c}\left(\mathrm{MWP}\right)\left\{\begin{array}{cc}max\hfill & f\left(u\right),\hfill \\ \mathrm{s}.\mathrm{t}.\hfill & {\displaystyle 0\in \sum _{i=1}^m{\lambda }_i\left({\partial }_T{f}_i\left(u\right)+{\rho }_i\mathbb{B}\right)}\hfill \\ \hfill & {\displaystyle +\sum _{j=1}^p{\mu }_j{\partial }_T{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(u\right)+N(\Omega ,u),}\hfill \\ \hfill & {\displaystyle \sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)\ge 0,}\hfill \\ \hfill & {\displaystyle u\in \Omega ,\lambda \in {\mathbb{R}}_{+}^m,\sum _{i=1}^m{\lambda }_i=1,\mu \in {\mathbb{R}}_{+}^p,\nu \in {\mathbb{R}}^q,\rho \in \mathrm{int}\left({\mathbb{R}}_{+}^m\right),}\hfill \end{array}\right.\hfill \end{array}$$

which has a feasible set given by

$$\begin{array}{cc}\hfill {\Pi }_D=& \{(u,\lambda ,\mu ,\nu ,\rho )\in \Omega \times {\mathbb{R}}_{+}^m\times {\mathbb{R}}_{+}^p\times {\mathbb{R}}^q\times \mathrm{int}\left({\mathbb{R}}_{+}^m\right):\sum _{i=1}^m{\lambda }_i=1,\hfill \\ \hfill & 0\in \sum _{i=1}^m{\lambda }_i\left({\partial }_T{f}_i\left(u\right)+{\rho }_i\mathbb{B}\right)+\sum _{j=1}^p{\mu }_j{\partial }_T{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(u\right)+N(\Omega ,u),\hfill \\ \hfill & \sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(u\right)\ge 0\}.\hfill \end{array}$$

The following theorem presents a weak Mond–Weir duality result.

Theorem 7.

(Weak duality) If $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_D$ and $(f,g,h)$ is generalized convex on Ω at u, then, for each $\xi \ge \rho$, we can find $r>0$ such that, for each $x\in B(u,r)\cap \Pi$, the inequality $f\left(x\right)<f\left(u\right)-\xi \parallel x-u\parallel$ does not hold.

Proof. 

Since $(u,\lambda ,\mu ,\nu ,\rho )$ is a feasible point for (MWP), there exist $x_i^{*}\in {\partial }_T{f}_i\left(u\right)$, $i=1,\dots ,m$, $y_j^{*}\in {\partial }_T{g}_j\left(u\right)$, $j=1,\dots ,p$, $z_k^{*}\in {\partial }_T{h}_k\left(u\right)$, $k=1,\dots ,q$, $b^{*}\in \mathbb{B}$ and $w^{*}\in N(\Omega ,u)$ satisfying

$$\begin{array}{c}\hfill \sum _{i=1}^m{\lambda }_i{x}_i^{*}+\sum _{j=1}^p{\mu }_j{y}_j^{*}+\sum _{k=1}^q{\nu }_k{z}_k^{*}+\sum _{i=1}^m{\lambda }_i{\rho }_i{b}^{*}+w^{*}=0,\end{array}$$

(15)

and

$$\begin{array}{c}\hfill \sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)\ge 0.\end{array}$$

(16)

By contradiction, assume that there is $\xi \ge \rho$ such that, for each $r>0$, there is $x\in B(u,r)\cap \Pi$ such that $f\left(x\right)<f\left(u\right)-\xi \parallel x-u\parallel$. Then,

$$\begin{array}{c}\hfill \sum _{i=1}^m{\lambda }_i\left(f_i\left(x\right)-f_i\left(u\right)+{\xi }_i\parallel x-u\parallel \right)<0.\end{array}$$

(17)

Because $(f,g,h)$ is generalized convex on $\Omega$ at u, then, by Definition 4 and Remark 4, there is $\upsilon \in T(\Omega ,u)$ such that

$$\begin{array}{ccc}{f}_i\left(x\right)-f_i\left(u\right)& \ge & \langle x_i^{*},\upsilon \rangle ,\forall i\in I,\\ g_j\left(x\right)-g_j\left(u\right)& \ge & \langle y_j^{*},\upsilon \rangle ,\forall j\in J,\\ h_k\left(x\right)-h_k\left(u\right)& =& \langle z_k^{*},\upsilon \rangle ,\forall k\in K,\\ \langle b^{*},\upsilon \rangle & \le & \parallel x-u\parallel ,\forall b^{*}\in \mathbb{B}.\end{array}$$

The combination of the above inequalities with (15), while taking into account that $g_j\left(x\right)\le 0$ for any $j\in J$ and $h_k\left(x\right)=0$ for any $k\in K$, tells us that

$$\begin{array}{cc}\hfill 0\le -\langle w^{*},\upsilon \rangle =& \sum _{i=1}^m{\lambda }_i\langle x_i^{*},\upsilon \rangle +\sum _{j=1}^p{\mu }_j\langle y_j^{*},\upsilon \rangle +\sum _{k=1}^q{\nu }_k\langle z_k^{*},\upsilon \rangle +\sum _{i=1}^m{\lambda }_i{\rho }_i\langle b^{*},\upsilon \rangle \hfill \\ \hfill \le & \sum _{i=1}^m{\lambda }_i\left(f_i\left(x\right)-f_i\left(u\right)+{\xi }_i\parallel x-u\parallel \right)-\left(\sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)\right),\hfill \end{array}$$

From (16), we see that

$$\sum _{i=1}^m{\lambda }_i\left(f_i\left(x\right)-f_i\left(u\right)+{\xi }_i\parallel x-u\parallel \right)\ge 0,$$

which stands in contradiction to (17). □

Next, we establish the following strong Mond–Weir duality result.

Theorem 8.

(Strong duality) Let u be a local ρ-quasi efficient solution of (NMOP), D be closed and (WZCQ) be satisfied at u. Then, there exist $\lambda \in {\mathbb{R}}_{+}^m$, $\mu \in A\left(u\right)$ and $\nu \in {\mathbb{R}}^q$ such that $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_D$. Moreover, if $(f,g,h)$ is generalized convex on Ω at u, then, for each $\xi \ge \rho$, one has $(u,\lambda ,\mu ,\nu ,\rho )$ is a local weak ξ-quasi efficient solution of (MWP), and the objective values of (NMOP) and (MWP) are equal.

Proof. 

According to Theorem 1, there exist $\lambda \in {\mathbb{R}}_{+}^m$, $\mu \in A\left(u\right)$ and $\nu \in {\mathbb{R}}^q$ satisfying

$$\begin{array}{c}\hfill 0\in \sum _{i=1}^m{\lambda }_i{\partial }_T{f}_i\left(u\right)+\sum _{j=1}^p{\mu }_j{\partial }_T{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{\partial }_T{h}_k\left(u\right)+\sum _{i=1}^m{\lambda }_i{\rho }_i\mathbb{B}+N(\Omega ,u).\end{array}$$

From ${\mu }_j{g}_j\left(u\right)=0$ for any $j\in J$ and $h_k\left(u\right)=0$ for all $k\in K$, we see that

$${\displaystyle \sum _{j=1}^p{\mu }_j{g}_j\left(u\right)+\sum _{k=1}^q{\nu }_k{h}_k\left(u\right)=0.}$$

Consequently $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_D$. As $(f,g,h)$ is generalized convex on $\Omega$ at u, Theorem 7 tells us that, for each $\xi \ge \rho$, there exists $r>0$ such that, for each $x\in B(u,r)\cap \Pi$, we do not have $f\left(x\right)<f\left(u\right)-\xi \parallel x-u\parallel .$ Hence, we obtain that $(u,\lambda ,\mu ,\nu ,\rho )$ is a local weak $\xi$-quasi efficient solution of (WP). It is easy to see that the objective values of (NMOP) and (MWP) are equal to $f\left(u\right)$. □

Remark 9. 

Again, we can easily verify that, if we require the triplet $(f,g,h)$ to be strictly generalized convex in Theorem 7, then we will obtain, for each $\xi \ge \rho$, that there exists $r>0$ such that, for each $x\in B(u,r)\cap \Pi$, the inequality $f\left(x\right)\le f\left(u\right)-\xi \parallel x-u\parallel$ does not hold. This stronger assumption when added in Theorem 8 will yield that, for each $\xi \ge \rho$, one has $(u,\lambda ,\mu ,\nu ,\rho )$ is a local ξ-quasi efficient solution of (MWP).

Using a proof similar to that of Theorem 5, we can show the following theorem.

Theorem 9.

(Converse duality) Assume that $(u,\lambda ,\mu ,\nu ,\rho )\in {\Pi }_D$ such that $u\in \Pi$. If $(f,g,h)$ is strictly generalized convex on Ω at u, then u is a ρ-quasi efficient solution of (NMOP).

The proof of the following theorem follows on the lines of Theorem 6 and hence is omitted.

Theorem 10.

(Strict converse duality) Assume that u is a ρ-quasi efficient solution of (NMOP) and $(\overline{u},\lambda ,\mu ,\nu ,\rho )$ is a ρ-quasi efficient solution of (MWP) such that (13) is fulfilled. If $(f,g,h)$ is strictly generalized convex on Ω at $\overline{u}$, then $u=\overline{u}$.

6. Conclusions

In this paper, in terms of tangential subdifferentials, we established necessary and sufficient optimality conditions as well as Wolfe and Mond–Weir type duality results for a nonsmooth multiobjective optimization problem with mixed constraints using the concept of (local/weak) quasi efficiency and generalized convexity. Our results include two lines of generalization regarding optimality conditions for (NMOP):

• Since the notions of quasi efficiency and generalized convexity are respectively more general than those of efficiency and Dini-convexity, our results improve and extend the corresponding recent results in [11,12] when the number of constraints is finite.
• In view of Remarks 1 and 2, our results extend and improve those of Golestani et al. [8] and Gupta et al. [4] as we obtained similar optimality conditions and duality results, but we dropped the local Lipschitz assumption at the local quasi efficient solution of (NMOP). Furthermore, instead of a Clarke subdiffierential, we have used the tangential subdifferential, which yields more generalized optimality conditions for a large class of nonconvex and nondifferentiable optimization problems. Finally, our work also includes many additional duality results of Wolfe and Mond–Weir type which are not found in the aforementioned papers.

For future research, the same approach could be used to investigate optimality conditions and duality results for multiobjective semi-infinite programming.