1 Introduction

Karush–Kuhn–Tucker (KKT) optimality conditions are one of the most important results in optimization theory. However, KKT optimality conditions do not need to be fulfilled at local minimum points unless some constraint qualifications are satisfied; see, for example [4, p. 97], [5, Sect. 3.1] and [12, p. 78]. In Andreani et al. [2] introduced the so-called complementary approximate Karush–Kuhn–Tucker (CAKKT) condition for scalar optimization problems with smooth data. Then, the authors proved that this condition is necessary for a point to be a local minimizer without any constraint qualification. Moreover, they also showed that the augmented Lagrangian method with lower-level constraints introduced in [1] generates sequences converging to CAKKT points under certain conditions. Optimality conditions of CAKKT-type have been recognized to be useful in designing algorithms for finding approximate solutions of optimization problems; see, for example [3, 5, 8, 10, 11].

Recently, Giorgi et al. [9] extended the results in [2] to multiobjective problems with continuously differentiable data. The authors proposed the so-called approximate Karush–Kuhn–Tucker (AKKT) condition for multiobjective optimization problems. Then, they proved that the AKKT condition holds for local weak efficient solutions without any additional requirement. Under the convexity of the related functions, an AKKT-type sufficient condition for global weak efficient solutions is also established.

An interesting question arises: How does one obtain AKKT-type optimality conditions for weak efficient solutions of locally Lipschitz multiobjective optimization problems? This paper is aimed at solving the problem. We hope that our results will be useful in finding approximate efficient solutions of nonsmooth multiobjective optimization problems.

The paper is organized as follows. In Sect. 2, we recall some basic definitions and preliminaries from variational analysis, which are widely used in the sequel. Section 3 is devoted to presenting the main results.

2 Preliminaries

We use the following notation and terminology. Fix \(n \in {{\mathbb {N}}}:=\{1, 2, \ldots \}\). The space \({\mathbb {R}}^n\) is equipped with the usual scalar product and Euclidean norm. The topological closure and the topological interior of a subset S of \({\mathbb {R}}^n\) are denoted, respectively, by \(\mathrm {cl}\,{S}\) and \(\mathrm {int}\,{S}\). The closed unit ball of \({\mathbb {R}}^n\) is denoted by \({\mathbb {B}}^n.\)

Definition 2.1

(See [13]) Given \({\bar{x}}\in \text{ cl }\,S\). The set

$$\begin{aligned} N({\bar{x}}; S):=\left\{ z^*\in {\mathbb {R}}^n:\exists x^k{\mathop {\longrightarrow }\limits ^{S}}{\bar{x}}, \varepsilon _k\rightarrow 0^+, z^*_k\rightarrow z^*, z^*_k\in {{\widehat{N}}_{\varepsilon _k}}(x^k; S), \forall k \in {\mathbb {N}}\right\} , \end{aligned}$$

is called the Mordukhovich/limiting normal cone of S at \({\bar{x}}\), where

$$\begin{aligned} {\hat{N}}_\varepsilon (x; S):= \left\{ {z^* \in {{\mathbb {R}}^n}: \limsup _{u\overset{S}{\rightarrow }x} \frac{{\langle z^* , u - x\rangle }}{{\parallel u - x\parallel }} \leqq \varepsilon } \right\} \end{aligned}$$

is the set of \(\varepsilon \)-normals of S at x and \(u\xrightarrow {{S}} x\) means that \(u \rightarrow x\) and \(u \in S\).

Let \(\varphi :{\mathbb {R}}^n \rightarrow \overline{{\mathbb {R}}}\) be an extended-real-valued function. The epigraph and domain of \(\varphi \) are denoted, respectively, by

$$\begin{aligned} \text{ epi } \varphi&:=\{(x, \alpha )\in {\mathbb {R}}^n\times {\mathbb {R}}:\alpha \geqq \varphi (x) \}, \\ \text{ dom } \varphi&:= \{x\in {\mathbb {R}}^n\ \ :|\varphi (x)|<+\infty \}. \end{aligned}$$

Definition 2.2

(See [13]) Let \({\bar{x}}\in \text{ dom } \varphi \). The set

$$\begin{aligned} \partial \varphi ({\bar{x}}):=\{x^*\in {\mathbb {R}}^n: (x^*, -1)\in N(({\bar{x}}, \varphi ({\bar{x}})); \text{ epi } \varphi )\}, \end{aligned}$$

is called the Mordukhovich/limiting subdifferential of \(\varphi \) at \({\bar{x}}\). If \({\bar{x}}\notin \text{ dom } \varphi \), then we put \(\partial \varphi ({\bar{x}})=\emptyset \).

Recall that \(\varphi : {\mathbb {R}}^n\rightarrow {\mathbb {R}}^m\) is strictly differentiable at \({\bar{x}}\) iff there is a linear continuous operator \(\nabla \varphi ({\bar{x}}) : {\mathbb {R}}^n\rightarrow {\mathbb {R}}^m\), called the Fréchet derivative of \(\varphi \) at \({\bar{x}}\), such that

$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow {\bar{x}}\\ u\rightarrow {\bar{x}} \end{array}}\dfrac{\varphi (x)-\varphi (u)-\nabla \varphi ({\bar{x}})( x-u)}{\Vert x-u\Vert }=0. \end{aligned}$$

As is well-known, any function \(\varphi \) that is continuously differentiable in a neighborhood of \({\bar{x}}\) is strictly differentiable at \({\bar{x}}\). We now summarize some properties of the Mordukhovich subdifferential that will be used in the next section.

Proposition 2.1

(See [14, Proposition 6.17(d)]) Let \(\varphi :{\mathbb {R}}^n\rightarrow \overline{{\mathbb {R}}}\) be lower semicontinuous around \({\bar{x}}\). Then, for all \(\lambda \geqq 0\), one has \(\partial (\lambda \varphi )({\bar{x}})=\lambda \partial \varphi ({\bar{x}})\).

Proposition 2.2

(See [13, Corollary 1.81]) If \(\varphi :{\mathbb {R}}^n\rightarrow \overline{{\mathbb {R}}}\) is Lipschitz continuous around \({\bar{x}}\) with modulus \(L>0\), then \(\partial \varphi ({\bar{x}})\) is a nonempty compact set in \({\mathbb {R}}^n\) and contained in \(L{\mathbb {B}}^n\).

Proposition 2.3

(See [13, Theorem 3.36]) Let \(\varphi _l:{\mathbb {R}}^n\rightarrow \overline{{\mathbb {R}}}\), \(l=1, \ldots , p\), \(p\geqq 2\), be lower semicontinuous around \({\bar{x}}\) and let all but one of these functions be locally Lipschitz around \({\bar{x}}\). Then we have the following inclusion

$$\begin{aligned} \partial (\varphi _1+\cdots +\varphi _p) ({\bar{x}})\subset \partial \varphi _1 ({\bar{x}}) +\cdots +\partial \varphi _p ({\bar{x}}). \end{aligned}$$

Proposition 2.4

(See [13, Theorem 3.46]) Let \(\varphi _l:{\mathbb {R}}^n\rightarrow \overline{{\mathbb {R}}}\), \(l=1, \ldots , p\), be locally Lipschitz around \({\bar{x}}\). Then the function \( \phi (\cdot ):=\max \{\varphi _l(\cdot ):l=1, \ldots , p\}\) is also locally Lipschitz around \({\bar{x}}\) and one has

$$\begin{aligned} \partial \phi ({\bar{x}})\subset \bigcup \left\{ \partial \left( \sum _{l=1}^{p}\lambda _l\varphi _l\right) ({\bar{x}}): (\lambda _1, \ldots , \lambda _p)\in \Lambda ({\bar{x}})\right\} , \end{aligned}$$

where \(\Lambda ({\bar{x}}):=\{(\lambda _1, \ldots , \lambda _p): \lambda _l\geqq 0, \sum _{l=1}^{p}\lambda _l=1, \lambda _l[\varphi _l({\bar{x}})-\phi ({\bar{x}})]=0\}.\)

Proposition 2.5

(See [13, Theorem 3.41]) Let \(g:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^m\) be locally Lipschitz around \({\bar{x}}\) and \(\varphi :{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) be locally Lipschitz around \(g({\bar{x}})\). Then one has

$$\begin{aligned} \partial (\varphi \circ g)({\bar{x}})\subset \bigcup _{y\in \partial \varphi (g({\bar{x}}))} \partial \langle y, g\rangle ({\bar{x}}). \end{aligned}$$

In particular, if \(m=1\) and \(\varphi \) is strictly differentiable at \(g({\bar{x}})\), then

$$\begin{aligned} \partial (\varphi \circ g)({\bar{x}})\subset \partial (\nabla \varphi (g({\bar{x}}))g)({\bar{x}}). \end{aligned}$$

Proposition 2.6

(See [13, Proposition 1.114]) Let \(\varphi :{\mathbb {R}}^n\rightarrow \overline{{\mathbb {R}}}\) be finite at \({\bar{x}}\). If \(\varphi \) has a local minimum at \({\bar{x}}\), then \( 0\in \partial \varphi ({\bar{x}}).\)

Proposition 2.7

(See [6, Proposition 5.2.28]) Let \(\varphi :{\mathbb {R}}^n\rightarrow \overline{{\mathbb {R}}}\) be a lower semicontinuous function. Then the set-valued mapping \(\partial \varphi :{\mathbb {R}}^n\rightrightarrows {\mathbb {R}}^n\) is closed.

3 Main results

Let \({\mathcal {L}}:=\{1, \ldots , p\}\), \({\mathcal {I}}:=\{1, \ldots , m\}\) and \({\mathcal {J}}:=\{1, \ldots , r\}\) be index sets. Suppose that \(f=(f_1, \ldots , f_p):{\mathbb {R}}^n\rightarrow {\mathbb {R}}^p\), \(g=(g_1, \ldots , g_m):{\mathbb {R}}^n\rightarrow {\mathbb {R}}^m\), and \(h=(h_1, \ldots , h_r):{\mathbb {R}}^n\rightarrow {\mathbb {R}}^r\) are vector-valued functions with locally Lipschitz components defined on \({\mathbb {R}}^n\). Let \({\mathbb {R}}^p_+\) be the nonnegative orthant of \({\mathbb {R}}^p\). For \(a, b\in {\mathbb {R}}^p\), by \(a\leqq b\), we mean \(a-b\in -{\mathbb {R}}^p_+\); by \(a\le b\), we mean \(a-b\in -{\mathbb {R}}^p_+\setminus \{0\}\); and by \(a<b\), we mean \(a-b\in -\text {int}\,{\mathbb {R}}^p_+\).

We focus on the following constrained multiobjective optimization problem:

$$\begin{aligned} \min \,_{{\mathbb {R}}^p_+} f(x)\ \ \text {subject to}\ \ x\in {\mathcal {F}}, \end{aligned}$$
(MOP)

where \({\mathcal {F}}\) is the feasible set given by \( {\mathcal {F}}:=\{x\in {\mathbb {R}}^n\,:\, g(x)\leqq 0, h(x)=0\}. \)

Definition 3.1

Let \({\bar{x}}\in {\mathcal {F}}\). We say that:

  1. (i)

    \({\bar{x}}\) is a (global) weak efficient solution of (MOP) iff there is no \(x\in {\mathcal {F}}\) satisfying \(f(x)<f({\bar{x}})\).

  2. (ii)

    \({\bar{x}}\) is a local weak efficient solution of (MOP) iff there exists a neighborhood U of \({\bar{x}}\) such that \({\bar{x}}\) is a weak efficient solution on \(U\cap {\mathcal {F}}\).

We now introduce the concept of approximate Karush–Kuhn–Tucker condition for (MOP) inspired by the work of Giorgi et al. [9].

Definition 3.2

We say that the approximate Karush–Kuhn–Tucker condition (AKKT) is satisfied for (MOP) at a feasible point \({\bar{x}}\) iff there exist sequences \(\{x^k\}\subset {\mathbb {R}}^n\) and \(\{(\lambda ^k, \mu ^k, \tau ^k)\}\subset {\mathbb {R}}^p_+\times {\mathbb {R}}^m_+\times {\mathbb {R}}^r\) such that

  1. (A0)

    \(x^k\rightarrow {\bar{x}}\),

  2. (A1)

    \({\mathfrak {m}}(x^k; \lambda ^k, \mu ^k, \tau ^k)\rightarrow 0\) as \(k\rightarrow \infty \), where

  3. (A2)

    \(\sum _{l=1}^{p}\lambda ^k_l=1\),

  4. (A3)

    \(g_i({\bar{x}})<0\Rightarrow \mu ^k_i=0\) for sufficiently large k and \(i\in {\mathcal {I}}\).

We are now ready to state and prove our main results.

Theorem 3.1

If \({\bar{x}}\in {\mathcal {F}}\) is a local weak efficient solution of (MOP), then there exist sequences \(\{x^k\}\) and \(\{(\lambda ^k, \mu ^k, \tau ^k)\}\) satisfying the AKKT condition at \({\bar{x}}\). Furthermore, we can choose these sequences such that the following conditions hold:

  1. (E1)

    \(\mu _i^k=b_k\max (g_i(x^k), 0)\geqq 0,\)\(\forall i\in {\mathcal {I}}\), and \(\tau _j^k=c_kh_j(x^k)\geqq 0,\)\(\forall j\in {\mathcal {J}}\), where \(b_k, c_k>0,\)\(\forall k\in {\mathbb {N}}\),

  2. (E2)

    \(f_l(x^k)-f_l({\bar{x}})+\frac{1}{2}\left[ \sum _{i=1}^{m}\mu _i^kg_i(x^k)+ \sum _{j=1}^{r}\tau ^k_jh_j(x^k)\right] \leqq 0,\)\(\forall k\in {\mathbb {N}}\), \(l\in {\mathcal {L}}\).

Proof

Since \({\bar{x}}\) is a local weak efficient solution of (MOP), \(f_l\), \(g_i\) and \(h_j\) are locally Lipschitz functions, we can choose \(\delta >0\) such that these functions are Lipschitz on \(B({\bar{x}}, \delta ):=\{x\in {\mathbb {R}}^n: \Vert x-{\bar{x}}\Vert \leqq \delta \}\) and \({\bar{x}}\) is a global weak efficient solution of f on \({\mathcal {F}}\cap B({\bar{x}}, \delta )\). It is easily seen that \({\bar{x}}\) is also a global minimum solution of the function \(\phi (\cdot ):=\max \{f_l(\cdot )-f_l({\bar{x}}): l\in {\mathcal {L}}\}\) on \({\mathcal {F}}\cap B({\bar{x}}, \delta )\).

For each \(k\in {\mathbb {N}}\), we consider the following problem

figure a

where

$$\begin{aligned} \varphi _k(x):=\phi (x)+\frac{k}{2}\left[ \sum _{i=1}^{m}(\max (g_i(x),0))^2+\sum _{j=1}^{r}(h_j(x))^2\right] +\frac{1}{2}\Vert x-{\bar{x}}\Vert ^2. \end{aligned}$$

Clearly, \(\varphi _k\) is continuous on the compact set \(B({\bar{x}}, \delta )\). Thus, by the Weierstrass theorem, the problem (\(\hbox {P}_k\)) admits an optimal solution, say \(x^k\). This and the fact that \(\varphi _k({\bar{x}})=0\) imply that

$$\begin{aligned} \phi (x^k)+\frac{k}{2}\left[ \sum _{i=1}^{m}(\max (g_i(x^k),0))^2+\sum _{j=1}^{r}(h_j(x^k))^2\right] +\frac{1}{2}\Vert x^k-{\bar{x}}\Vert ^2\leqq 0, \end{aligned}$$
(1)

or, equivalently,

$$\begin{aligned} \left[ \sum _{i=1}^{m}(\max (g_i(x^k),0))^2+\sum _{j=1}^{r}(h_j(x^k))^2\right] \leqq -\frac{1}{k}\left[ 2\phi (x^k)+\Vert x^k-{\bar{x}}\Vert ^2\right] . \end{aligned}$$
(2)

By the continuity of \(\phi \) and \(\Vert x^k-{\bar{x}}\Vert \leqq \delta \), the right-hand-side of (2) tends to zero when k tends to infinity. Hence,

$$\begin{aligned} \max (g_i(x^k),0)\rightarrow 0, \forall i\in {\mathcal {I}}, h_j(x^k) \rightarrow 0,\;\;\forall j\in {\mathcal {J}}, \text {as} k\rightarrow \infty . \end{aligned}$$

This and the continuity of the functions \(\max (g_i(\cdot ),0)\) and \(h_j\) imply that every accumulation point of \(\{x^k\}\) must belongs to \({\mathcal {F}}\). Since \(\{x^k\}\subset B({\bar{x}}, \delta )\), the sequence has at least an accumulation point, say \({\tilde{x}}\in {\mathcal {F}}\). By (1), one has

$$\begin{aligned} \phi (x^k)+\frac{1}{2}\Vert x^k-{\bar{x}}\Vert ^2\leqq 0,\quad \forall k\in {\mathbb {N}}. \end{aligned}$$

Passing the last inequality to the limit as \(k\rightarrow \infty \), we get

$$\begin{aligned} \phi ({\tilde{x}})+\frac{1}{2}\Vert {\tilde{x}}-{\bar{x}}\Vert ^2\leqq 0. \end{aligned}$$

This and \(\phi ({\tilde{x}})\geqq 0\) imply that \({\tilde{x}}={\bar{x}}\). This means that the sequence \(\{x^k\}\) has a unique accumulation point \({\bar{x}}\), thus converges. Consequently, \(x^k\) belongs to the interior of \(B({\bar{x}}, \delta )\) for k large enough. Thanks to Proposition 2.6, we have

$$\begin{aligned} 0\in \partial \varphi _k(x^k). \end{aligned}$$
(3)

By Propositions 2.1–2.5, one has

$$\begin{aligned} \partial \varphi _k(x^k) \subset \partial \phi (x^k)+\frac{k}{2} \sum _{i=1}^{m}\partial (\max (g_i(\cdot ),0))^2(x^k)+\frac{k}{2}\sum _{j=1}^{r}\partial (h_j)^2(x^k)+(x^k-{\bar{x}}), \end{aligned}$$

where

$$\begin{aligned} \partial \phi (x^k)&\subset \bigcup \left\{ \partial \left( \sum _{l=1}^{p}\lambda _l(f_l(\cdot )-f_l({\bar{x}}))\right) (x^k): (\lambda _1, \ldots , \lambda _p)\in \Lambda (x^k)\right\} \\&\subset \bigcup \left\{ \sum _{i=1}^{p}\lambda _l\partial f_l (x^k): (\lambda _1, \ldots , \lambda _p)\in \Lambda (x^k)\right\} , \end{aligned}$$

with

$$\begin{aligned} \Lambda (x^k)=\left\{ (\lambda _1, \ldots , \lambda _p):\lambda _l\geqq 0, \sum _{l=1}^{p}\lambda _l=1, \lambda _l[(f_l(x^k)-f_l({\bar{x}}))-\phi (x^k)]=0\right\} , \end{aligned}$$

and

$$\begin{aligned} \partial (\max (g_i(\cdot ),0))^2(x^k)&\subset \partial (2\max (g_i(x^k), 0)\max (g_i(\cdot ), 0))(x^k) \\&=2\max (g_i(x^k), 0)\,\partial (\max (g_i(\cdot ), 0))(x^k) \\&= 2\max (g_i(x^k), 0)\,\partial g_i(x^k), \\ \partial (h_j)^2(x^k)&\subset \partial (2h_j(x^k)h_j)(x^k) \\&\subset 2|h_j(x^k)|\left[ \partial h_j(x^k)\cup \partial (-h_j)(x^k)\right] . \end{aligned}$$

Hence, (3) implies that

$$\begin{aligned} 0&\in \bigcup \left\{ \sum _{i=1}^{p}\lambda _l\partial f_l (x^k): (\lambda _1, \ldots , \lambda _p)\in \Lambda (x^k)\right\} +k\sum _{i=1}^{m}\max (g_i(x^k), 0) \partial g_i(x^k) \\&\quad \quad +k\sum _{j=1}^{r} |h_j(x^k)|\left[ \partial h_j(x^k)\cup \partial (-h_j)(x^k)\right] +(x^k-{\bar{x}}). \end{aligned}$$

This means that there exist \((\lambda _1^k, \ldots , \lambda _p^k)\in \Lambda (x^k)\), \(\xi _l^k\in \partial f_l (x^k)\), \(\eta _i^k\in \partial g_i(x^k)\) and \(\gamma _j^k\in [\partial h_j(x^k)\cup \partial (-h_j)(x^k)]\) such that

$$\begin{aligned} \sum _{i=1}^{p}\lambda _l^k\xi _l^k+k\sum _{i=1}^{m}\max (g_i(x^k), 0)\eta _i^k+k\sum _{j=1}^{r} |h_j(x^k)|\gamma _j^k+(x^k-{\bar{x}})=0. \end{aligned}$$

Hence,

$$\begin{aligned} \left\| \sum _{i=1}^{p}\lambda _l^k\xi _l^k+k\sum _{i=1}^{m}\max (g_i(x^k), 0)\eta _i^k+k\sum _{j=1}^{r} |h_j(x^k)|\gamma _j^k\right\| =\Vert x^k-{\bar{x}}\Vert . \end{aligned}$$

Setting \( \lambda ^k=(\lambda _1^k, \ldots , \lambda _p^k), \mu ^k=(\mu _1^k, \ldots , \mu _m^k), \tau ^k=(\tau _1^k, \ldots , \tau _r^k),\) where

$$\begin{aligned} \mu _i^k:=k\max (g_i(x^k), 0)\geqq 0,\quad \forall i\in {\mathcal {I}},\quad \tau _j^k:=k|h_j(x^k)|\geqq 0, \quad \forall j\in {\mathcal {J}}. \end{aligned}$$

For each \(k\in {\mathbb {N}}\), we have

$$\begin{aligned} 0\leqq {\mathfrak {m}}(x^k; \lambda ^k, \mu ^k, \tau ^k)&\leqq \left\| \sum _{i=1}^{p}\lambda _l^k\xi _l^k+\sum _{i=1}^{m}\mu _i^k\eta _i^k+\sum _{j=1}^{r} \tau _j^k\gamma _j^k\right\| \\&=\Vert x^k-{\bar{x}}\Vert . \end{aligned}$$

This and \(\lim \nolimits _{k\rightarrow \infty }x^k={\bar{x}}\) imply that \(\lim \nolimits _{k\rightarrow \infty }{\mathfrak {m}}(x^k; \lambda ^k, \mu ^k, \tau ^k)=0.\) Thus, \({\bar{x}}\) satisfies conditions (A0)–(A2). If \(g_j({\bar{x}})<0\), then \(g_j(x^k)<0\) for k large enough. Consequently, \(\mu _i^k=0\) for k large enough and we therefore get condition (A3).

For each \(j\in {\mathcal {J}}\), by passing to a subsequence if necessary, we may assume that \(h_j(x^k)\geqq 0\) for all \(k\in {\mathbb {N}}\), or \(h_j(x^k)< 0\) for all \(k\in {\mathbb {N}}\). For the last case, by replacing \(h_j\) by \({\bar{h}}_j:=-h_j\), one has

$$\begin{aligned}&\left\{ x\in {\mathbb {R}}^n:g_i(x)\leqq 0, i\in {\mathcal {I}}, h_k(x)=0, k\in {\mathcal {J}}, k\ne j, {\bar{h}}_j(x)=0\right\} ={\mathcal {F}}, \\&\partial h_j(x^k)\cup \partial (-~h_j)(x^k)=\partial {\bar{h}}_j(x^k)\cup \partial (-~{\bar{h}}_j)(x^k), \end{aligned}$$

and \({\bar{h}}_j(x^k)\geqq 0\) for all \(k\in {\mathbb {N}}\). Hence we may assume that \(h_j(x^k)\geqq 0\) for all \(k\in {\mathbb {N}}\) and \(j\in {\mathcal {J}}\). This means that \(\tau ^k_j=kh_j(x^k)\geqq 0\) for all \(k\in {\mathbb {N}}\) and \(j\in {\mathcal {J}}\) and we therefore get condition (E1). Moreover, we see that

$$\begin{aligned} \mu _i^kg_i(x^k)=k(\max (g_i(x^k), 0))^2\;\;\text { and }\;\;\tau ^k_jh_j(x^k)=k(h_j(x^k))^2. \end{aligned}$$

Thus, (1) can be rewrite as

$$\begin{aligned} \phi (x^k)+\frac{1}{2}\left[ \sum _{i=1}^{m}\mu _i^kg_i(x^k)+\sum _{j=1}^{r}\tau _j^kh_j(x^k)\right] +\frac{1}{2}\Vert x^k-{\bar{x}}\Vert ^2\leqq 0 \end{aligned}$$

and condition (E2) follows. The proof is complete. \(\square \)

Remark 3.1

If \(h_j\), \(j\in {\mathcal {J}}\), are continuously differentiable functions, then

$$\begin{aligned}&\partial (h_j)^2(x^k)=2h_j(x^k)\nabla h_j(x^k)\quad \text {and} \\&\partial h_j(x^k)\cup \partial (-~h_j)(x^k)= \left\{ \nabla h_j(x^k), -~\nabla h_j(x^k) \right\} . \end{aligned}$$

In this case, we can choose \(\gamma _j^k=\nabla h_j(x^k)\) for all \(j\in {\mathcal {J}}\) and \(k\in {\mathbb {N}}\). Thus, the conclusions of Theorem 3.1 still hold if condition (A1) is replaced by the following condition:

(A1)\(^\prime \):

\({\mathfrak {m}}^\prime (x^k; \lambda ^k, \mu ^k, \tau ^k)\rightarrow 0\) as \(k\rightarrow \infty \), where

Conditions (A0), (A1)\(^\prime \), (A2), (A3) are called by the AKKT\(^\prime \) condition. In case the problem (MOP) has no equality constraints, then conditions (A1) and (A1)\(^\prime \) coincide. In general, condition (A1)\(^\prime \) is stronger than condition (A1) because

$$\begin{aligned} {\mathfrak {m}}(x^k; \lambda ^k, \mu ^k, \tau ^k)\leqq {\mathfrak {m}}^\prime (x^k; \lambda ^k, \mu ^k, \tau ^k). \end{aligned}$$

Thus if \({\bar{x}}\) satisfies the AKKT\(^\prime \) condition with respect to sequences \(\{x^k\}\) and \(\{(\lambda ^k, \mu ^k, \tau ^k)\}\), then so does the AKKT one.

Definition 3.3

(See [9, Remark 3.2]) Let \({\bar{x}}\) be a feasible point of (MOP). We say that:

  1. (i)

    \({\bar{x}}\) satisfies the sign condition (SGN) with respect to sequences \(\{x^k\}\subset {\mathbb {R}}^n\) and \(\{(\lambda ^k, \mu ^k, \tau ^k)\}\subset {\mathbb {R}}^p_+\times {\mathbb {R}}^m_+\times {\mathbb {R}}^r\) iff, for every \(k\in {\mathbb {N}}\), one has

    $$\begin{aligned} \mu _i^k g_i(x^k)\geqq 0, i\in {\mathcal {I}},\;\;\text {and}\;\; \tau _j^k h_j(x^k)\geqq 0, j\in {\mathcal {J}}. \end{aligned}$$
  2. (ii)

    \({\bar{x}}\) satisfies the sum converging to zero condition (SCZ) with respect to sequences \(\{x^k\}\subset {\mathbb {R}}^n\) and \(\{(\lambda ^k, \mu ^k, \tau ^k)\}\subset {\mathbb {R}}^p_+\times {\mathbb {R}}^m_+\times {\mathbb {R}}^r\) iff

    $$\begin{aligned} \sum _{i=1}^{m}\mu _i^kg_i(x^k)+\sum _{j=1}^{r} \tau _j^kh_j(x^k)\rightarrow 0\;\;\text {as}\;\;k\rightarrow \infty . \end{aligned}$$

Remark 3.2

Clearly, if condition (E1) holds at \({\bar{x}}\), then so does condition SGN. Moreover, thanks to [9, Remark 3.2], conditions (A0), SGN and (E2) imply condition SCZ. The converse does not hold in general; see [9, Remark 3.4].

The following result gives sufficient optimality conditions for (global) weak efficient solutions of convex problems.

Theorem 3.2

Assume that \(f_l\)\((l=1, \ldots , p)\) and \(g_i\)\((i=1, \ldots , m)\) are convex and \(h_j\)\((j=1, \ldots , r)\) are affine. If \({\bar{x}}\) satisfies conditions \(AKKT^\prime \) and SCZ with respect to sequences \(\{x^k\}\subset {\mathbb {R}}^n\) and \(\{(\lambda ^k, \mu ^k, \tau ^k)\}\subset {\mathbb {R}}^p_+\times {\mathbb {R}}^m_+\times {\mathbb {R}}^r\), then \({\bar{x}}\) is a weak efficient solution of (MOP).

Proof

On the contrary, suppose that \({\bar{x}}\) is not a weak efficient solution of (MOP). Then, there exists \({{\hat{x}}}\in {\mathcal {F}}\) such that

$$\begin{aligned} f_l({\hat{x}})<f_l({\bar{x}})\quad \text {for all}\;\; l\in {\mathcal {L}}. \end{aligned}$$
(4)

By condition (A2), without any loss of generality, we may assume that \(\lambda ^k\rightarrow \lambda \) with \(\lambda \ge 0\) and \(\sum _{l=1}^{p}\lambda _l=1\). For k large enough, the sets \(\partial f_l(x^k)\) and \(\partial g_i(x^k)\) are compact. Hence, there exist \(\xi ^k_l\in \partial f_l(x^k)\) and \(\eta ^k_i\in \partial g_i(x^k)\) such that

$$\begin{aligned} {\mathfrak {m}}^\prime (x^k; \lambda ^k, \mu ^k, \tau ^k)=\left\| \sum _{l=1}^{p}\lambda _l^k\xi ^k_l+\sum _{i=1}^{m}\mu ^k_i\eta ^k_i+\sum _{j=1}^{r}\tau ^k_j\nabla h_j(x^k)\right\| . \end{aligned}$$

As \(f_l\) and \(g_i\) are convex and \(h_j\) are affine, for each \(k\in {\mathbb {N}}\), we have

$$\begin{aligned} f_l({\hat{x}})\geqq & {} f_l(x^k)+\langle \xi _l^k, {\hat{x}}-x^k\rangle ,\quad \forall l\in {\mathcal {L}}, \end{aligned}$$
(5)
$$\begin{aligned} g_i({\hat{x}})\geqq & {} g_i(x^k)+\langle \eta _i^k, {\hat{x}}-x^k\rangle ,\quad \forall i\in {\mathcal {I}}, \end{aligned}$$
(6)
$$\begin{aligned} h_j({\hat{x}})= & {} h_j(x^k)+\langle \nabla h_j(x^k), {\hat{x}}-x^k\rangle , \ \ \forall j\in {\mathcal {J}}. \end{aligned}$$
(7)

Multiplying (5) by \(\lambda _l^k\), (6) by \(\mu _i^k\), (7) by \(\tau ^k_j\) and adding up, we obtain

$$\begin{aligned} \sum _{l=1}^{p}\lambda _l^kf_l({\hat{x}})&\geqq \sum _{l=1}^{p}\lambda _l^kf_l({\hat{x}})+\sum _{i=1}^{m}\mu _i^kg_i({\hat{x}})+\sum _{j=1}^{r}\tau _j^kh_j({\hat{x}})\nonumber \\&\geqq \sum _{l=1}^{p}\lambda _l^kf_l(x^k)+\sum _{i=1}^{m}\mu _i^kg_i(x^k)+\sum _{j=1}^{r}\tau _j^kh_j(x^k)+\sigma _k, \end{aligned}$$
(8)

where \(\sigma _k:=(\sum _{l=1}^{p}\lambda _l^k\xi ^k_l+\sum _{i=1}^{m}\mu ^k_i\eta ^k_i+\sum _{j=1}^{r}\tau ^k_j\nabla h_j(x^k))({\hat{x}}-x^k).\) Since \(x^k\rightarrow {\bar{x}}\) and \({\mathfrak {m}}^\prime (x^k; \lambda ^k, \mu ^k, \tau ^k)\rightarrow 0\) as \(k\rightarrow \infty \), and

$$\begin{aligned} \Vert \sigma _k\Vert&\leqq \left\| \sum _{l=1}^{p}\lambda _l^k\xi ^k_l+\sum _{i=1}^{m}\mu ^k_i\eta ^k_i+\sum _{j=1}^{r}\tau ^k_j\nabla h_j(x^k)\right\| \Vert {\hat{x}}-x^k\Vert \\&= {\mathfrak {m}}^\prime (x^k; \lambda ^k, \mu ^k, \tau ^k)\Vert {\hat{x}}-x^k\Vert , \end{aligned}$$

we have \(\lim \limits _{k\rightarrow \infty }\sigma _k= 0\). By condition SCZ, taking the limit in (8), we obtain

$$\begin{aligned} \sum _{l=1}^{p}\lambda _lf_l({\hat{x}})\geqq \sum _{l=1}^{p}\lambda _lf_l({\bar{x}}). \end{aligned}$$
(9)

Moreover, since \(\lambda \ge 0\) and (4), we have \( \sum _{l=1}^{p}\lambda _lf_l({\hat{x}})<\sum _{l=1}^{p}\lambda _lf_l({\bar{x}}),\) contrary to (9). The proof is complete. \(\square \)

Clearly, if f, g and h are continuously differentiable, then Theorems 3.1 and 3.2 reduce to [9, Theorems 3.1, 3.2], respectively.

We now show that, under the additional that the quasi-normality constraint qualification and condition (E1) hold at a given feasible solution \({\bar{x}}\), an AKKT condition is also a KKT one.

Definition 3.4

We say that \({\bar{x}}\in {\mathcal {F}}\) satisfies the KKT optimality condition iff there exists a multiplier \((\lambda , \mu , \tau )\) in \({\mathbb {R}}^p_+\times {\mathbb {R}}^m_+\times {\mathbb {R}}^r\) such that

  1. (i)

    \(\lambda \ge 0\),

  2. (ii)

    \(0\in \sum _{l=1}^{p}\lambda _l\partial f_l({\bar{x}})+\sum _{i=1}^{m}\mu _i\partial g_i({\bar{x}})+\sum _{j=1}^{r}\tau _j[\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}})]\),

  3. (iii)

    \(\mu _i g_i({\bar{x}})=0\), \(i\in {\mathcal {I}}\).

Definition 3.5

We say that \({\bar{x}}\in {\mathcal {F}}\) satisfies the quasi-normality constraint qualification (QNCQ) if there is not any multiplier \((\mu , \tau )\in {\mathbb {R}}^m_+\times {\mathbb {R}}^r\) satisfying

  1. (i)

    \((\mu , \tau )\ne 0\),

  2. (ii)

    \(0\in \sum _{i=1}^{m}\mu _i\partial g_i({\bar{x}})+\sum _{j=1}^{r}\tau _j[\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}})]\),

  3. (iii)

    in every neighborhood of \({\bar{x}}\) there is a point \(x\in {\mathbb {R}}^n\) such that \(g_i(x)>0\) for all i having \(\mu _i> 0\), and \(\tau _jh_j(x)>0\) for all j having \(\tau _j\ne 0\).

Theorem 3.3

Let \({\bar{x}}\in {\mathcal {F}}\) be such that conditions AKKT and (E1) are satisfied with respect to sequences \(\{x^k\}\) and \(\{(\lambda ^k, \mu ^k, \tau ^k)\}\). If the QNCQ holds at \({\bar{x}}\), then so does the KKT optimality condition.

Proof

For each \(k\in {\mathbb {N}}\), put \(\delta _k=\Vert (\lambda ^k, \mu ^k, \tau ^k)\Vert \). By condition (A2), we have

$$\begin{aligned} \delta _k \geqq \bigg (\sum _{l=1}^{p}(\lambda _l^k)^2\bigg )^{\frac{1}{2}} \geqq \frac{1}{\sqrt{p}}>0. \end{aligned}$$
(10)

Since \(\Vert \frac{1}{\delta _k}(\lambda ^k, \mu ^k, \tau ^k)\Vert =1\) for all \(k\in {\mathbb {N}}\), we may assume that the sequence \(\{\frac{1}{\delta _k}(\lambda ^k, \mu ^k, \tau ^k)\}\) converges to \((\lambda , \mu , \tau )\in ({\mathbb {R}}^p_+\times {\mathbb {R}}^m_+\times {\mathbb {R}}^r){\setminus }\{0\}\) as k tends to infinity. By condition (A0) and Proposition 2.2, for k large enough, the sets \(\partial f_l(x^k)\), \(\partial g_i(x^k)\) and \([\partial h_j(x^k)\cup \partial (-h_j)(x^k)]\) are compact. Thus, there exist \(\xi _l^k\in \partial f_l(x^k)\), \(\eta _i^k\in \partial g_i(x^k)\) and \(\gamma _j^k\in [\partial h_j(x^k)\cup \partial (-h_j)(x^k)]\) such that

$$\begin{aligned} {\mathfrak {m}}(x^k; \lambda ^k, \mu ^k, \tau ^k)=\left\| \sum _{l=1}^{p}\lambda _l^k\xi ^k_l+\sum _{i=1}^{m}\mu ^k_i\eta ^k_i+\sum _{j=1}^{r}\tau ^k_j\gamma _j^k\right\| \end{aligned}$$
(11)

for k large enough. Since \(f_l, g_i\) and \(h_j\) are locally Lipschitz around \({\bar{x}}\), without any loss of generality, we may assume that these functions are locally Lipschitz around \({\bar{x}}\) with the same modulus L. Again by condition (A0) and Proposition 2.2, for k large enough, one has \( (\xi _l^k, \eta _i^k, \gamma _j^k)\in L {\mathbb {B}}^n\times L{\mathbb {B}}^n\times L{\mathbb {B}}^n.\) By replacing \(\{(\xi _l^k, \eta _i^k, \gamma _j^k)\}\) by a subsequence if necessary, we may assume that this sequence converges to some \((\xi _l, \eta _i, \gamma _j)\in {\mathbb {R}}^n\times {\mathbb {R}}^n\times {\mathbb {R}}^n\). By Proposition 2.7, we have

$$\begin{aligned} (\xi _l, \eta _i, \gamma _j)\in \partial f_l({\bar{x}})\times \partial g_i({\bar{x}})\times [\partial h_j({\bar{x}})\cup \partial (-h_j)({\bar{x}})]. \end{aligned}$$

From conditions (A1) and (10), dividing the both sides of (11) by \(\delta _k\) and taking the limits, we have

$$\begin{aligned} \sum _{l=1}^{p}\lambda _l\xi _l+\sum _{i=1}^{m}\mu _i\eta _i+\sum _{j=1}^{r}\tau _j\gamma _j=0. \end{aligned}$$

Thanks to condition (A3), one has \(\mu _i g_i({\bar{x}})=0 \ \ \text {for all} \ \ i\in {\mathcal {I}}\). We claim that \(\lambda \ne 0\). Indeed, if otherwise, one has \((\mu , \tau )\ne 0\) and

$$\begin{aligned} \sum _{i=1}^{m}\mu _i\eta _i+\sum _{j=1}^{r}\tau _j\gamma _j=0. \end{aligned}$$

By condition (10) and \(\mu _i^k\rightarrow \mu _i\) as \(k\rightarrow \infty \), we see that if \(\mu _i>0\), then \(\mu _i^k>0\) for k large enough. Hence, due to condition (E1), we obtain \(g_i(x^k)>0\) for all k large enough. Similarly, if \(\tau _j\ne 0\), then \(\tau _j h_j(x^k)>0\) for k large enough. Thus, the multiplier \((\mu , \tau )\) satisfies conditions (i)–(iii) in Definition 3.5, contrary to the fact that \({\bar{x}}\) satisfies the QNCQ. The proof is complete. \(\square \)

We finish this section with the following remarks.

Remark 3.3

  1. (i)

    It is well known that if \({\bar{x}}\) is a weak efficient solution of (MOP) and satisfies the QNCQ, then the KKT condition holds at this point; see [7, Theorem 3.3]. This fact may not hold if \({\bar{x}}\) is not a weak efficient solution.

  2. (ii)

    If condition (E1) does not hold, then the AKKT condition and the QNCQ do not guarantee the correctness of KKT optimality conditions even for smooth scalar optimization problems; see [4, Example 4] for more details.

  3. (iii)

    Analysis similar to that in the proof of Theorem 3.3 shows that Theorems 4.1–4.5 in [9] can be extended to multiobjective optimization problems with locally Lipschitz data. We leave the details to the reader.