1 Introduction

In this paper, we consider the following multi-objective semi-infinite programming problem (MOSIP in brief):

$$\begin{aligned} \text {(P)} \quad \inf \;\big (f_1(x), f_2(x), \ldots , f_p(x)\big ) \\ \text {s.t.} \quad g_t(x)\le 0\quad \ t\in T,\\ x \in {\mathbb {R}}^n, \end{aligned}$$

where, \(f_i\) for \(i\in I:=\{1,2,\ldots ,p\}\) and \(g_t\) for \(t\in T\) are locally Lipschitz functions from \({\mathbb {R}}^n\) to \({\mathbb {R}}\cup \{+\infty \}\), and the index set T is arbitrary, not necessarily finite (but nonempty).

If T is a finite set, then MOSIP coincides to the multi-objective programming problem, denoted by MPP. First, for differentiable MPP, necessary optimality conditions of Karush–Kuhn–Tucker (KKT, shortly) type have been established under various constraint qualifications (CQ, briefly) by Maeda (1994). Sufficient conditions for optimality of differentiable MPP were studied by several authors, too; see Kaul et al. (1994), Osuna-Gomez et al. (1998), Bigi and Pappalardo (1999), Hanson et al. (2001). Also, for non-differentiable MPP, several CQs and optimality conditions are presented in many articles see, e.g., Hiriart-Urruty and Lemarechal (1991), Yang and Jeyakumar (1997) in convex cases, and Clarke (1983); Nobakhtian (2006a, b, c, 2008, 2009); Chuong and Kim (2014b) in non-convex cases.

At the other hand, a semi-infinite program, which denotes by SIP, is a single-objective optimization problem on a feasible set described by infinitely many inequality (and\(\backslash\)or equality) constraints, i.e., \(p=1,\ |T|\nless \infty\). Necessary and sufficient optimality conditions of SIP have been studied by many authors; see for example Goberna and López (1998), López and Still (2007) in linear case, Fajardo and López (1999), Goberna and López (1998), Mordukhovich and Nghia (2011), Mordukhovich and Nghia (2012) in convex case, Hettich and Kortanek (1993) in smooth case, and Kanzi (2011, 2014), Kanzi and Nobakhtian (2008, 2009), Kanzi and Soleimani-damaneh (2009), Zheng and Yang (2007) in locally Lipshitz case.

It can be easy to see that MOSIP is a mixture of MMP and SIP. There are only a few works available dealing with optimality conditions for MOSIP. For instance, for differentiable MOSIPs, some optimality conditions in Fritz–John (FJ, briefly) type have been presented by Caristi et al. (2010). Glover et al. (1999) considered a non-differentiable convex MOSIP, and presented some necessary and sufficient optimality conditions for it. For a non-smooth MOSIP, the “basic CQ” has been studied by Chuong and Yao (2014), who have given optimality and duality conditions of KKT type for the problem which involves the notion of Mordukhovich subdifferential. Also, Gao presented some sufficient and duality results for MOSIPs under the various generalized convexity assumptions in Gao (2012, 2013). Recently, the MOSIP with locally Lipshitz data considered in Kanzi (2015), Kanzi and Nobakhtian (2013), who presented some necessary and sufficient optimality conditions under several CQs.

The first aim of this paper is to obtain some necessary conditions for (weakly) efficient points of problem (P). In this way, we should introduce two new CQs, named Zangwill and strong Zangwill CQs, using Clarke subdifferential. For giving some new sufficient conditions, we shall define the non-smooth version of (\(\Phi\), \(\rho\))-invexity which defined by Caristi et al. (2010).

This paper is structured as follows: Sect. 2 contains some needed notations and preliminaries. In Sect. 3, we propose a FJ type necessary condition. Also, we derive some (weak and strong types) KKT necessary condition for optimality of (P) under several suitable CQs, in Sect. 3. In Sect. 4, we introduce an extension of a Caristi–Ferrara–Stefanescu result for the (\(\Phi\), \(\rho\))-invexity, and apply this notion to obtain such sufficient optimality conditions for (P).

2 Preliminaries

In this section, we briefly overview some notions of convex analysis and non-smooth analysis widely used in formulations and proofs of main results of the paper. For more details, discussion, and applications see Rockafellar (1970), Clarke (1983) and Hiriart-Urruty and Lemarechal (1991).

Given a nonempty set \(A \subseteq {\mathbb {R}}^{n}\), we denote by \(\overline{A}\), ri(A), conv(A), and cone(A), the closure of A, the relative interior of A, convex hull and convex cone (containing the origin) generated by A, respectively. The polar cone and strict polar cone of A are defined respectively by

$$\begin{aligned}& A^{-}:=\{x\in {\mathbb {R}}^{n}\mid \langle x,a \rangle \le 0,~~~ \forall a\in A\}, \\ &A^{s}:=\{x\in {\mathbb {R}}^{n}\mid \langle x,a \rangle &<;0, ~~~\forall a\in A\}. \end{aligned}$$

It is easy to show that if \(A^{s}\not =\phi\) then \(\overline{A^{s}}=A^{-}\). The bipolar theorem states that (Hiriart-Urruty and Lemarechal 1991)

$$\begin{aligned} A^{--}=\overline{cone}(A):=\overline{cone(A)}. \end{aligned}$$
(1)

The cone of feasible direction of A at \(\hat{x} \in A\) is defined by

$$\begin{aligned} D(A,\hat{x}):=\{v\in {\mathbb {R}}^{n}\mid \exists \delta >0,\ \hat{x}+\varepsilon v\in A \quad \ \forall ~ \varepsilon \in (0,\delta )\}. \end{aligned}$$

It is worth to observe that if \(\hat{x}\) is a minimizer of convex function \(\phi :\mathbb {R}^n \rightarrow \mathbb {R}\) on a convex set \(C \subseteq \mathbb {R}^n\), then

$$\begin{aligned} 0\in \partial \phi (\hat{x})+N(C,\hat{x}), \end{aligned}$$
(2)

where \(N(C,\hat{x})\) and \(\partial \phi (\hat{x})\) denote respectively the normal cone of C at \(\hat{x}\) and the convex subdifferential of \(\phi\) at \(\hat{x}\), i.e.,

$$\begin{aligned}N(C,\hat{x}):= \big \{y\in \mathbb {R}^n \mid \langle y,x-\hat{x} \rangle \le 0 \ \ \ \ \ \forall x\in C \big \},\\ \partial \phi (\hat{x}):=\big \{\xi \in {\mathbb {R}}^{n} \mid \phi (x)\ge \phi (\hat{x})+\big <\xi ,x-\hat{x} \big >\ \ \ \ \ \ \forall \; x\in {\mathbb {R}}^{n} \big \}. \end{aligned}$$

We recall from (Hiriart-Urruty and Lemarechal 1991, pp. 137) that if \(K\subseteq \mathbb {R}^n\) is an arbitrary set, then

$$\begin{aligned} N(K^-,0)=K^{--}. \end{aligned}$$
(3)

If \(\{A_{\alpha } \mid \alpha \in \Lambda \}\) is a collection of convex sets in \(\mathbb {R}^n\), and \(B:=\bigcup _{\alpha \in \Lambda } A_{\alpha }\), then it is easy to see that

$$\begin{aligned}&conv (B)=\bigg \{\sum _{j=1}^k \lambda _{\alpha _j} a_{\alpha _j} \mid a_{\alpha _j} \in A_{\alpha _j},\ k\in {\mathbb {N}},\ \lambda _{\alpha _j}\ge 0,\ \sum _{j=1}^k \lambda _{\alpha _j}=1\bigg \}, \end{aligned}$$
(4)
$$\begin{aligned}&cone (B)=\bigg \{\sum _{j=1}^k \lambda _{\alpha _j} a_{\alpha _j} \mid a_{\alpha _j} \in A_{\alpha _j},\ k\in {\mathbb {N}},\ \lambda _{\alpha _j} \ge 0 \bigg \}. \end{aligned}$$
(5)

Let \(\hat{x} \in \mathbb {R}^n\), and let \(\varphi :\mathbb {R}^n \rightarrow \mathbb {R}\) be a locally Lipschitz function. The Clarke directional derivative of \(\varphi\) at \(\hat{x}\) in the direction \(v \in \mathbb {R}^n\), and the Clarke subdifferential of \(\varphi\) at \(\hat{x}\) are respectively defined by

$$\begin{aligned} \varphi ^0(\hat{x};v):=\limsup _{y\rightarrow \hat{x},\ t\downarrow 0} \frac{\varphi (y+tv)- \varphi (y)}{t}, \end{aligned}$$
$$\begin{aligned} \partial _c \varphi (\hat{x}):=\big \{\xi \in \mathbb {R}^n \mid \left<\xi ,v\right> \le \varphi ^0 (\hat{x};v) \ \ \ \ \ \text {for\,\,all}\ v\in \mathbb {R}^n \big \}. \end{aligned}$$

The Clarke subdifferential is a natural generalization of the classical derivative since it is known that when function \(\varphi\) is continuously differentiable at \(\hat{x}\), \(\partial _c \varphi (\hat{x})=\{\nabla \varphi (\hat{x})\}\). Moreover when a function \(\varphi\) is convex, the Clarke subdifferential coincides with the subdifferential in the sense of convex analysis.

In the following theorem, we summarize some important properties of the Clarke directional derivative and the Clarke subdifferential from (Clarke 1983) which are widely used in what follows.

Theorem 1

Let \(\varphi\) and \(\phi\) be functions from \(\mathbb {R}^n\) to \(\mathbb {R}\) which are Lipschitz near \(\hat{x}\). Then,

  1. (i)

    the function \(v \rightarrow \varphi ^0(\hat{x};v)\) is finite, positively homogeneous, and subadditive on \(\mathbb {R}^n\) , and

    $$\begin{aligned}&\varphi ^0(\hat{x};v)=\max \big \{\left<\xi ,v\right> \mid \xi \in \partial _c \varphi (\hat{x}) \big \}, \end{aligned}$$
    (6)
    $$\begin{aligned}&\partial \big (\varphi ^0 (\hat{x};.)\big )(0)=\partial _c \varphi (\hat{x}). \end{aligned}$$
    (7)
  2. (ii)

    \(\partial _c \varphi (\hat{x})\) is a nonempty, convex, and compact subset of \(\mathbb {R}^n\).

  3. (iii)

    \(\varphi ^0 (x;v)\) is upper semicontinuous as a function of (x, v).

3 Necessary Conditions

As a starting point of this section, we denote by M the feasible region of problem (P), i.e.,

$$\begin{aligned} M:=\big \{x\in \mathbb {R}^n \mid g_t(x)\le 0, \ \ \ \ \ \ \forall t\in T \big \}. \end{aligned}$$

For a given \(\hat{x} \in M\), let \(T(\hat{x})\) denotes the index set of all active constraints at \(\hat{x}\),

$$\begin{aligned} T(\hat{x}):=\big \{ t\in T \mid g_t(\hat{x})=0 \big \}. \end{aligned}$$

A feasible point \(\hat{x}\) is said to be an efficient solution [resp. weakly efficient solution] to problem (P) iff there is no \(x\in M\) satisfying \(f_i(x)\le f_i(\hat{x}),\ i\in I\) and \(\big (f_1(x),\ldots ,f_p(x) \big ) \ne \big (f_1(\hat{x}),\ldots ,f_p(\hat{x}) \big )\) [resp. \(f_i(x)< f_i(\hat{x}),\ i\in I\)]. For each \(\hat{x} \in M\), set

$$\begin{aligned} F_{\hat{x}}:=\bigcup _{i\in I} \partial _c f_i(\hat{x}) \ \ \ \text {and}\ \ \ \ G_{\hat{x}}:=\bigcup _{t\in T(\hat{x})}\partial _c g_t({\hat{x}}). \end{aligned}$$

Let

$$\begin{aligned} \Psi (x):=\sup _{t\in T} g_t(x),\ \ \ \ \ \ \forall x\in M. \end{aligned}$$

Recall the following definition from Li et al. (2000) and Kanzi (2014).

Definition 1

We say that (P) has the Pshenichnyi–Levin–Valadire (PLV) property at \(x \in M\), if \(\Psi (.)\) is Lipschitz around x, and

$$\begin{aligned} \partial _c \Psi (x)\subseteq conv \left(\bigcup _{t\in T(x)} \partial _c g_t(x) \right)=conv (G_x ). \end{aligned}$$

The following condition is standard in very literatures, even in differentiable cases; see, e.g., Caristi et al. (2010), Kanzi (2015).

Assumption [A] The index set T is a nonempty compact subset of \(\mathbb {R}^l\), the function \((x,t)\rightarrow g_t(x)\) is upper semicontinuous on \(\mathbb {R}^n \times T\), and \(\partial _c g_t(x)\) is an upper semicontinuous mapping in t for each x.

The following lemma, from Theorem 5 in Kanzi (2015), will be used in sequel.

Lemma 1

Suppose that assumption [A] satisfies at \(\hat{x}\). Then,

  1. 1.

    \(G_{\hat{x}}\, {is\, a\, compact\, set}\).

  2. 2.

    \({the\,\,PLV\,\,property\,\,holds\,\,at\,\,} {\hat{x}}\).

The following result is an extension of Theorem 4 in Caristi et al. (2010).

Theorem 2

(FJ necessary condition) Let \(\hat{x}\) be a weakly efficient solution of problem (P). If assumption [A] holds at \(\hat{x}\), then there exist \(\alpha _{i}\ge 0\) (for \(i\in I\)), and \(\beta _{t}\ge 0\) (for \(t\in T(\hat{x}))\), with \(\beta _{t} \ne 0\) for finitely many indexes, such that

$$\begin{aligned}&0\in \displaystyle \sum _{i= 1} ^ p \alpha _i \partial _c f_i(\hat{x})+\displaystyle \sum _{t\in T(\hat{x})}\beta _{t}\partial _c g_{t}(\hat{x}),&\end{aligned}$$
(8)
$$\begin{aligned}&\displaystyle \sum _{i=1}^p \alpha _i+\displaystyle \sum _{t\in T(\hat{x})} \beta _t=1.&\end{aligned}$$
(9)

Proof

We know from Lemma 1 that \(G_{\hat{x}}\) is a compact set. Thus, \(F_{\hat{x}} \cup G_{\hat{x}}\) is also a compact set (since \(F_{\hat{x}}\) is obviously compact), and hence, \(conv \big (F_{\hat{x}} \cup G_{\hat{x}} \big )\) is closed.

If \(0\notin conv \big (F_{\hat{x}} \cup G_{\hat{x}} \big )\), by strict separation theorem we find a nonzero vector \(q\in \mathbb {R}^n\) such that \(\langle q,u \rangle <0\) for all \(u\in conv \big (F_{\hat{x}} \cup G_{\hat{x}} \big )\). This implies that

$$\begin{aligned} q\in \Big (conv \big (F_{\hat{x}} \cup G_{\hat{x}} \big )\Big )^s= \big (F_{\hat{x}} \cup G_{\hat{x}} \big )^s=F_{\hat{x}}^s \cap G_{\hat{x}}^s. \end{aligned}$$

Owing to Lemma 1 we conclude that the PLV property is satisfied at \(\hat{x}\). Thus, (6) and \(d\in G_{\hat{x}}^s\) imply that

$$\begin{aligned} q\in \big (conv(G_{\hat{x}})\big )^s \subseteq \big (\partial _c \Psi (\hat{x}) \big )^s \ \Longrightarrow \ \Psi ^0(\hat{x};q)<0. \end{aligned}$$

Hence, there exists a \(\delta >0\) such that \(\Psi (\hat{x}+\varepsilon q)-\Psi (\hat{x})<0\) for all \(\varepsilon \in (0,\delta )\). The last inequality and the fact that \(\Psi (\hat{x})\le 0\) (since \(\hat{x} \in M\)) conclude that \(\Psi (\hat{x}+\varepsilon q)<0\), and hence

$$\begin{aligned} g_t(\hat{x}+\varepsilon q)<0, \ \ \ \ \ \ \forall \varepsilon \in (0,\delta ),\ \forall t\in T. \end{aligned}$$
(10)

On the other hand, owing to the

$$\begin{aligned} q\in F_{\hat{x}}^s =\Big (\bigcup _{i=1}^p \partial _c f_i(\hat{x})\Big )^s=\bigcap _{i=1}^p \big (\partial _c f_i(\hat{x})\big )^s \ \Longrightarrow \ f_i^0(\hat{x};q)<0 \ \ \ \ \forall i\in I, \end{aligned}$$

for each \(i\in I\) we find \(\delta _i >0\) such that

$$\begin{aligned} f_i(\hat{x}+\varepsilon q)-f_i(\hat{x})<0, \ \ \ \ \ \ \forall \varepsilon \in (0,\delta _i). \end{aligned}$$
(11)

Take \(\hat{\delta }:=\min \{\delta ,\delta _1,\ldots ,\delta _p \}\). By (10) and (11), for each \(\varepsilon \in (0,\hat{\delta })\) we have

$$\begin{aligned} \Big (f_1(\hat{x}+\varepsilon q),\ldots ,f_p(\hat{x}+\varepsilon q)\Big )<\Big (f_1(\hat{x}),\ldots ,f_p(\hat{x})\Big ) \ \ \ \ \text {and} \ \ \ \ {\hat{x}}+\varepsilon q \in M, \end{aligned}$$

which contradicts the weak efficiency of \({\hat{x}}\). This contradiction implies that

$$\begin{aligned} 0\in conv \big (F_{\hat{x}} \cup G_{\hat{x}} \big ). \end{aligned}$$

Now, (4) proves the result. \(\square\)

The necessary conditions of FJ type can be viewed as being degenerate when the multiplier corresponding to the objective function vanishes, since then the function being minimized is not involved. In the next theorem we derive a KKT type necessary condition for optimality of (P) under a suitable qualification condition.

Definition 2

Let \(\hat{x} \in M\). We say that (P) satisfies the Zangwill CQ (ZCQ) at \(\hat{x},\) if

$$\begin{aligned} G^- _{\hat{x}}\subseteq \overline{D(M,\hat{x})}. \end{aligned}$$

Remark 1

When T is finite and each \(g_t\) is a differentiable function, Definition 2 is the classical definition of Zangwill CQ introduced in Zangwill (1969). When T is finite and each \(g_t\) is an affine function, the ZCQ is satisfied at each \(\hat{x} \in M\); see, e.g., Hiriart-Urruty and Lemarechal (1991). When each \(g_t\) is affine, for semi-infinite system of linear inequalities, the ZCQ may not hold as shown by the following example.

Example 1

Let \(T:=\mathbb {N}\cup \{0\}\) and \(n:=2\). Also, for each \(x:=(x_1,x_2)\) take

$$\begin{aligned}&\&g_0(x):=x_2-1, \\&\&g_1(x):=-x_1, \\&\&g_2(x):=x_1-1, \\&\&g_t(x):=-x_2-\frac{1}{t}, \ \ \ \forall t\in \{3,4,\ldots \}. \end{aligned}$$

Then \(M=[0,1]\times [0,1]\) and \(T(\hat{x})=\{1\}\), where \(\hat{x}:=(0,0)\). It is easy to see that \(G^- _{\hat{x}} =\{ (-1,0)\}^-=[0,+\infty )\times \mathbb {R}\), whence \(\overline{D(M,\hat{x})}=[0,+\infty ) \times [0,+\infty )\). Thus, the ZCQ does not hold at \(\hat{x}\).

Theorem 3

(KKT necessary condition) Let \(\hat{x}\) be a weakly efficient solution of (P), ZCQ holds at \(\hat{x}\), and \(cone\big (G_{\hat{x}}\big )\) be a closed cone. Then there exist \(\alpha _{i}\ge 0\) (for \(i\in I\)), and \(\beta _{t}\ge 0,\) (for \(t\in T(\hat{x})\)), with \(\beta _{t} \ne 0\) for finitely many indexes, such that

$$\begin{aligned}&0\in \displaystyle \sum _{i= 1} ^ p \alpha _i \partial _c f_i(\hat{x})+\displaystyle \sum _{t\in T(\hat{x})}\beta _{t}\partial _c g_{t}(\hat{x}),&\end{aligned}$$
(12)
$$\begin{aligned}&\displaystyle \sum _{i=1}^p \alpha _i=1.&\end{aligned}$$
(13)

Proof

We first claim that

$$\begin{aligned} \max _{i\in I}f_i^0(\hat{x};d)\ge 0, \ \ \ \ \ \ \ \ \ \forall d\in D(M,\hat{x}). \end{aligned}$$
(14)

On the contrary, suppose that there exists a vector \(d\in D(M,\hat{x})\) such that \(f_i^0(\hat{x};d)< 0\) for all \(i\in I\). Thus, there exist positive scalars \(\delta ,\delta _1,\ldots ,\delta _p,\) such that

$$\begin{aligned} \left\{ \begin{array}{ll} \hat{x}+\varepsilon d \in M&{}~~~~~~ \forall \varepsilon \in (0,\delta ),\\ \ \ &{} ~ \ \ \\ f_i(\hat{x}+\varepsilon d )-f_i(\hat{x})<0&{}~~~~~~ \forall \varepsilon \in (0,\delta _i). \end{array} \right. \end{aligned}$$

Take \(\hat{\delta }:=\min \{\delta ,\delta _1,\ldots ,\delta _p \}\). Under consideration above inequalities, for each \(\varepsilon \in (0,\hat{\delta })\) we have

$$\begin{aligned} \Big (f_1(\hat{x}+\varepsilon d),\ldots ,f_p(\hat{x}+\varepsilon d)\Big )<\Big (f_1(\hat{x}),\ldots ,f_p(\hat{x})\Big ) \ \ \ \ \text {and} \ \ \ \ \hat{x}+\varepsilon d \in M, \end{aligned}$$

which contradicts the weak efficiency of \(\hat{x}\). Thus, (14) is true.

If \(\hat{d}\in \overline{D(M,\hat{x})}\), there exists a sequence \(\{d_k\}_{k=1}^\infty\) in \(D(M,\hat{x})\) converging to \(\hat{d}\). Owning to (14) and continuity of function \(\varphi (d):=\max _{i\in I}f_i^0(\hat{x};d)\), we deduce that

$$\begin{aligned} \varphi (\hat{d})=\lim _{k\rightarrow \infty }\varphi (d_k) \ge 0. \end{aligned}$$

We thus proved that (by assumption of ZCQ at \(\hat{x}\))

$$\begin{aligned} \varphi (d)=\max _{i\in I}f_i^0(\hat{x}; d)\ge 0, \ \ \ \ \ \ \ \ \ \forall d \in G_{\hat{x}}. \end{aligned}$$

Since \(0\in G^- _{\hat{x}}\) and \(\varphi (0)=0\), the last relation implies that the following convex problem has a minimum at \(\bar{d}:=0\),

$$\begin{aligned} \min \varphi (d), \ \ \ \ \ \ \text {subject\,\,to} \ \ \ \ \ d\in G_{\hat{x}}. \end{aligned}$$

Hence, by (2), (3), and (7) we obtain that

$$\begin{aligned} 0\in \partial \varphi (0)+N\big ( G^- _{\hat{x}}, 0\big )=conv \Big ( \bigcup _{i=1}^p \partial f_i^0(\hat{x};.)(0) \Big )+G^{--} _{\hat{x}}=conv(F_{\hat{x}})+\overline{cone}(G_{\hat{x}}). \end{aligned}$$

Now, the closedness of \(cone\big (G_{\hat{x}}\big )\), (1), (4), and (5) prove the results. \(\square\)

In almost all example, we could not obtain positive KKT multipliers associated with the vector-valued objective function, namely, some of the multipliers may be equal to zero. This means that the components of the vector-valued objective function do not play the role in the necessary conditions for weakly efficiency. To avoid the case where some of the KKT multipliers associated with the objective function vanish for the problem (P), an approach has been developed in Kanzi (2015), and strong KKT necessary optimality conditions have been obtained. We say that strong KKT condition holds for a multi-objective optimization problem, when the KKT multipliers are positive for all components of the objective function. In the below theorem, we establish the strong KKT necessary conditions for efficient solution (not weakly efficient solution) of (P) under a suitable qualification condition. We should observe that in Kanzi (2015) presented a strong KKT necessary condition for weakly efficient solution of (P).

Let \(\hat{x} \in S\). On the lines of Maeda (1994), for each \(i\in I\), define the set

$$\begin{aligned} &Q^i(\hat{x}):=\Big \{x\in M \mid f_l(x)\le f_l(\hat{x}) \ \ \ \forall l\in I{\setminus} \{i\} \Big \}, \ \ \text {if}\ \ p& >1,\\&Q^i(\hat{x}):=M,\ \ {\text {if}}\ \ p=1. \end{aligned}$$

For the sake of the simplicity, we denote \(Q^i(\hat{x})\) by \(Q^i\) in this paper.

Definition 3

Let \(\hat{x} \in M\). We say that (P) satisfies the strong Zangwill CQ (SZCQ) at \(\hat{x},\) if

$$\begin{aligned} G^- _{\hat{x}}\subseteq \bigcap _{i=1}^p\overline{D(Q^i,\hat{x})}. \end{aligned}$$

Remark 2

When T is finite and each \(g_t\) is a differentiable function, the SZCQ reduces to the CQ introduced in Maeda (1994). We observe that the following implication holds at each \(\hat{x} \in M\):

$$\begin{aligned} \text {SZCQ} \ \Longrightarrow \ \text {ZCQ}. \end{aligned}$$

The converse of above implication is true when \(p=1\). In the following example, a non-smooth semi-infinite problem with \(p=2\) is considered that satisfies SZCQ.

Example 2

Let \(T:=[0,1],\) \(\hat{x}:=(0,0)\), \(f_1(x_1,x_2):=-x_1,\) and \(f_2(x)\) is the support function of \(V:=\left\{ (y_1,y_2)\in \mathbb {R}^2\mid y_1^2 + (y_2 + 1)^2 \le 1\right\}\), i.,e.,

$$\begin{aligned} f_2(x):=\sup _{y\in V} \left<y,x\right>. \end{aligned}$$

Take also

$$\begin{aligned}& g_0(x_1,x_2):=-|x_1|, \\& g_t(x_1,x_2):=-x_1-t,\ \ \ \forall t\in (0,1]. \end{aligned}$$

A short calculation shows that:

  • \(M=[0,+\infty )\times \mathbb {R}\),

  • \(Q^1=\{0\} \times \mathbb {R}=\overline{D(Q^1,\hat{x})}\),

  • \(Q^2=[0,+\infty )\times \mathbb {R}=\overline{D(Q^2,\hat{x})}\),

  • \(G_{\hat{x}}^- =\big ([-1,1] \times \{ 0 \} \big )^-=\{0 \}\times \mathbb {R}\),

and hence, \(G_{\hat{x}}^- =\overline{D(Q^1,\hat{x})}\cap \overline{D(Q^2,\hat{x})}\). Thus, SZCQ holds at \(\hat{x}\).

Theorem 4

(Strong KKT necessary condition) Let \(\hat{x}\) be an efficient solution of (P). If in addition, (SZCQ) and the condition

$$\begin{aligned} (\mathfrak {A}): \ \ \ \ \ \Big (\bigcup _{i=1}^p\partial _c f_i(\hat{x}) \Big )^- {\setminus} {\{0\}}\subseteq \bigcup _{i=i}^p \big ( \partial _c f_i(\hat{x})\big )^s \end{aligned}$$

hold at \(\hat{x}\), then there exist \(\alpha _{i}>0\) (for \(i\in I\)), and \(\beta _{t}\ge 0,\) (for \(t\in T(\hat{x})\)), with \(\beta _{t} \ne 0\) for finitely many indexes, such that

$$\begin{aligned}&0\in \displaystyle \sum _{i= 1} ^ p \alpha _i \partial _c f_i(\hat{x})+\displaystyle \sum _{t\in T(\hat{x})}\beta _{t}\partial _c g_{t}(\hat{x}),&\\&\displaystyle \sum _{i=1}^p \alpha _i=1.&\end{aligned}$$

Proof

We present the proof in four steps.

Step 1. We claim that

$$\begin{aligned} \Big ( \bigcup _{i=1}^p \big ( \partial _c f_i(\hat{x})\big )^s \Big ) \cap \Big (\bigcap _{i=1}^p D(Q^i,\hat{x})\Big )= \emptyset . \end{aligned}$$

It suffices only to prove that

$$\begin{aligned} \big ( \partial _c f_l(\hat{x})\big )^s \cap D(Q^l,\hat{x})= \emptyset , \ \ \ \ \ \ \ \forall l\in I. \end{aligned}$$
(15)

On the contrary, suppose that for some \(l\in I\) there is a vector d such that

$$\begin{aligned} d\in \big ( \partial _c f_l(\hat{x})\big )^s \cap D(Q^l,\hat{x}). \end{aligned}$$
(16)

By the definition of \(D(Q^l,\hat{x})\), there exists a \(\delta >0\) such that \(\hat{x} + \varepsilon d \in Q^l\) for each \(\varepsilon \in (0,\delta )\). Thus, owing to the definition of \(Q^l\), we obtain that

$$\begin{aligned} \left\{ \begin{array}{ll} f_i(\hat{x} + \varepsilon d) \le f_i(\hat{x}), & \forall i\in I \setminus \{l\},\ \forall \varepsilon \in (0,\delta ),\\ \hat{x} + \varepsilon d \in M,& \forall \varepsilon \in (0,\delta ). \\ \end{array}\right. \end{aligned}$$
(17)

On the other hand, (16) leads to \(f_l^0(\hat{x};d) <0\). This means that there exists a \(\delta _l >0\) satisfying

$$\begin{aligned} f_l(\hat{x} + \varepsilon d) - f_l(\hat{x})<0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \forall \varepsilon \in (0,\delta _l). \end{aligned}$$

The above inequality with (17) imply that for each \(\varepsilon \in (0,\widehat{\delta })\) with \(\widehat{\delta }:=\min \{\delta ,\delta _l\}\), we have:

$$\begin{aligned} \left\{ \begin{array}{ll} f_i(\hat{x} + \varepsilon d) \le f_i(\hat{x}), & \forall i\in I \setminus \{l\},\\ f_l(\hat{x} + \varepsilon d)< f_l(\hat{x}), &\\ \hat{x} + \varepsilon d \in M. & \end{array}\right. \end{aligned}$$

This contradicts the efficiency of \(\hat{x}\). Therefore, our claim holds.

Step 2. Let \(\widehat{d}\in \overline{D(Q^l,\hat{x})}\) for some \(l\in I\). Then, there exists a sequence \(\{d_k \}_{k=1}^\infty\) in \(D(Q^l,\hat{x})\) converging to \(\widehat{d}\). By (15) and continuity of \(f_l^0(\hat{x}; .)\) concluded that

$$\begin{aligned} f_l^0(\hat{x};\widehat{d})=\lim _{k\rightarrow \infty }f_l^0(\hat{x}; d_k) \ge 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \big ( \partial _c f_l(\hat{x})\big )^s \cap \overline{D(Q^l,\hat{x})}= \emptyset , \ \ \ \ \ \ \ \forall l\in I, \end{aligned}$$

and hence

$$\begin{aligned} \Big ( \bigcup _{i=1}^p \big ( \partial _c f_i(\hat{x})\big )^s \Big ) \cap \Big (\bigcap _{i=1}^p \overline{D(Q^i,\hat{x})}\Big )= \emptyset . \end{aligned}$$
(18)

Step 3. We claim that

$$\begin{aligned} 0 \in ri\big (conv ( F_{\hat{x}}) \big )+cone (G_{\hat{x}} ). \end{aligned}$$
(19)

On contradiction we suppose that (19) does not hold. Then

$$\begin{aligned} ri\big (conv ( F_{\hat{x}} ) \big )\cap \big (-cone ( G_{\hat{x}} )\big ) =\emptyset . \end{aligned}$$

Thus, by the strong convex separation Theorem 11.3 in Rockafellar (1970) and noting that \(\big (-cone( G_{\hat{x}})\big )\) is a convex cone, it follows that there is a hyperplane

$$\begin{aligned} H:=\Big \{x \mid \big <x,d \big >=0\ \ \text {for some}\ d \in \mathbb {R}^n {\setminus} \{0\} \Big \} \end{aligned}$$

separating \(conv(F_{\hat{x}})\) and \(\big (-cone ( G_{\hat{x}})\big )\) properly. Hence, there exists a vector \(d \in \mathbb {R}^n\) satisfying

$$\begin{aligned} 0 \ne d\in \big ( conv ( F_{\hat{x}}) \big )^- \cap \big (cone (G_{\hat{x}})\big )^-= F_{\hat{x}} ^- \cap G_{\hat{x}} ^-. \end{aligned}$$

Thus, owing to (SZCQ) and (\(\mathfrak {A}\)), we conclude that

$$\begin{aligned} d\in \Big ( \bigcup _{i=1}^p \big ( \partial _c f_i(\hat{x})\big )^s \Big ) \cap \Big (\bigcap _{i=1}^p \overline{D(Q^i,\hat{x})}\Big ), \end{aligned}$$

which contradicts (18).

Step 4. The result is immediate from (5), (19), and the following fact from Theorem 6.9 in Rockafellar (1970)

$$\begin{aligned} ri\big (conv (F_{\hat{x}}) \big )\subseteq \Big \{\sum _{i=1}^p \alpha _i \xi _i \mid \xi _i \in \partial _c f_i(\hat{x}),\ \alpha _i >0,\ \sum _{i=1} ^p \alpha _i =1 \Big \}. \end{aligned}$$

\(\square\)

4 Sufficient Conditions

Similar to Caristi et al. (2010), let \(x^* \in \mathbb {R}^n\) be a point, and let \(\Phi :\mathbb {R}^n \times \mathbb {R}^n \times \mathbb {R}^{n+1} \rightarrow \mathbb {R}\) and \(\rho :\mathbb {R}^n \times \mathbb {R}^n \rightarrow \mathbb {R}\) be given functions satisfying

$$\begin{aligned} \Phi \big (x,x^*, (0,r)\big ) \ge 0 \ \ \ \ \ \text {for\,\,all }\, x\in \mathbb {R}^n \,\,\text { and }\,\,r \ge 0. \end{aligned}$$
(20)

Observe that an element of \(\mathbb {R}^{n+1}\) is represented as the order pair (y, r) with \(y\in \mathbb {R}^n\) and \(r\in \mathbb {R}\).

In Caristi et al. (2010) a differentiable function \(h:\mathbb {R}^n \rightarrow \mathbb {R}\) was named \((\Phi ,\rho )\)-invex at \(x^*\) with respect to \(\mathbb {A} \subseteq \mathbb {R}^n\) if, for each \(x\in \mathbb {A}\),

$$\begin{aligned}&\Phi \Big (x,x^*,\big (\nabla h(x^*),\rho (x,x^*)\big ) \Big ) \le h(x)-h(x^*), \\&\Phi (x,x^*,.)\, \text { is\,\,convex\,\,on }\, \mathbb {R}^{n+1}. \end{aligned}$$

We extend this result as below.

Definition 4

The locally Lipschitz function \(\hbar :\mathbb {R}^n \rightarrow \mathbb {R}\), at \(x^*\in \mathbb {R}^n\), is called generalized \((\Phi ,\rho )\)-invex at \(x^*\) with respect to \(\mathbb {A} \subseteq \mathbb {R}^n\) if, for each \(x\in \mathbb {A}\), it satisfies

$$\begin{aligned}&\Phi \Big (x,x^*,\big (\xi ,\rho (x,x^*)\big ) \Big ) \le \hbar (x)-\hbar (x^*),\ \ \ \ \forall \xi \in \partial _c \hbar (x^*), \\&\Phi (x,x^*,.)\, \text { is \,convex\, on }\, \mathbb {R}^{n+1}. \end{aligned}$$

Remark 3

We observe that, for \(\Phi (x,z,u,r)=F(x,z,u)+r \Vert x-z\Vert ^2\), where F(x, z, .) is sublinear on \(\mathbb {R}^n\), the definition of \((\Phi ,\rho )-\)invexity reduces to the definition of \((F,\rho )-\)convexity introduced in Preda (1992), which is turn generalizes the concepts of \(F-\)convexity (Jeyakumar 1985) and \(\rho -\)invexity (Vial 1983). Also, if \(\Lambda\) is an arbitrary index set and \(\widetilde{\hbar }(x):=\left( \hbar _\nu (x)\right) _{\nu \in \Lambda }\) is a generalized convex at \(\hat{x}\) introduced in Chuong and Kim (2014a) and Chuong and Yao (2014), then \(\widetilde{\hbar }(.)\) is \((\Phi ,\rho )-\)invex at \(\hat{x}\) with \(\Phi (x,z,u,r)=\left<u,y \right>\) for some \(y\in \mathbb {R}^n\) satisfying (Chuong and Yao (2014), Definition 3.3).

In the rest of this paper, we will always assume that \(\mathbb {A}\) to be equal with the set M of feasible solution of (P).

Theorem 5

(Sufficient KKT condition) Suppose that there exist a feasible solution \(\hat{x} \in M\), and scalars \(\alpha _i \ge 0\) (for \(i\in I\)) with \(\sum _{i=1}^p \alpha _i =1\), and a finite set \(T^*:=\{t_1,t_2,\ldots ,t_m\} \subseteq T(\hat{x})\), and scalars \(\beta _{j_s} \ge 0\) (for \(s\in \{1,2,\ldots ,m\}\)) such that

$$\begin{aligned} 0\in \sum _{i=1}^p \alpha _i \partial _c f_i(\hat{x})+\sum _{s=1}^m \beta _{t_s}\partial _c g_{t_s}(\hat{x}). \end{aligned}$$
(21)

Moreover, if the \(f_i\) functions and the \(g_t\) functions (for \((i,t)\in I \times T(\hat{x})\)) are generalized \((\Phi ,\rho )\)-invex at \(\hat{x}\), and \(\sum _{i=1}^p \alpha _i \rho _i(x,\hat{x})+\sum _{s=1}^m \beta _{t_s}\rho _{t_s}(x,\hat{x})\ge 0\) for all \(x\in M\), then \(\hat{x}\) is a weakly efficient solution for (P).

Proof

The inclusion (21) implies that there exist some \(\xi _i \in \partial _c f_i(\hat{x})\) (for \(i\in I\)) and \(\zeta _{t_s} \in \partial _c g_{t_s}(\hat{x})\) (for \(s\in \{1,\ldots ,m \}\)) satisfying

$$\begin{aligned}&\sum _{i=1}^p \alpha _i \xi _i+\sum _{s=1}^m \beta _{t_s}\zeta _{t_s}=0, \\&\sum _{i=1}^p \alpha _i +\sum _{s=1}^m \beta _{t_s}=b>0. \end{aligned}$$

Taking \(\widehat{\alpha }_i:=\frac{\alpha _i}{b}\) and \(\widehat{\beta }_{t_s}:=\frac{\beta _{t_s}}{b}\), we conclude that

$$\begin{aligned}&\sum _{i=1}^p \widehat{\alpha }_i \xi _i+\sum _{s=1}^m \widehat{\beta }_{t_s}\zeta _{t_s}=0, \\&\sum _{i=1}^p \widehat{\alpha }_i +\sum _{s=1}^m \widehat{\beta }_{t_s}=1. \end{aligned}$$

Owing to these equalities, (20), \(\sum _{i=1}^p \widehat{\alpha }_i \rho _i(x,\hat{x})+\sum _{s=1}^m \widehat{\beta }_{t_s}\rho _{t_s}(x,\hat{x})\ge 0\), and convexity of \(\Phi (x,x^*,.)\), we obtain that

$$\begin{aligned} 0 & \le \Phi \Big (x,\hat{x}, \sum _{i=1}^p \widehat{\alpha }_i \xi _i+\sum _{s=1}^m \widehat{\beta }_{t_s}\zeta _{t_s} , \sum _{i=1}^p \widehat{\alpha }_i \rho _i(x,\hat{x})+\sum _{s=1}^m \widehat{\beta }_{t_s}\rho _{t_s}(x,\hat{x}) \Big ) \nonumber \\ & = \Phi \Big (x,\hat{x}, \sum _{i=1}^p \widehat{\alpha }_i \big ( \xi _i,\rho _i(x,\hat{x})\big ) +\sum _{s=1}^m \widehat{\beta }_{t_s} \big (\zeta _{t_s} , \rho _{t_s}(x,\hat{x})\big ) \Big ) \nonumber \\\le & {} \sum _{i=1}^p \widehat{\alpha }_i \Phi \big ( x,\hat{x}, \xi _i, \rho (x,\hat{x}) \big ) + \sum _{s=1}^m \widehat{\beta }_{t_s} \Phi \big (x, \hat{x}, \zeta _{t_s}, \rho _{t_s}(x,\hat{x}) \big ). \end{aligned}$$
(22)

Now, if \(\hat{x}\) is not a weakly efficient of (P), there exists a point \(x\in M\) such that \(f_i(x) \,<\, f_i(\hat{x})\) for all \(i\in I\). Hence, by generalized \((\Phi ,\rho )\)-invexity of \(f_i\) functions we have

$$\begin{aligned} \Phi \big ( x,\hat{x}, \xi _i, \rho _i (x,\hat{x}) \big ) \,\le\, f_i(x)-f_i(\hat{x})<0. \end{aligned}$$
(23)

Similarly, for each \(s=1,\ldots ,m\), we have

$$\begin{aligned} \Phi \big (x, \hat{x}, \zeta _{t_s}, \rho _{t_s}(x,\hat{x}) \big )\le g_{t_s}(x)-g_{t_s}(\hat{x})=g_{t_s}(x)\le 0, \end{aligned}$$
(24)

the last equality holds since \(t_s \in T(\hat{x})\).

From (23), (24), and \(\sum _{i=1}^p \widehat{\alpha }_i >0\), we conclude that

$$\begin{aligned} \sum _{i=1}^p \widehat{\alpha }_i \Phi \big ( x,\hat{x}, \xi _i, \rho (x,\hat{x}) \big ) + \sum _{s=1}^m \widehat{\beta }_{t_s} \Phi \big (x, \hat{x}, \zeta _{t_s}, \rho _{t_s}(x,\hat{x}) \big )<0, \end{aligned}$$

which contradicts (22). This contradiction shows that \(\hat{x}\) is a weakly efficient for (P). \(\square\)

Strengthening the assumptions concerning \(\alpha _i\)s we obtain sufficient conditions for efficiency.

Theorem 6

(Strong sufficient KKT condition) Suppose that there exist a feasible solution \(\hat{x} \in M\), and scalars \(\alpha _i > 0\) (for \(i\in I\)), and a finite set \(T^*:=\{t_1,t_2,\ldots ,t_m\} \subseteq T(\hat{x})\), and scalars \(\beta _{j_s} \ge 0\) (for \(s\in \{1,2,\ldots ,m\}\)) such that

$$\begin{aligned} 0\in \sum _{i=1}^p \alpha _i \partial _c f_i(\hat{x})+\sum _{s=1}^m \beta _{t_s}\partial _c g_{t_s}(\hat{x}). \end{aligned}$$

Moreover, if the \(f_i\) functions and the \(g_t\) functions (for \((i,t)\in I \times T(\hat{x})\)) are generalized \((\Phi ,\rho )\)-invex at \(\hat{x}\), and \(\sum _{i=1}^p \alpha _i \rho _i(x,\hat{x})+\sum _{s=1}^m \beta _{t_s}\rho _{t_s}(x,\hat{x})\ge 0\) for all \(x\in M\), then \(\hat{x}\) is an efficient solution for (P).

Proof

Similar to (23) and (24), if \(\hat{x}\) is not efficient, we find \(x\in M\) and \(l\in I\) such that:

$$\begin{array}{ll} \Phi \big ( x,\hat{x}, \xi _i, \rho _i (x,\hat{x}) \big )\le 0, \ \ \ \ \ \ \ \ \forall i\in I \setminus \{l\},\\ \Phi \big ( x,\hat{x}, \xi _l, \rho _l (x,\hat{x}) \big )&<0, \\ \Phi \big (x, \hat{x}, \zeta _{t_s}, \rho _{t_s}(x,\hat{x}) \big )\le 0, \ \ \ \ \ \ \forall s\in \{1,\ldots ,m\}. \end{array}$$

These inequalities and \(\widehat{\alpha }_i >0\) (for all \(i\in I\)) and \(\widehat{\beta }_{t_s}\ge 0\) (for all \(s\in \{1,\ldots ,m\}\)), imply that

$$\begin{aligned} \sum _{i=1}^p \widehat{\alpha }_i \Phi \big ( x,\hat{x}, \xi _i, \rho (x,\hat{x}) \big ) + \sum _{s=1}^m \widehat{\beta }_{t_s} \Phi \big (x, \hat{x}, \zeta _{t_s}, \rho _{t_s}(x,\hat{x}) \big )<0, \end{aligned}$$

which contradicts (22). \(\square\)

Remark 4

Similar to Caristi et al. (2010), we can define some weaker \((\Phi ,\rho )\)-invexiy assumption for the function \(\hbar\), and then, we can prove some weaker sufficient conditions for optimality of (P). Since the proof of these extensions are similar to previous theorems, we omit them.