1 Introduction

The study of quasiconvex functions and their properties has attracted great attention due to their applications in various scientific and technological areas such as mathematics, economics, image processing, and machine learning; see [2, 19, 25, 28, 32,33,34] and the references therein. The characterization of quasiconvex functions in terms of subdifferentials is a noteworthy issue in the generalized convex analysis and nonsmooth analysis. By introducing the notion of quasimonotonicity, Hassouni [18] proved the equivalence between quasiconvexity of a locally Lipschitz function and the quasimonotonicity of its subdifferential on a separable Banach space. Luc [24] characterized lower semi-continuous quasiconvex functions in terms of generalized Clarke–Rockafellar subdifferential and upper and lower Dini directional derivatives in topological vector spaces. In the same way but independently, an extension of Hassouni’s characterization for lower semi-continuous quasiconvex functions on arbitrary Banach spaces using Clarke–Rockafellar subdifferential was investigated by Aussel et al. [4]. Clarke subdifferential was used by Ellaia and Hassouni [15] to study the characterization of locally Lipschitz quasiconvex functions. Soleimani-damaneh [30] gave some characterizations of nonsmooth locally Lipschitz quasiconvex and pseudoconvex functions using limiting subdifferentials. The relationship between quasimonotonicity and quasiconvexity in terms of convexificators was considered by Jeyakumar and Luc [20]. By defining some concepts such as semi-strict quasimonotonicity and strict quasimonotonicity for multi-valued maps, Daniilidis and Hadjisavvas [10] found some characterizations for locally Lipschitz (semi) strictly quasiconvex functions in terms of Clarke subdifferential. Two characterizations for the robust quasiconvexity of lower semi-continuous functions were obtained by Bui et al. [8] by use of Fréchet subdifferential.

Another well-known and important notion in nonsmooth analysis is quasidifferentiability. This notion was firstly introduced by Pshenichnyi [29] as a generalization of subdifferentiability notion in convex analysis. In the sense of Pshenichnyi [29], a directional differentiable function \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) with directional derivative \(f'({\bar{x}};d)\) at \({\bar{x}}\in {\mathbb {R}}^n\) in the direction \(d\in {\mathbb {R}}^n\) is quasidifferentiable at \({\bar{x}}\) if there exists a nonempty convex compact set \(C\subseteq {\mathbb {R}}^n\) such that

$$\begin{aligned} f'({\bar{x}};d)=\max _{\zeta \in C}\langle \zeta , d\rangle , \quad \forall d\in {\mathbb {R}}^n. \end{aligned}$$

Demyanov and Rubinov [11] extend the quasidifferentiability notion. A directional differentiable function \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is quasidifferentiable at \({\bar{x}}\) in the sense of Demyanov and Rubinov [11] if there are two nonempty convex compact sets \(C, D\subseteq {\mathbb {R}}^n\) such that

$$\begin{aligned} f'({\bar{x}};d)=\max _{\eta \in C}\langle \eta , d\rangle +\min _{\zeta \in D}\langle \zeta , d\rangle , \quad \forall d\in {\mathbb {R}}^n. \end{aligned}$$

This notion has been considered in nonsmooth optimization by many authors specially to obtain optimality conditions [1, 3, 5,6,7, 12, 13, 16, 23]. Sutti [31] has studied the quasidifferentiability of quasiconvex functions in the sense of Demyanov and Rubinov [11]. However, the quasiconvex functions are not necessary directionally differentiable or continuous. For example, the quasiconvex function \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) where \(f(x)=1\) for \(x> 0\) and \(f(x)=0\) for \(x\le 0\) is not quasidifferentiable. Also, see Example 4.1 in the following for a continuous quasiconvex function that is not quasidifferentiable. Recently, Kabgani and Soleimani-damaneh (semi-quasidifferentiability in nonsmooth nonconvex multi-objective optimization, submitted) have introduced semi-quasidifferentiability notion as a generalization of quasidifferentiability to cover a wider class of functions. With respect to this notion, the main object of this paper is twofold: on the one hand, the relationships between semi-quasidifferential notion and some well-known subdifferentials for quasiconvex functions are investigated. On the other hand, some characterizations for quasiconvexity in terms of semi-quasidifferentials are given. Kabgani and Soleimani-damaneh have shown that each locally Lipschitz function is semi-quasidifferentiable and, moreover, under some conditions, Clarke subdifferential and the singleton set \(\{0\}\) are parts of a semi-quasidifferential for a locally Lipschitz function. Thus, the obtained results generalize some well-known classical results in the literature and are new for quasidifferentiability notion, as well.

The structure of this paper is as follows. Some preliminaries, notations and definitions are provided in Section 2. The relationships between some well-known subdifferentials for quasiconvex functions and semi-quasidifferentials of them are investigated in Section 3. Section 4 is devoted to some characterizations of quasiconvex functions in terms of semi-quasidifferentials. Section 5 concludes the paper.

2 Preliminaries

Throughout this paper, \({\mathbb {R}}^n\) stands for an n-dimensional Euclidean space with Euclidean norm \(\Vert \cdot \Vert \), and \(\langle \cdot , \cdot \rangle \) for the standard inner product. Given a set \(C\subseteq {\mathbb {R}}^n\), \(\mathrm{co}\, C\), \(\mathrm{cone}\, C\), \(\mathrm{int}\, C\), and \(\mathrm{cl}\, C\) denote the convex hull, convex cone hull, the interior, and the closure of C, respectively. We use the convention \(\infty -\infty =\infty \). C is called a cone if \(\lambda x\in C\), for each \(x\in C\) and nonnegative scalar \(\lambda \). The polar cone of C is defined by

$$\begin{aligned} C^\circ :=\{d \in {\mathbb {R}}^n : \langle d, x\rangle \le 0, ~~\forall x\in C\}. \end{aligned}$$

Let K be a nonempty set in \({\mathbb {R}}^{n}\). Then, the cone of feasible directions and the tangent cone of K at \({\bar{x}} \in \mathrm{cl} \,K\), denoted by \(D_K({\bar{x}})\) and \(T_K({\bar{x}})\), respectively, are defined by

$$\begin{aligned}&D_K ({\bar{x}}) := \left\{ d \in {\mathbb {R}}^{n}: ~ \exists ~ \delta > 0, ~ \forall ~ \lambda \in (0, \delta ), ~ {\bar{x}} + \lambda d \in K \right\} , \\&T_K({\bar{x}}) := \left\{ d \in {\mathbb {R}}^{n}: ~ \exists ~ t_{n} \downarrow 0, ~ \exists ~ \{d_{n}\} \subseteq {\mathbb {R}}^{n}, ~ d_{n} \rightarrow d, ~ {\bar{x}} + t_{n} d_{n} \in K\right\} . \end{aligned}$$

If K is convex, then the normal cone of K at \({\bar{x}} \in \mathrm{cl}\, K\), is defined by

$$\begin{aligned} N_K({\bar{x}}) := \{ d \in {\mathbb {R}}^{n}: ~ \langle d, x - {\bar{x}} \rangle \le 0, ~ \forall ~ x \in K\}. \end{aligned}$$

If K is convex, then \(\mathrm{cl}\,D_K({\bar{x}}) = T_K({\bar{x}})\) and \(N_K({\bar{x}})=T_K^\circ ({\bar{x}})\).

A function \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is said to be quasiconvex if for all \(x, y\in {\mathbb {R}}^n\) and \(\lambda \in [0, 1]\),

$$\begin{aligned} f(\lambda x+(1-\lambda ) y)\le \max \{f(x), f(y)\}. \end{aligned}$$

The strict sublevel set of f at \({\bar{x}}\in {\mathbb {R}}^n\) is defined as

$$\begin{aligned} S^s_{{\bar{x}}}(f):=\{x\in {\mathbb {R}}^n : f(x)< f({\bar{x}})\}. \end{aligned}$$

If f is a quasiconvex function, then for each \(x\in {\mathbb {R}}^n\), \(S^s_{{\bar{x}}}(f)\) is a convex set.

The upper Dini directional derivative of \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) at \(x\in {\mathbb {R}}^n\) in direction \(d\in {\mathbb {R}}^n\) is defined by

$$\begin{aligned} f^+(x; d):= \displaystyle \limsup _{t\downarrow 0} \frac{f(x+td) - f(x)}{t}. \end{aligned}$$

The directional derivative of f at \(x\in {\mathbb {R}}^n\) in direction \(d\in {\mathbb {R}}^n\), denoted by \(f'(x;d)\), is defined as

$$\begin{aligned}f'(x; d):= \displaystyle \lim _{t\downarrow 0} \frac{f(x+td) - f(x)}{t}.\end{aligned}$$

Definition 2.1

[20] Let \(f: {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) and \({{\bar{x}}}\in {\mathbb {R}}^n\). A closed set \(\partial ^{JL} f({{\bar{x}}})\subseteq {\mathbb {R}}^n\) is called an upper regular convexificator (URC) of f at \({{\bar{x}}}\) if for each \(d\in {\mathbb {R}}^n\),

$$\begin{aligned} f^+({\bar{x}}; d)= \sup _{\zeta \in \partial ^{JL} f({\bar{x}})}\langle \zeta , d\rangle . \end{aligned}$$

Definition 2.2

A function \(h:{\mathbb {R}}^n\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is said to be quasimonotone if for all \(x, y\in {\mathbb {R}}^n\),

$$\begin{aligned} \min \{h(x,y-x), h(y, x-y)\}\le 0. \end{aligned}$$

Theorem 2.1

[24] A lower semi-continuous function f is quasiconvex if and only if \(f^+(\cdot ; \cdot )\) is quasimonotone.

Let \(f: {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be locally Lipschitz at \({\bar{x}}\in {\mathbb {R}}^n\). The Clarke generalized directional derivative of f at \({\bar{x}}\) in the direction \(d\in {\mathbb {R}}^n\), is defined by

$$\begin{aligned} f^{Cl}({\bar{x}}; d) := \limsup _{y\rightarrow {\bar{x}}, t\downarrow 0}\frac{f(y+td) - f(y)}{t}. \end{aligned}$$

The Clarke subdifferential of the function f at \({{\bar{x}}}\), denoted by \(\partial ^{Cl} f({{\bar{x}}})\), is defined as

$$\begin{aligned} \partial ^{Cl} f({{\bar{x}}}) := \{ \zeta \in {\mathbb {R}}^n : f^{Cl}({{\bar{x}}};d)\ge \langle \zeta , d\rangle , ~~ \forall d\in {\mathbb {R}}^n\}. \end{aligned}$$

If f is locally Lipschitz at \({\bar{x}}\), then \(\partial ^{Cl} f(\bar{x})\) is a nonempty compact convex set [9].

Kabgani and Soleimani-damaneh (semi-quasidifferentiability in nonsmooth nonconvex multi-objective optimization, submitted) have introduced semi-quasidifferentiability as a generalization of quasidifferentiability in the sense of Demyanov and Rubinov [11] and URC notion [20].

Definition 2.3

The function \(f: {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) is said to be semi-quasidifferentiable at \({\bar{x}}\in {\mathbb {R}}^n\) if there are two nonempty closed sets \(\partial _*f({\bar{x}}), \partial ^*f({\bar{x}})\subseteq {\mathbb {R}}^n\) such that

$$\begin{aligned} f^+({\bar{x}};d)=\sup _{\eta \in \partial _*f({\bar{x}})}\langle \eta , d\rangle +\inf _{\zeta \in \partial ^*f({\bar{x}})}\langle \zeta , d\rangle , ~~~\forall d\in {\mathbb {R}}^n. \end{aligned}$$

The pair of closed sets \((\partial _*f({\bar{x}}), \partial ^*f({\bar{x}}))\) is said to be a semi-quasidifferential (SQD) of f at \({\bar{x}}\).

In Definition 2.3, in contrast to quasidifferentiability in the sense of Pshenichnyi and Demyanov-Rubinov, f is not necessarily directional differentiable at \({\bar{x}}\) and \(\partial _*f({\bar{x}})\) and \(\partial ^*f({\bar{x}})\) are not necessarily convex compact sets. However, the quasidifferentiable functions in the sense of Pshenichnyi and Demyanov-Rubinov are semi-quasidifferentiable. Also, a function which admits a URC is semi-quasidifferentiable.

3 Relationships Between Greenberg–Pierskalla’s Subdifferentials and SQDs

Generalized subdifferentials for nondifferentiable locally Lipschitz functions are developed well in the nonsmooth analysis literature; see [9, 20, 26] and the references therein. However, quasiconvex functions are not necessary locally Lipschitz or even continuous. Hence, several attempts have been done to develop subdifferentials for this class of functions [17, 27, 32]. Greenberg–Pierskalla’s subdifferential [17] and some variants of it introduced by Penot [27] are important class of subdifferentials devoted to quasiconvex functions based on normal cone to sublevel sets. These subdifferentials are applied to obtain optimality conditions for quasiconvex programming; see [22, 32, 33] and the references therein. The mentioned subdifferentials are cones and therefore are the singleton set \(\{0\}\) or unbounded. However, under some conditions one may characterize these cones by other well-known compact subdifferentials [21]. On the other hand, characterization of these subdifferentials in terms of other subdifferentials may help to obtain sharper optimality conditions for quasiconvex programming. Dutta and Chandra [14] and Kabgani and Soleimani-damaneh [21] have studied the relationships between Greenberg–Pierskalla’s subdifferential [17] and some variants of it with convexificator notion for quasiconvex functions. Moreover, Kabgani and Soleimani-damaneh [21] have investigated the connection between Clarke [9] and Mordukhovich [26] subdifferentials with these subdifferentials for locally Lipschitz quasiconvex functions. In this section, the relationships between Greenberg–Pierskalla’s subdifferential and some well-known subdifferentials of quasiconvex functions introduced by Penot [27] and SQDs of them are investigated.

Let \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) and \({\bar{x}}\in {\mathbb {R}}^n\). The Greenberg–Pierskalla’s subdifferential [17] of f at \({\bar{x}}\) is defined as

$$\begin{aligned}\partial ^{G}f({\bar{x}}):=\{\eta \in {\mathbb {R}}^n: \langle \eta , x - {\bar{x}}\rangle \ge 0 \Rightarrow f(x)\ge f({\bar{x}})\}.\end{aligned}$$

The star subdifferential [27] as a variant of Greenberg–Pierskalla’s subdifferential at \({\bar{x}}\) is defined as

$$\begin{aligned}\partial ^\star f({\bar{x}}):= \{\eta \in {\mathbb {R}}^n\setminus \{0\} : \langle \eta , x - {\bar{x}}\rangle >0 \Rightarrow f(x)\ge f({{\bar{x}}})\},\end{aligned}$$

when \({\bar{x}}\) is not a minimizer of f and \(\partial ^\star f({\bar{x}}):={\mathbb {R}}^n\) when \({\bar{x}}\) is a minimizer of f. We have \(\partial ^{G}f({\bar{x}})\subseteq \partial ^\star f({\bar{x}})\) and \(\partial ^\star f({\bar{x}})=N_{S^s_{{\bar{x}}}(f)}({\bar{x}})\setminus \{0\}\) when \({\bar{x}}\) is not a minimizer of f. Another variant of the Greenberg–Pierskalla’s subdifferential of f at \({\bar{x}}\) is defined as

$$\begin{aligned} \partial ^\nu f({\bar{x}}):= \{\eta \in {\mathbb {R}}^n: \langle \eta , x - {\bar{x}}\rangle>0 \Rightarrow f(x)> f({{\bar{x}}})\}. \end{aligned}$$

If there is no local minimizer of f in \(f^{-1}(f({\bar{x}}))\), then \(\partial ^\nu f({\bar{x}})=\partial ^\star f({\bar{x}})\cup \{0\}\) [27, Proposition 8].

Theorem 3.1

Let \(f: {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be quasiconvex and has a SQD at \({\bar{x}}\) as \((\partial _*f({\bar{x}}),\partial ^*f({\bar{x}}))\). If \(S^s_{{\bar{x}}}(f)\ne \emptyset \) and is an open set, \(0\notin \mathrm{cl\, co}\, \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) \), and

$$\begin{aligned} \sup _{\zeta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})}\langle \zeta , x-{\bar{x}}\rangle \le 0,\qquad \forall x\in S^s_{{\bar{x}}}(f), \end{aligned}$$
(1)

then

$$\begin{aligned} \partial ^G f({\bar{x}})\cup \{0\}=\partial ^\star f({\bar{x}})\cup \{0\}=\mathrm{cl\,cone} (\partial _*f({\bar{x}})+\partial ^*f({\bar{x}})). \end{aligned}$$
(2)

Moreover, if there is no local minimizer of f in \(f^{-1}(f({\bar{x}}))\), then,

$$\begin{aligned} \partial ^{\nu } f({\bar{x}})=\partial ^G f({\bar{x}})\cup \{0\}=\partial ^\star f({\bar{x}})\cup \{0\}=\mathrm{cl\,cone} (\partial _*f({\bar{x}})+\partial ^*f({\bar{x}})). \end{aligned}$$
(3)

Proof

By (1),

$$\begin{aligned} \mathrm{cl\, cone\,} \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) \subseteq N_{S^s_{{\bar{x}}}(f)}({\bar{x}}). \end{aligned}$$
(4)

On the other hand, we claim that \({\bar{x}}\) is not a local minimizer of f. If \({\bar{x}}\) is a local minimizer of f, for each \(d\in {\mathbb {R}}^n\), \(f^+({\bar{x}};d)\ge 0\). Thus,

$$\begin{aligned} \sup _{\eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})} \langle \eta , d\rangle \ge 0, \quad \forall d\in {\mathbb {R}}^n. \end{aligned}$$

Therefore,

$$\begin{aligned} 0\in \mathrm{cl\, co\,}(\partial _*f({\bar{x}})+\partial ^*f({\bar{x}})), \end{aligned}$$

which is a contradiction with the assumption. Thus, (4) implies,

$$\begin{aligned} \mathrm{cl\, cone\,} \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) \subseteq \partial ^\star f({\bar{x}})\cup \{0\}. \end{aligned}$$
(5)

Let \(x\in S^s_{{\bar{x}}}(f)\). There exists some scalar \(\varepsilon >0\) such that \(x+\varepsilon d\in S^s_{{\bar{x}}}(f)\) for each \(d\in {\mathbb {R}}^n\) with \(\Vert d\Vert \le 1\). This implies

$$\begin{aligned}\left\langle \eta , \varepsilon \frac{d}{\Vert d \Vert }+ x- {\bar{x}}\right\rangle \le 0,~~~ \forall d\in {\mathbb {R}}^n, \forall \eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}}).\end{aligned}$$

Thus,

$$\begin{aligned}\langle \eta , x- {\bar{x}}\rangle \le -\varepsilon \left\langle \eta , \frac{d}{\Vert d \Vert }\right\rangle ,~~~ \forall d\in {\mathbb {R}}^n, \forall \eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}}).\end{aligned}$$

Set \(d:=\eta \). We have,

$$\begin{aligned}\langle \eta , x- {\bar{x}}\rangle \le -\varepsilon \Vert \eta \Vert ,~~~ \forall \eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}}).\end{aligned}$$

Since \(0\notin \mathrm{cl\, co}\, \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) \),

$$\begin{aligned} \sup _{\eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})}\langle \eta , x- {\bar{x}}\rangle \le -\varepsilon \inf _{\eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})}\Vert \eta \Vert <0. \end{aligned}$$

Let \(d\in \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) ^\circ \) be arbitrary. We have

$$\begin{aligned}&f^+({\bar{x}}; d+ t(x - {\bar{x}}))\nonumber \\&{}\qquad =\sup _{\eta \in \partial _*f({\bar{x}})}\langle \eta ,d+ t(x - {\bar{x}})\rangle +\inf _{\zeta \in \partial ^*f({\bar{x}})}\langle \zeta ,d+ t(x - {\bar{x}})\rangle \nonumber \\&{}\qquad \le \sup _{\eta \in \partial _*f({\bar{x}})}\langle \eta , d+ t(x - {\bar{x}})\rangle +\sup _{\zeta \in \partial ^*f({\bar{x}})}\langle \zeta , d+ t(x - {\bar{x}})\rangle \nonumber \\&{}\qquad =\sup _{\eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})}\langle \eta , d+ t(x - {\bar{x}})\rangle \nonumber \\&{}\qquad \le \sup _{\eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})}\langle \eta , d\rangle \nonumber \\&{}\qquad ~~~+t\sup _{\eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})}\langle \eta , x-{\bar{x}}\rangle <0, ~~\forall t>0. \end{aligned}$$

Therefore, for each \(t>0\), there exists some \(\delta >0\) such that for each \(\lambda \in (0, \delta )\),

$$\begin{aligned} f({\bar{x}}+ \lambda (d + t(y -{\bar{x}})))< f({\bar{x}}). \end{aligned}$$

Thus, \(d + t(x -{\bar{x}})\in D_{S^s_{{\bar{x}}}(f)}({\bar{x}})\), for each \(t>0\), which implies \(d\in \mathrm{cl}\,D_{S^s_{{\bar{x}}}(f)}({\bar{x}})=T_{S^s_{{\bar{x}}}(f)}({\bar{x}})\). Therefore \(\left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) ^\circ \subseteq T_{S^s_{{\bar{x}}}}({\bar{x}}).\) So

$$\begin{aligned}&\partial ^\star f({\bar{x}})\cup \{0\}=N_{S^s_{{\bar{x}}}(f)}({\bar{x}})=\left( T_{S^s_{{\bar{x}}}(f)}({\bar{x}})\right) ^{\circ }\nonumber \\&{} \subseteq \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) ^{\circ \circ }=\mathrm{cl\, cone\,} \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) . \end{aligned}$$
(6)

From (5) and (6),

$$\begin{aligned} \partial ^\star f({\bar{x}})\cup \{0\}=\mathrm{cl\, cone\,} \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) . \end{aligned}$$
(7)

Now, let \(f(x)<f({\bar{x}})\). Since \(S^s_{{\bar{x}}}(f)\) is an open set, similar to the first part of the proof, for each \(\eta \in \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\), \(\langle \eta , x- {\bar{x}}\rangle <0\). Hence, from definition of \(\partial ^G f({\bar{x}})\),

$$\begin{aligned} \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\subseteq \partial ^G f({\bar{x}}). \end{aligned}$$
(8)

Since \(\partial ^G f({\bar{x}})\cup \{0\}\) is a closed convex cone and \(\partial ^G f({\bar{x}})\subseteq \partial ^\star f({\bar{x}})\), the equalities in (2) hold from (7) and (8).

Now, if there is no local minimizer of f in \(f^{-1}(f({\bar{x}}))\), then, \(\partial ^{\nu } f({\bar{x}})=\partial ^\star f({\bar{x}})\cup \{0\}\), and (3) holds from (7). \(\square \)

Remark 3.1

  1. (i)

    If in Theorem 3.1, we have \(\partial ^*f({\bar{x}})=\{0\}\), then \(int\, S^s_{{\bar{x}}}(f)\ne \emptyset \) implies\(\sup _{\zeta \in \partial _*f({\bar{x}})}\langle \zeta , x-{\bar{x}}\rangle \le 0\), for each \(x\in S^s_{{\bar{x}}}(f)\). Thus, Theorem 3.1 is a generalization of [21, Theorem 3.1 (iv), (v)].

  2. (ii)

    If \(f: {\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is an upper semi-continuous quasiconvex function and has an URC as \(\partial ^{JL} f({\bar{x}})\) at \({\bar{x}}\) such that \(0\notin \partial ^{JL} f({\bar{x}})\), then \(S^s_{{\bar{x}}}(f)\ne \emptyset \) is an open set and Theorem 3.1 reduces to [14, Proposition 2.1].

In Example 3.1, an application of Theorem 3.1 is shown. Note that, in this example, the given function f does not have any URC or quasidifferential in the sense of Demyanov and Rubinov [11] at the given \({\bar{x}}\).

Example 3.1

Consider

$$\begin{aligned} f(x)=\left\{ \begin{array}{ll} -\sqrt{x},~~&{} x\ge 0\\ -x, &{} x<0. \end{array}\right. \end{aligned}$$

Let \({\bar{x}}=0\). Then, \(\left( \partial _*f({\bar{x}}):=\{0\},\partial ^*f({\bar{x}}):= (-\infty , -1]\right) \) is a SQD of f at \({\bar{x}}\), \(S^s_{{\bar{x}}}(f)=(0,+\infty )\) is an open set, \(0\notin \mathrm{cl\, co}\, \left( \partial _*f({\bar{x}})+\partial ^*f({\bar{x}})\right) \), and (1) holds. Thus,

$$\begin{aligned} \partial ^{\nu } f({\bar{x}})=\partial ^G f({\bar{x}})\cup \{0\}=\partial ^\star f({\bar{x}})\cup \{0\}=\mathrm{cl\,cone} (\partial _*f({\bar{x}})+\partial ^*f({\bar{x}}))=(-\infty , 0]. \end{aligned}$$

4 Characterization of Quasiconvexity in Terms of SQDs

In this section, some characterizations for quasiconvex functions in terms of SQDs are given. We also consider the connection between quasimonotonicity and quasiconvexity. In the some results, we assume that a given function f has an SQD as \((\partial _*f({\bar{x}}),\partial ^*f({\bar{x}}))\) at a given \({\bar{x}}\in {\mathbb {R}}^n\) such that \(\partial ^*f({\bar{x}})\) is compact. This condition clearly holds if f is quasidifferentiable in the sense of Demyanov and Rubinov or has a URC at \({\bar{x}}\). However, no compactness condition is assumed on \(\partial _*f({\bar{x}})\).

Theorem 4.1

Let \(f: {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) for each \(x\in {\mathbb {R}}^n\), has an SQD as \((\partial _*f(x),\partial ^*f(x))\). Then,

  1. (i)

    If f is lower semi-continuous and for each \(x, y\in {\mathbb {R}}^n\) with \(f(x)\le f(y)\), there exists some \(\bar{\zeta }\in \partial ^*f(y)\) such that

    $$\begin{aligned} \sup _{\eta \in \partial _*f(y)+\bar{\zeta }}\langle \eta ,x-y\rangle \le 0, \end{aligned}$$
    (9)

    then f is quasiconvex.

  2. (ii)

    If f is quasiconvex and \(\partial ^*f(x)\) is compact for all x, then for each \(x, y\in {\mathbb {R}}^n\) with \(f(x)\le f(y)\), there exists some \(\bar{\zeta }\in \partial ^*f(y)\) such that (9) holds.

Proof

(i) Assume that f is not quasiconvex. Thus, by Theorem 2.1, there exist some \(x, y\in {\mathbb {R}}^n\) such that

$$\begin{aligned} \min \{f^+(x; y-x), f^+(y; x-y)\}> 0. \end{aligned}$$
(10)

Without loss of generality, assume that \(f(x)\le f(y)\). By (10), \(f^+(y; x-y)>0\) that implies

$$\begin{aligned} \sup _{\eta \in \partial _*f(y)+\bar{\zeta }}\langle \eta ,x-y\rangle > 0, \quad \forall \bar{\zeta }\in \partial ^*f(y). \end{aligned}$$

This contradiction with the assumptions proves the claim.

(ii) If f is quasiconvex, then \(f(x)\le f(y)\) implies \(f^+(y; x-y)\le 0\). Thus,

$$\begin{aligned} \sup _{\eta \in \partial _* f(y)}\langle \eta , x-y\rangle +\inf _{\zeta \in \partial ^* f(y)}\langle \zeta , x-y\rangle \le 0. \end{aligned}$$

Since \(\partial ^* f(y)\) is compact, then there exists some \(\bar{\zeta }\in \partial ^* f(y)\) such that

$$\begin{aligned} \sup _{\eta \in \partial _* f(y)}\langle \eta , x-y\rangle +\langle \bar{\zeta }, x-y\rangle \le 0. \end{aligned}$$

Considering this \(\bar{\zeta }\in \partial ^*f(y)\), (9) holds. \(\square \)

Example 4.1 shows an application of Theorem 4.1 (i). In this example, the given function does not have any URC and is not quasidifferentiable in the sense of Demyanov and Rubinov.

Example 4.1

Consider the following function

$$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 0,~~~&{} x\le 0,\\ -\sqrt{x},~~~ &{} x>0. \end{array}\right. \end{aligned}$$

Then, \(\left( \{0\}, (-\infty , 0]\right) \) is a SQD of f at \(x_1=0\), \(\left( \{-\frac{1}{2\sqrt{x}}\}, \{0\}\right) \) is a SQD of f at \(x_2>0\), and \(\left( \{0\}, \{0\}\right) \) is a SQD of f at \(x_3<0\). If \(y\le 0\), for each x with \(f(x)\le f(y)\), \(\sup _{\eta \in \{0\}+0}\langle \eta , x-y\rangle \le 0\). If \(y>0\), \(f(x)\le f(y)\) implies \(x-y\ge 0\). Thus,

$$\begin{aligned} \sup _{\eta \in \left\{ -\frac{1}{2\sqrt{y}}\right\} +0}\langle \eta , x-y\rangle \le 0. \end{aligned}$$

Thus, by Theorem 4.1(i), f is quasiconvex.

Remark 4.1

Ellaia and Hassouni [15, Theorem 3.1] shown that if \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is a locally Lipschitz function, then f is quasiconvex if and only if,

$$\begin{aligned} x, y\in {\mathbb {R}}^n, ~~f(x)<f(y)\Rightarrow \forall \zeta \in \partial ^{Cl}f(y),~~\langle \zeta , x- y\rangle \le 0. \end{aligned}$$

They also remark that [15, Remark 3.3], the strict inequality could not be replaced by inequality. However, in Theorem 4.1 we shown a similar result without strict inequality. We apply Theorem 4.1 for the example in [15, Remark 3.3]. Consider the following function

$$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 0,~~~&{} x\ge 0,\\ x, &{} x<0. \end{array}\right. \end{aligned}$$

Set \({\bar{x}}=0\). We have \(\partial ^{Cl} f({\bar{x}})=[0, 1]\) and indeed, for \(x=1\) while \(f(x)\le f({\bar{x}})\), with \(\zeta =1\), \(\langle \zeta , x-{\bar{x}}\rangle >0\). However, \(\left( \{0\}, [0,1]\right) \) is a SQD of f at \({\bar{x}}\) and it is easy to check that for each \(x, y\in {\mathbb {R}}^n\) with \(f(x)\le f(y)\), (9) holds.

Proposition 4.1 gives another characterization of quasiconvexity similar to Theorem 4.1.

Proposition 4.1

Let \(f: {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) be quasiconvex function and for each \(x\in {\mathbb {R}}^n\), has an SQD as \((\partial _*f(x),\partial ^*f(x))\) such that \(\partial ^*f(x)\) is compact for all x. Then, for each \(x, y\in {\mathbb {R}}^n\),

$$\begin{aligned} \sup _{\eta \in \partial _*f(y)+\zeta }\langle \eta ,x-y\rangle > 0,~~\forall \zeta \in \partial ^*f(y) \Rightarrow \exists \xi \in \partial ^*f(x),~~\inf _{\zeta \in \partial _*f(x)+\xi }\langle \zeta ,x-y\rangle \ge 0. \end{aligned}$$

Proof

Since \(\partial ^*f(x)\) is compact and for each \(\zeta \in \partial ^*f(y)\), \( \sup _{\eta \in \partial _*f(y)+\zeta }\langle \eta ,x-y\rangle > 0\), then \(f^+(y; x-y)>0\). Now, quasiconvexity of f implies \(f(x)>f(y)\). By Theorem 4.1(ii), there exists some \(\bar{\zeta }\in \partial ^*f(x)\) such that

$$\begin{aligned} \sup _{\eta \in \partial _*f(x)+\bar{\zeta }}\langle \eta ,y-x\rangle \le 0, \end{aligned}$$

which implies

$$\begin{aligned} \inf _{\eta \in \partial _*f(x)+\bar{\zeta }}\langle \eta ,x-y\rangle \ge 0. \end{aligned}$$

\(\square \)

In the rest of this section, we investigate the relation between quasimonotonicity of SQDs and quasiconvexity.

Definition 4.1

Assume that \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) has an SQD mapping on \({\mathbb {R}}^n\) as \((\partial _*f(\cdot ),\partial ^*f(\cdot ))\). This set-valued mapping is called quasimonotone if for each \(x, y\in {\mathbb {R}}^n\) and for each \(\eta \in \partial _*f(x)\) and \(\zeta \in \partial _*f(y)\), and for some \(\mu \in \partial ^*f(x)\) and \(\xi \in \partial ^*f(y)\),

$$\begin{aligned} \min \{\langle \eta +\mu , y- x\rangle , \langle \zeta +\xi , x-y\rangle \}\le 0. \end{aligned}$$

Theorem 4.2

Assume that \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) for each \(x\in {\mathbb {R}}^n\) has an SQD at x as \((\partial _*f(x),\partial ^*f(x))\).

  1. (i)

    If \((\partial _*f(\cdot ),\partial ^*f(\cdot ))\) is quasimonotone, then \(f^+(\cdot ; \cdot )\) is quasimonotone.

  2. (ii)

    If \(\partial ^*f(x)\) is compact and \(f^+(\cdot ; \cdot )\) is quasimonotone then \((\partial _*f(\cdot ),\partial ^*f(\cdot ))\) is quasimonotone.

Proof

(i) Assume that \((\partial _*f(\cdot ),\partial ^*f(\cdot ))\) is quasimonotone. Then, for some arbitrary \(x, y\in {\mathbb {R}}^n\), if for each \(\eta \in \partial _*f(x)\) and for some \(\mu \in \partial ^*f(x)\), \(\langle \eta +\mu , y-x\rangle \le 0\), then

$$\begin{aligned}&f^+(x; y-x)=\sup _{\eta \in \partial _*f(x)}\langle \eta , y-x\rangle +\inf _{\nu \in \partial ^*f(x)}\langle \nu , y-x\rangle \\&\le \sup _{\eta \in \partial _*f(x)}\langle \eta , y-x\rangle +\langle \mu , y-x\rangle \le 0, \end{aligned}$$

and similarly, if for each \(\zeta \in \partial _*f(y)\) and for some \(\xi \in \partial ^*f(y)\), \(\langle \zeta +\xi , x-y\rangle \le 0\), then \(f^+(y; x-y)\le 0\). Thus,

$$\begin{aligned} \min \{f^+(x; y-x), f^+(y; x-y)\}\le 0, \quad \forall x, y\in {\mathbb {R}}^n. \end{aligned}$$

Thus, \(f^+(\cdot ; \cdot )\) is quasimonotone.

(ii) Assume that \(f^+(\cdot ; \cdot )\) is quasimonotone. If \((\partial _*f(\cdot ),\partial ^*f(\cdot ))\) is not quasimonotone, then there exist \(x, y\in {\mathbb {R}}^n\) such that for some \(\eta \in \partial _*f(x)\) and \(\zeta \in \partial _*f(y)\) and for each \(\mu \in \partial ^*f(x)\) and \(\xi \in \partial ^*f(y)\),

$$\begin{aligned} \min \{\langle \eta +\mu , y- x\rangle , \langle \zeta +\xi , x-y\rangle \}> 0. \end{aligned}$$

Thus,

$$\begin{aligned}&f^+(x; y-x)=\sup _{\eta \in \partial _*f(x)}\langle \eta , y- x\rangle +\inf _{\mu \in \partial ^*f(x)}\langle \mu , y- x\rangle>0,\\&f^+(y; x-y)=\sup _{\zeta \in \partial _*f(y)}\langle \zeta , x-y\rangle +\inf _{\xi \in \partial ^*f(y)}\langle \mu , x- y\rangle >0, \end{aligned}$$

which is a contradiction with quasimonotonicity of \(f^+(\cdot ; \cdot )\). \(\square \)

Theorem 4.3

Assume that \(f:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) is lower semi-continuous and for each \(x\in {\mathbb {R}}^n\) has an SQD at x as \((\partial _*f(x),\partial ^*f(x))\).

  1. (i)

    If \((\partial _*f(\cdot ),\partial ^*f(\cdot ))\) is quasimonotone, then f is quasiconvex.

  2. (ii)

    If f is quasiconvex and \(\partial ^*f(x)\) is compact for all x, then \((\partial _*f(\cdot ),\partial ^*f(\cdot ))\) is quasimonotone.

Proof

It is obtained from Theorems 2.1 and 4.2. \(\square \)

Remark 4.2

If f for each \(x\in {\mathbb {R}}^n\) has an SQD at x as \((\partial _*f(x),\{0\})\), then \(\partial _*f(x)\) is an upper regular convexificator of f at \({\bar{x}}\). Thus, Theorem 4.3 is a generalization of [20, Theorem 6.2].

5 Conclusions

Using semi-quasidifferentiability notion, which is a generalization of quasidifferentiability in the sense of Pshenichnyi [29] and Demyanov and Rubinov [11] and also upper regular convexificator notion [20], some characterizations are obtained for Greenberg–Pierskalla’s subdifferential and some well-known subdifferentials of quasiconvex functions introduced by Penot [27]. Moreover, the connection between quasimonotonicity of semi-quasidifferentials and quasiconvexity is investigated. With some example, it is shown that semi-quasidifferentiability implies sharper results than other well-known generalized gradient even for locally Lipschitz function.