Characterization of quadratic growth of extendedrealvalued functions
 Jin jiang Wang^{1, 2} and
 Wen Song^{1, 3}Email author
https://doi.org/10.1186/s1366001609774
© Wang and Song 2016
Received: 11 October 2015
Accepted: 15 January 2016
Published: 27 January 2016
Abstract
This paper shows that the sharpest possible bound in the secondorder growth condition of a proper lower semicontinuous function can be attained under some assumptions. We also establish a relationship among strong metric subregularity, quadratic growth, the positivedefiniteness property of the secondorder subdifferential/generalized Hessian, the strong metric regularity, and tilt stability in a finitedimensional setting.
Keywords
quadratic growth Mordukhovich (limiting) subdifferential strong metric regularity strong metric subregularity proxregularMSC
49J53 14P10 54C60 65K101 Introduction
This work is devoted to studying the relationships among strong metric subregularity, the quadratic growth condition, strong metric regularity, and the positivedefiniteness property of the secondorder subdifferential/generalized Hessian in finite and infinitedimensional settings.
In other lines of development, in [8], the secondorder subdifferential/generalized Hessian is used to characterize the tiltstable local minimizers introduced by Poliquin and Rockafellar [8] for proxregular and subdifferentially continuous functions on \(\mathbb {R}^{n}\). Later, Mordukhovich and Nghia [5] extended this characterization to the settings of Hilbert spaces by using a new notion of the combined secondorder subdifferential. Quite recently, Drusvyatskiy et al. [7] extended the known equivalences between metric regularity and the strong metric regularity properties of the gradient mappings for \(\mathcal{C}^{2}\)smooth functions, characterized via the classical Hessian, to the general class of proxregular and subdifferentially continuous functions on \(\mathbb {R}^{n}\) in terms of the secondorder subdifferential/generalized Hessian.
The current work aims to achieve new results in the aforementioned directions by improving the known characterizations. After recording in Section 2 the necessary definitions and facts that will be needed throughout the manuscript, we devote Section 3 to characterizations of the secondorder growth condition of a proper lower semicontinuous function. Theorem 3.1 provides a sufficient condition which allows strong metric subregularity with modulus κ to imply the secondorder growth condition with the constant \(c=1/\kappa\). Theorem 3.2 gives a sufficient condition for metric subregularity in a nonconvex setting. In Section 4, in finitedimensional setting, we establish an equivalence among strong metric subregularity, quadratic growth, the positivedefiniteness property of the secondorder subdifferential/generalized Hessian, the strong metric regularity and tilt stability, and we show that the constant c in the quadratic growth condition can attain \(1/\kappa\) (Theorem 4.2).
2 Preliminaries
Our study is focused on two key notions: metric subregularity and strong metric subregularity. They plays an important role in stability analysis. For more details, one can refer to the references [13–17]. They are defined as follows.
Definition 2.1
Definition 2.2
We notice that the strong metric subregularity of F at x̄ for ȳ is equivalent to the metric subregularity if x̄ is an isolated point of \(F^{1}(\bar{y})\).
The fundamental tools for studying general nonsmooth functions are subdifferentials. The following two subdifferential notions are used in this paper.
Definition 2.3
(Subdifferentials of functions)

The regular/Fréchet subdifferential of f at x̄ is$$ \hat{\partial}f(\bar{x}):=\biggl\{ x^{*}\in X^{*} \Bigm \liminf_{x\rightarrow\bar{x}} \frac{f(x)f(\bar{x})\langle x^{*},x\bar {x}\rangle}{\x\bar{x}\}\ge0\biggr\} . $$(5)

The (basic, limiting, Mordukhovich) subdifferential of f at x̄ is defined via (2) bywhere the symbol \(x\stackrel{f}{\rightarrow}\bar{x}\) means that \(x \rightarrow \bar{x}\) with \(f(x) \rightarrow f(\bar{x})\).$$ \partial f(\bar{x}):=\mathop{\operatorname{Lim}\sup}_{x\stackrel{f}{\rightarrow}\bar{x}}\hat {\partial}f(x), $$(6)
When f is convex, \(\partial f(x)\) is the usual subgradient set of convex analysis. When f is smooth, \(\partial f(x)\) is the singleton \({\nabla f(x)}\).
Definition 2.4
Definition 2.5
(Normal cone)

The regular/Fréchet normal cone of K at x̄ is defined bywhere \(x\stackrel{K}{\rightarrow}\bar{x}\) signifies the convergence \(x \rightarrow\bar{x}\) with \(x \in K\).$$ \hat{N}_{K}(\bar{x}):=\biggl\{ x^{*}\in X^{*} \Bigm \limsup _{x\stackrel{K}{\rightarrow}\bar {x}}\frac{\langle x^{*},x\bar{x}\rangle}{\x\bar{x}\}\leq0\biggr\} , $$(7)

The limiting normal cone of f at x̄ is defined via (2) by$$ N_{K}(\bar{x}):=\mathop{\operatorname{Lim}\sup}_{x\stackrel{K}{\rightarrow}\bar{x}} \hat{N}_{K}(x). $$(8)
In order to characterize the relationships between the quadratic growth conditions and metric subregularity of the subdifferentials in nonsmooth and nonconvex settings, we need the following two notions, which were introduced in [19] and were further studied in [20] and [21].
Definition 2.6
Definition 2.7
A function \(f : X \rightarrow\bar{\mathbb {R}}\) is subdifferentially continuous at \(\bar{x} \in\operatorname{dom} f\) for \(\bar{v}\in\partial f(\bar{x})\) if for every \(\varepsilon> 0\) there exists \(\delta> 0\) such that \(f(x)f(\bar{x})<\varepsilon\) whenever \(x\in \mathbb {B}_{\delta}(\bar{x})\) and \(\v\bar{v}\ <\delta\) with \(v \in\partial f(x)\). If this occurs for all \(\bar {v}\in\partial f(\bar{x})\), we say that f is subdifferentially continuous at x̄.
3 Characterizing quadratic growth properties
The main focus of this section is relating the quadratic growth properties of a function f to the subregularity of the subdifferential ∂f.
It will be showed in the next theorem that a condition similar to (13) guarantees that the upper bound \(\frac{1}{\kappa }\) in (1) can be attained.
Theorem 3.1
 (i)there are real numbers \(r\in(0,\kappa^{1})\) and \(\delta >0\) such that$$ f(x)\geq f(\bar{x})+\bigl\langle \bar{x}^{*},x\bar{x}\bigr\rangle  \frac {r}{2}\x\bar{x}\^{2} \quad \textit{for all } x\in \mathbb {B}_{\delta}(\bar{x}), $$(14)
 (ii)for any real number \(\alpha\in(0,\kappa^{1})\), there is a real number \(\eta>0\) such that$$ f(x)\geq f(\bar{x})+\bigl\langle \bar{x}^{*},x\bar{x}\bigr\rangle + \frac {\alpha}{2}\x\bar{x}\^{2}\quad \textit{for all } x\in \mathbb {B}_{\eta }(\bar{x}). $$(15)
 (iii)there is a real number \(\theta>0\) such that$$ f(x)\geq f(\bar{x})+\bigl\langle \bar{x}^{*},x\bar{x}\bigr\rangle + \frac {1}{2\kappa}\x\bar{x}\^{2} \quad \textit{for all } x\in \mathbb {B}_{\theta }(\bar{x}). $$(17)
Proof
Remark 3.1
 (i)The condition (16) is a sufficient condition for (17). For example, for \(x=(x_{1},x_{2})\in \mathbb {R}^{2}\), consider the functionObviously, f is not convex, and the subdifferential ∂f is strongly metrically subregular at \((\bar{x}, \bar{x}^{*})=(\mathbf{0}, \mathbf{0})\in\operatorname{gph} \partial f\) with modulus \(\kappa=1\). Furthermore, we observe that, for any neighborhood U of \(\bar{x}=(0,0)\) and \(\lambda\in(0, 1)\), the condition (16) holds. On the other hand, we see that (17) holds at \((\bar{x}, \bar {x}^{*})=(\mathbf{0}, \mathbf{0})\).$$f(x):= \textstyle\begin{cases} +\infty,&x_{1}=x_{2}\neq0, \\ \frac{1}{2}\x\^{2},&\text{otherwise}. \end{cases} $$
 (ii)
The condition (16) is only a sufficient condition for (17). Consider the real function \(f(x)=x\). We observe that the subgradient mapping ∂f is a strongly metrically subregular at \(\bar{x}=0\) for \(\bar{x}^{*}=0\) with any \(\kappa>0\), and inequality (17) is satisfied at \((0,0)\). However, the inequality (16) does not hold.
 (iii)
We observe that the bound \(\alpha= \kappa^{1}\) in (15) is the sharpest possible bound. To illustrate this, consider the real function \(f(x)=\frac{1}{2}x^{2}\). We observe that the subgradient mapping ∂f is strongly metrically subregular at \(\bar{x}=0\) for \(\bar{x}^{*}=0\) with modulus \(\kappa=1\), and inequality (15) is satisfied at \((0,0)\) for \(\alpha= \kappa^{1}\). However, inequality (15) is false at \((0,0)\) for any \(\alpha>\kappa^{1}\).
 (iv)Notice that condition (16) is weaker than the strong convexity of f in U, because the strong convexity entails the condition that there exists a constant c such that, for all \(x_{1}, x_{2} \in U\) and \(\lambda\in(0, 1)\),$$ f\bigl(\lambda x_{1}+ (1\lambda)x_{2}\bigr) \leq\lambda f(x_{1})+(1\lambda) f(x_{2})c\lambda(1\lambda) \x_{1}x_{2}\^{2}. $$(23)
Theorem 3.2
Proof
Furthermore, it is easy to observe the following consequence of Theorem 3.2 concerning the strong metric subregularity of the subdifferential.
Corollary 3.1
The conditions (25) and (28) can be weakened, if we strengthen slightly the condition (26). More specifically, we have the following result.
Theorem 3.3
Proof
4 Secondorder characterizations of quadratic growth properties
In this section, we characterize the quadratic growth properties by the secondorder subdifferential (or generalized Hessian) defined in (10). To motivate the subsequent discussion, we first review the standard secondorder optimality condition and restate it in terms of the gradient itself. The proof of the following proposition is similar to [22], Proposition 5.1.
Proposition 4.1
 (i)
the gradient mapping ∇f is strongly metrically subregular at \((\bar{x},0)\),
 (ii)
the Hessian matrix \(\nabla^{2} f(\bar{x})\) is positive definite.
Proof
In a nonsmooth setting, it is natural to use a secondorder subdifferential construction. The next theorem establishes a relationship among the strong metric subregularity, the quadratic growth condition, and the positivedefiniteness property of the combined secondorder subdifferential for a proxregular and subdifferentially continuous function. Its proof technique follows [6], Corollary 3.7.
Definition 4.1
Let there be given \(f: \mathbb {R}^{n} \rightarrow\bar{\mathbb {R}}\) and \((\bar{x}, \bar{x}^{*}) \in\operatorname{gph} \partial f\). We say that f is contingentnormal at x̄ for \(\bar{x}^{*}\) if the following property holds: for a vector \(u\in \mathbb {R}^{n}\), if there exists a vector \(w^{*}\) such that \((u,w^{*})\in T_{\operatorname{gph}\partial f}(\bar{x}, \bar{x}^{*})\), then there is a vector \(u^{*}\) such that \((u^{*},u)\in\hat{N}_{\operatorname{gph}\partial f}(\bar{x}, \bar{x}^{*})\).
Definition 4.2
Let there be given \(f: \mathbb {R}^{n} \rightarrow\bar{\mathbb {R}}\) and \((\bar{x}, \bar{x}^{*}) \in\operatorname{gph} \partial f\). We say that f is normalcontingent at x̄ for \(\bar{x}^{*}\) if the following property holds: for a vector \(u\in \mathbb {R}^{n}\), if there exists a vector \(u^{*}\) such that \((u^{*},u)\in\hat{N}_{\operatorname {gph}\partial f}(\bar{x}, \bar{x}^{*})\), then there is a vector \(w^{*}\) such that \((u,w^{*})\in T_{\operatorname{gph}\partial f}(\bar{x}, \bar{x}^{*})\).
Remark 4.1
Theorem 4.1
 (i)
the subdifferential ∂f is strongly metrically subregular at \((\bar{x}, \bar{x}^{*})\) with modulus \(\kappa>0\),
 (ii)there exist a neighborhood U of x̄ and a positive constant \(\alpha>r\) such that$$ f(x)\geq f(\bar{x})+\bigl\langle \bar{x}^{*},x\bar{x}\bigr\rangle + \frac {\alpha}{2}\x\bar{x}\^{2}\quad \textit{for all } x\in U, $$(46)
 (iii)there is a constant \(c>0\) such that the combined secondorder subdifferential \(\breve{\partial}^{2} f(\bar{x}, \bar {x}^{*})\) is positivedefinite in the sense that$$ \bigl\langle u^{*},u\bigr\rangle \geq c\u\^{2} \quad \textit{for all } u\in X \textit{ and } u^{*}\in\breve{\partial}^{2} f \bigl(\bar{x}, \bar{x}^{*}\bigr) (u). $$(47)
Proof
In the following theorem, we establish an equivalence among the strong metric subregularity, the quadratic growth condition, the positivedefiniteness property of the secondorder subdifferential/generalized Hessian, the strong metric regularity, and tilt stability for a proper l.s.c. function acting in \(\mathbb {R}^{n}\).
Theorem 4.2
 (i)
the subgradient mapping ∂f is strongly metrically subregular at \((\bar{x}, 0)\) with \(\kappa>0\),
 (ii)there is a constant \(c>0\) such that the secondorder subdifferential \(\partial^{2} f(\bar{x}, 0)\) is positivedefinite in the sense that$$ \bigl\langle u^{*},u\bigr\rangle \geq c\u\^{2}\quad \textit{for all } u\in X \textit{ and } u^{*}\in\partial^{2} f(\bar{x}, 0) (u), $$(51)
 (iii)there exist neighborhoods U of x̄ and V of 0 such that for any \(u^{*}\in V\) there is a point \(u \in(\partial f)^{1}(u^{*}) \cap U\) satisfying the inequality$$ f(x)\geq f(u)+\bigl\langle u^{*},xu\bigr\rangle +\frac{1}{2\kappa} \xu\^{2}\quad \textit{for all } x\in U, $$(52)
 (iv)
the subgradient mapping ∂f is strongly metrically regular around \((\bar{x}, 0)\) with \(\kappa>0\),
 (v)the secondorder subdifferential \(\partial^{2} f(\bar{x}, 0)\) is positivedefinite in the sense that$$ \bigl\langle u^{*},u\bigr\rangle >0 \quad \textit{for all } u\in X \textit{ and } u^{*}\in \partial^{2} f(\bar{x}, 0) (u), u\ne0, $$(53)
 (vi)
the point x̄ is a tiltstable local minimizer of the function f.
Proof
Since \(\operatorname{gph} \partial f\) is convex, we have \(\hat {N}_{\operatorname{gph}\partial f}(\bar{x}, \bar{x}^{*})=N_{\operatorname {gph}\partial f}(\bar{x}, \bar{x}^{*})\). Thereby, \(\breve{\partial}^{2} f(\bar{x}, \bar{x}^{*})(u)=\partial^{2} f(\bar{x}, \bar{x}^{*})(u)\), which together with Theorem 4.1 shows that the implications (i) ⇒ (ii) holds. The equivalence among (iii), (iv), (v), and (vi) has been established by Drusvyatskiy et al. [7]. The implications (ii) ⇒ (v) and (iv) ⇒ (i) hold trivially. □
Remark 4.2
 (i)
Under the assumptions of Theorem 4.2, we see that the constant c in the quadratic growth condition (1) can attain to \(\frac{1}{\kappa}\), and the strong metric subregularity is equivalent to the strong metric regularity.
 (ii)
The assumption that \(\operatorname{gph} \partial f\) is convex is essential in the conclusion of Theorem 4.2. If this assumption fails, the equivalence between the strong metric subregularity and the strong metric regularity may be a failure. To illustrate it, take the realvalued function \(f(x)=x\), we observe that ∂f is strongly metrically subregular at \((0, 0)\) with any \(\kappa>0\) and f is normalcontingent at \(\bar{x}=0\) for \(\bar{x}^{*}=0\) as well as \(\operatorname{gph} \partial f\) is not convex. On the other hand, ∂f is not strongly metrically regular at \((0, 0)\).
Declarations
Acknowledgements
This work was supported by the National Natural Sciences Grant (No. 11371116).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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