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¹¹institutetext: Nguyen Huy Chieu, Corresponding author ²²institutetext: Department of Mathematics, Vinh University
Nghe An, Vietnam
[email protected] ³³institutetext: Nguyen Thi Quynh Trang ⁴⁴institutetext: Department of Mathematics, Vinh University
Nghe An, Vietnam
[email protected] ⁵⁵institutetext: Ha Anh Tuan ⁶⁶institutetext: Faculty of Basic Science, Ho Chi Minh city University of Transport
Ho Chi Minh city, Viet Nam
[email protected]

Quadratic Growth and Strong Metric Subregularity of the Subdifferential for a Class of Non-prox-regular Functions

Nguyen Huy Chieu Nguyen Thi Quynh Trang Ha Anh Tuan

(Received: date / Accepted: date)

Abstract

This paper mainly studies the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as the sum of a function twice differentiable in the extended sense and a subdifferentially continuous, prox-regular, twice epi-differentiable function. For such a function, which is not necessarily prox-regular, it is shown that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. In addition, other characterizations of the quadratic growth and the strong metric subregularity of the subdifferential are also given. Besides, properties of functions twice differentiable in the extended sense are examined.

Keywords:

Quadratic growth Strong metric subregularity Twice differentiability Twice epi-differentiability Prox-regularity

MSC:

49J5390C3190C46

1 Introduction

Quadratic growth is an important property of extended-real-valued functions, which plays a central role in optimization AG08 ; AG14 ; BLN18 ; BS00 ; CHNT21 ; DI15 ; DL18 ; DMN14 ; MMS20 ; MMS21 ; MS20 . It can be used for justifying the linear convergence of various optimization algorithms BLN18 ; DL18 ; OM21 as well as analyzing perturbations of optimization problems. Especially, for many favorable classes of functions, quadratic growth is closely related to critical point stability AG08 ; AG14 ; CHNT21 ; DI15 ; MMS20 ; MMS21 ; MS20 .

For a proper lower semicontinuous convex function, as shown by Aragón Artacho and Geoffroy AG08 , the quadratic growth and the strong metric subregularity of the subdifferential at a local minimizer are equivalent, and they can be characterized by the positive definiteness of the subgradient graphical derivative at a stationary point.

For an arbitrary proper lower semicontinuous function, Drusvyatskiy et al. DMN14 proved the validity of the quadratic growth under the strong metric subregularity of the subdifferential at a local minimizer. Drusvyatskiy and Ioffe DI15 established that the converse holds whenever the function under consideration is semi-algebraic, and it may fail if the function is not semi-algebraic. It is worth noting that the approach of DI15 is based on some facts from semi-algebraic geometry, which might not be available for functions that are not semi-algebraic.

Using tools of second-order variational analysis, Chieu et al. CHNT21 showed that for a proper lower semicontinuous function, the positive definiteness of the subgradient graphical derivative at a stationary point guarantees that the point is a local minimizer and the subdifferential is strongly metrically subregular, which implies by DMN14 that the quadratic growth holds. Furthermore, the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent whenever the function is either subdifferentially continuous, prox-regular, and twice epi-differentiable or variationally convex.

More recent developments in this direction can be found in MMS20 ; MMS21 ; MS20 ; OM21 , where the authors investigated composite models under certain assumptions on the component functions that make the composite function subdifferentially continuous, prox-regular, and twice epi-differentiable.

To the best of our knowledge, all known results on the equivalence relationship between the quadratic growth and the strong metric subregularity of the subdifferential, except for the one of Drusvyatskiy and Ioffe DI15 , are established only for subclasses of the class of subdifferentially continuous and prox-regular functions. This observation leads us to the question if such an equivalence relationship is valid for functions that are neither subdifferentially continuous and prox-regular nor semi-algebraic.

In the current work, we study the quadratic growth and the strong metric subregularity of the subdifferential of functions that can be represented as the sum of an extended twice differentiable function and a subdifferentially continuous, prox-regular, twice epi-differentiable function. This big class of functions encompasses subdifferentially continuous, prox-regular, twice epi-differentiable functions as well as twice differentiable functions.

For a function from the just mentioned class, which is not necessarily prox-regular, it is shown that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. In addition, other characterizations of quadratic growth as well as the strong metric subregularity of the subdifferential are also given. Besides, properties of functions that are twice differentiable in the extended sense are examined.

The rest of the paper is organized as follows. Section 2 collects notions from variational analysis that are needed in the sequel. Section 3 investigates functions that are twice differentiable in the extended sense. The focuses of this section are on sum rules and chain rules for second subderivative, parabolic subderivative, and subgradient graphical derivative, which are used for proving the main results reported in Section 4. Section 4 is devoted to the study of quadratic growth and strong metric subregularity of the subdifferential. Here we specially pay the attention to the relationship between these two properties. Besides, we are also interested in characterizations of quadratic growth and strong metric subregularity of the subdifferential via the second subderivative. Section 5 summarizes the main results of the paper and presents some remarks on this research direction.

2 Preliminaries

This section recalls some concepts and their properties from variational analysis M1 ; M18 ; RW98 , which are needed for our analysis. Unless otherwise stated, $\mathbb{R}^{n}$ is a Euclidean space with inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|,$ and $\overline{\mathbb{R}}:=\mathbb{R}\cup\{\infty\}.$ The closed ball with center $\bar{x}$ and radius $\varepsilon>0$ is denoted by $\mathbb{B}_{\varepsilon}(\bar{x}):=\{x\in\mathbb{R}^{n}\ |\ \|x-\bar{x}\|\leq\varepsilon\}.$

Definition 1

(M1 ; M18 ; RW98 ). Let $f:\mathbb{R}^{n}\rightarrow\overline{\mathbb{R}}$ and let $\bar{x}\in\mbox{\rm dom}\,f:=\big{\{}x\in\mathbb{R}^{n}|\;f(x)<\infty\big{\}}$ . The proximal subdifferential of $f$ at $\bar{x}\in\mbox{\rm dom}\,f$ is defined by

\partial_{p}f(\bar{x}):=\left\{v\in\mathbb{R}^{n}\ |\ \liminf\limits_{x\to\bar{x}}\frac{f(x)-f(\bar{x})-\langle v,x-\bar{x}\rangle}{\|x-\bar{x}\|^{2}}>-\infty\right\}.

The regular subdifferential (also called Fréchet subdifferential) of $f$ at $\bar{x}\in\mbox{\rm dom}\,f$ is given by

\widehat{\partial}f(\bar{x}):=\left\{v\in\mathbb{R}^{n}\ |\ \liminf\limits_{x\to\bar{x}}\frac{f(x)-f(\bar{x})-\langle v,x-\bar{x}\rangle}{\|x-\bar{x}\|}\geq 0\right\}.

The limiting subdifferential (also called Mordukhovich subdifferential) of $f$ at $\bar{x}\in\mbox{\rm dom}\,f$ is defined by

\partial f(\bar{x}):=\left\{v\in\mathbb{R}^{n}\ |\ \exists x_{k}\stackrel{{\scriptstyle f}}{{\to}}\bar{x},v_{k}\to v\ \mbox{with}\ v_{k}\in\widehat{\partial}f(x_{k})\right\}.

If $\bar{x}\not\in\mbox{\rm dom}\,f,$ one puts $\partial f(\bar{x})=\widehat{\partial}f(\bar{x})=\partial_{p}f(\bar{x}):=\emptyset.$

Definition 2

(RW98 ). A function $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is said to be prox-regular at $\bar{x}\in\mbox{\rm dom}\,f$ for $\bar{v}\in\partial f(\bar{x})$ if there exist $r,\varepsilon>0$ such that for all $x,u\in\mathbb{B}_{\varepsilon}(\bar{x})$ with $|f(u)-f(\bar{x})|<\varepsilon$ we have

f(x)\geq f(u)+\langle v,x-u\rangle-\frac{r}{2}\|x-u\|^{2}\quad\mbox{for all}\quad v\in\partial f(u)\cap\mathbb{B}_{\varepsilon}(\bar{v}).

(1)

Moreover, $f$ is said to be subdifferentially continuous at $\bar{x}$ for $\bar{v}$ if whenever $(x_{k},v_{k})\to(\bar{x},\bar{v})$ with $v_{k}\in\partial f(x_{k})$ , one has $f(x_{k})\to f(\bar{x})$ .

From (1) it follows that $\partial f(u)\cap\mathbb{B}_{\varepsilon}(\bar{v})\subset\partial_{p}f(x)$ whenever $\|u-\bar{x}\|<\varepsilon$ with $|f(u)-f(\bar{x})|<\varepsilon$ . Furthermore, if $f$ is subdifferentially continuous at $\bar{x}$ for $\bar{v}$ , then the inequality “ $|f(u)-f(\bar{x})|<\varepsilon$ ” in the definition of prox-regularity above can be removed.

The following result is a direct consequence of (1), which is very useful for us to verify the prox-regularity in the sequel.

Lemma 1

((RW98, , Theorem 13.36)). If $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is prox-regular and subdifferentially continuous at $\bar{x}$ for $\bar{v}$ then there exist $r,\epsilon>0$ such that

\langle v_{2}-v_{1},x_{2}-x_{1}\rangle\geq-r\left\|x_{2}-x_{1}\right\|^{2},

(2)

for every $x_{1},x_{2}\in\mathbb{B}_{\epsilon}(\bar{x}),\ v_{1}\in\partial f(x_{1})\cap\mathbb{B}_{\epsilon}(\bar{v}),\ v_{2}\in\partial f(x_{2})\cap\mathbb{B}_{\epsilon}(\bar{v}).$

Definition 3

(RW98 ). Given a function $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ with $f(\bar{x})\in\mathbb{R},$ the subderivative of $f$ at $\bar{x}$ is the function $df(\bar{x}):\mathbb{R}^{n}\to[-\infty,\infty]$ defined by

df(\bar{x})(w)=\liminf\limits_{w^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}w}\frac{f(\bar{x}+tw^{\prime})-f(\bar{x})}{t}\quad\mbox{for all}\ w\in\mathbb{R}^{n}.

The second subderivative of $f$ at $\bar{x}$ for $v\in\mathbb{R}^{n}$ and $w\in\mathbb{R}^{n}$ is given by

d^{2}f(\bar{x}|v)(w)=\liminf\limits_{\begin{subarray}{\quad}\quad t\downarrow 0\\ w^{\prime}\longrightarrow w\end{subarray}}\Delta_{t}^{2}f(\bar{x},v)(w^{\prime}),

(3)

where $\Delta_{t}^{2}f(\bar{x},v)(w^{\prime}):=\frac{f(\bar{x}+tw^{\prime})-f(\bar{x})-t\langle v,w^{\prime}\rangle}{\frac{1}{2}t^{2}}.$

Function $f$ is said to be twice epi-differentiable at $\bar{x}\in\mathbb{R}^{n}$ for $v\in\mathbb{R}^{n}$ if for every $w\in\mathbb{R}^{n}$ and choice of $t_{k}\downarrow 0$ there exists $w_{k}\to w$ such that

\Delta_{t_{k}}^{2}f(\bar{x},v)(w_{k})\to d^{2}f(\bar{x}|v)(w).

It is well-known that fully amenable functions RW98 , including the maximum of finitely many $C^{2}$ -functions, are subdifferentially continuous prox-regular and twice epi-differentiable lower semicontinuous proper functions (RW98, , Corollary 13.15 and Proposition 13.32).

Definition 4

(RW98 ). Let $\Omega$ be a nonempty subset of $\mathbb{R}^{n}$ and $\bar{x}\in\mathbb{R}^{n}.$

$(i)$ The (Bouligand-Severi) tangent cone to $\Omega$ at $\bar{x}\in\Omega$ is given by

T_{\Omega}(\bar{x}):=\big{\{}v\in\mathbb{R}^{n}|\,\exists t_{k}\downarrow 0,\ v_{k}\rightarrow v\ \mbox{ with }\ \bar{x}+t_{k}v_{k}\in\Omega\ \forall k\in\mathbb{N}\big{\}}.

If $\bar{x}\not\in\Omega$ then one puts $T_{\Omega}(\bar{x}):=\emptyset.$

$(ii)$ The second-order tangent set to $\Omega$ at $\bar{x}$ for $w\in T_{\Omega}(\bar{x})$ is defined by

T^{2}_{\Omega}(\bar{x},w)=\left\{u\in\mathbb{R}^{n}\ |\ \exists t_{k}\downarrow 0,u_{k}\rightarrow u\ \mbox{with}\ \bar{x}+t_{k}w+\frac{1}{2}t_{k}^{2}u_{k}\in\Omega\ \forall k\in\mathbb{N}\right\}.

$\Omega$ is called parabolically derivable at $\bar{x}$ for $w\in\mathbb{R}^{n}$ if $T^{2}_{\Omega}(\bar{x},w)\not=\emptyset,$ and for each $u\in T^{2}_{\Omega}(\bar{x},w)$ there exist $\varepsilon>0$ and a mapping $\xi:[0,\varepsilon]\to\Omega$ such that $\xi(0)=\bar{x},$ $\xi_{+}^{\prime}(0)=w$ and $\xi_{+}^{\prime\prime}(0)=u,$ where

\xi_{+}^{\prime}(0):=\lim\limits_{t\downarrow 0}\frac{\xi(t)-\xi(0)}{t}\quad\mbox{and}\quad\xi_{+}^{\prime\prime}(0):=\lim\limits_{t\downarrow 0}\frac{\xi(t)-\xi(0)-t\xi_{+}^{\prime}(0)}{\frac{1}{2}t^{2}}.

Definition 5

(RW98 ). The subgradient graphical derivative of $f$ at $\bar{x}$ for $\bar{v}\in\partial f(\bar{x})$ is the set-valued mapping $D(\partial f)(\bar{x}|\bar{v}):\mathbb{R}^{n}\rightrightarrows\mathbb{R}^{n}$ defined by

D(\partial f)(\bar{x}|\bar{v})(w):=\big{\{}z\,|\,(w,z)\in T_{{\rm gph}\,\partial f}(\bar{x},\bar{v})\big{\}}\quad\mbox{for all}\quad w\in\mathbb{R}^{n}.

If $f$ is twice epi-differentiable, prox-regular, subdifferentially continuous at $\bar{x}$ for $\bar{v}$ , then it is known from (RW98, , Theorem 13.40) that

D(\partial f)(\bar{x}|\bar{v})=\partial h\quad\mbox{with}\quad h=\frac{1}{2}d^{2}f(\bar{x}|\bar{v}).

(4)

Definition 6

(MS20 ). A function $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is said to be parabolically regular at $\bar{x}$ for $\bar{v}\in\mathbb{R}^{n}$ if $f(\bar{x})\in\mathbb{R}$ and for all $w$ with $d^{2}f(\bar{x},\bar{v})(w)<\infty$ there exist $t_{k}\downarrow 0$ and $w_{k}\to w$ such that

\lim\limits_{k\to\infty}\Delta_{t_{k}}^{2}f(\bar{x},\bar{v})(w_{k})=d^{2}f(\bar{x},\bar{v})(w)\quad\mbox{and}\quad\limsup\limits_{k\to\infty}\frac{\|w_{k}-w\|}{t_{k}}<\infty.

(5)

A nonempty set $\Omega\subset\mathbb{R}^{n}$ is called parabolically regular at $\bar{x}$ for $\bar{v}$ if its indicator function $\delta_{\Omega}$ is parabolically regular at $\bar{x}$ for $\bar{v}.$

Definition 7

(RW98 ). Let $f:\mathbb{R}^{n}\to\overline{\mathbb{R}},$ $\bar{x}\in\mbox{\rm dom}\,f,$ and $w\in\mathbb{R}^{n}$ with $df(\bar{x})(w)\in\mathbb{R}.$

$(i)$ The parabolic subderivative of $f$ at $\bar{x}$ for $w$ with respect to $z$ is

d^{2}f(\bar{x})(w|z):=\liminf\limits_{z^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}z}\frac{f(\bar{x}+tw+\frac{1}{2}t^{2}z^{\prime})-f(\bar{x})-tdf(\bar{x})(w)}{\frac{1}{2}t^{2}}.

$(ii)$ $f$ is said to be parabolically epi-differentiable at $\bar{x}$ for $w$ if

\mbox{\rm dom}\,d^{2}f(\bar{x})(w|\cdot)=\{z\in\mathbb{R}^{n}\ |\ d^{2}f(\bar{x})(w|z)<\infty\}\not=\emptyset,

and for every $z\in\mathbb{R}^{n}$ and every $t_{k}\downarrow 0$ there exists $z_{k}\to z$ such that

d^{2}f(\bar{x})(w|z):=\liminf\limits_{k\to\infty}\frac{f(\bar{x}+t_{k}w+\frac{1}{2}t_{k}^{2}z_{k})-f(\bar{x})-t_{k}df(\bar{x})(w)}{\frac{1}{2}t_{k}^{2}}.

(6)

As shown by Mohammadi and Sarabi (MS20, , Proposition 3.6), a function $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ with $\bar{v}\in\partial_{p}f(\bar{x})$ is parabolically regular at $\bar{x}$ for $\bar{v}$ if and only if

d^{2}f(\bar{x},\bar{v})(w)=\inf\limits_{z\in\mathbb{R}^{n}}\left\{d^{2}f(\bar{x})(w|z)-\langle z,\bar{v}\rangle\right\}\quad\mbox{for all}\ w\in K_{f}(\bar{x}|\bar{v}),

(7)

where $K_{f}(\bar{x}|\bar{v}):=\{w\in\mathbb{R}^{n}\ |\ df(\bar{x})(w)=\langle\bar{v},w\rangle\}$ is called the critical cone of $f$ at $(\bar{x},\bar{v}).$ Furthermore, if $f$ is parabolically regular at $\bar{x}$ for $\bar{v}$ and $w\in\mbox{\rm dom}\,d^{2}f(\bar{x},\bar{v})$ then there exists $\bar{z}\in\mbox{\rm dom}\,d^{2}f(\bar{x})(w|\cdot)$ such that

d^{2}f(\bar{x},\bar{v})(w)=d^{2}f(\bar{x})(w|\bar{z})-\langle\bar{z},\bar{v}\rangle.

(8)

3 Twice differentiability in the extended sense

The concept of twice differentiability of functions in the extended sense was introduced by Rockafellar and Wets (RW98, , Definition 13.1), which came from the desire to develop second order differentiability at $\bar{x}$ without having to assume the existence of the first partial derivatives at every point in some neighborhood of $\bar{x}$ .

This section investigates properties of functions that are twice differentiable in the extended sense, with special attention paying to sum rules and chain rules of the equality form for second subderivative, parabolic subderivative, and subgradient graphical derivative.

Definition 8

(RW98 ). Let $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ be finite at $\bar{x}.$ We say that

$(i)$ $f$ is differentiable (resp., strictly differentiable) at $\bar{x}$ if there exists an $(1\times n)$ -matrix $\nabla f(\bar{x}),$ called the Jacobian (matrix) of $f$ at $\bar{x},$ such that

\lim\limits_{x\to\bar{x}}\frac{f(x)-f(\bar{x})-\nabla f(\bar{x})(x-\bar{x})}{\|x-\bar{x}\|}=0\quad(\mbox{resp,}\lim\limits_{x,u\to\bar{x}}\frac{f(x)-f(u)-\nabla f(\bar{x})(x-u)}{\|x-u\|}=0);

$(ii)$ $f$ is twice differentiable at $\bar{x}$ (in the classical sense) if it is differentiable on a neighborhood $U$ of $\bar{x}$ and there exists a $n\times n$ matrix $\nabla^{2}f(\bar{x}),$ called the Hessian (matrix) of $f$ at $\bar{x}$ , such that

\lim\limits_{x\stackrel{{\scriptstyle U}}{{\rightarrow}}\bar{x}}\frac{\nabla f(x)-\nabla f(\bar{x})-H(x-\bar{x})}{\|x-\bar{x}\|}=0;

$(iii)$ $f$ is twice differentiable at $\bar{x}$ in the extended sense if it is differentiable at $\bar{x},$ and there exist a $n\times n$ matrix $A,$ a neighborhood $U$ of $\bar{x}$ and a subset $D$ of $U$ with $\mu(U\backslash D)=0$ such that $f$ is Lipschitz on $U,$ differentiable at every point in $D,$ and

\lim\limits_{x\stackrel{{\scriptstyle D}}{{\rightarrow}}\bar{x}}\frac{\nabla f(x)-\nabla f(\bar{x})-A(x-\bar{x})}{\|x-\bar{x}\|}=0,

where $\mu$ denotes the Lebesgue measure on $\mathbb{R}^{n}.$ This matrix $A$ , necessarily unique, is then called the Hesian (matrix) of $f$ at $\bar{x}$ in the extended sense and is likewise denoted by $\nabla^{2}f(\bar{x}).$

It is known (RW98, , Theorem 13.51) that a $C^{2}$ -lower function (and thus a $C^{1,1}$ -function) on an open set $\mathcal{O}\subset\mathbb{R}^{n}$ is twice differentiable in the extended sense almost everywhere in $\mathcal{O},$ with extended Hessian being symmetric where they exist.

It is easy to see that if $f$ is twice differentiable at $\bar{x}$ then it is twice differentiable at $\bar{x}$ in the extended sense, and the Hessian and the extended Hessian coincide.

The following example shows that there exists a function that is twice differentiable in the extended sense, but neither twice differentiable nor prox-regular.

Example 1

(Extended twice differentiability does not imply either twice differentiability or prox-regularity). Consider the function $g:\mathbb{R}\rightarrow\mathbb{R}$ given by

g(x)=\begin{cases}x^{10/3}\cos\frac{1}{x}+x^{4}\quad\ &\mbox{ if }x\geq 1,\\ x^{10/3}\cos\frac{1}{x}+\frac{(2n+1)(2n^{2}+2n+1)}{n^{3}(n+1)^{3}}x+\frac{1}{(n+1)^{3}}-\frac{1}{n^{3}}\quad\ &\mbox{ if }x\in\big{[}\frac{1}{n+1},\frac{1}{n}\big{)},\ n=1,2,...\\ 0\quad\ &\mbox{ if }x=0,\\ g(-x)\quad\ &\mbox{ if }x<0,\end{cases}

${\it Claim1:}$ $g$ is twice differentiable at $\bar{x}=0$ in the extended sense, but it is not twice differentiable at $\bar{x}$ in the classical sense. Indeed, we see that $g$ is differentiable at $\bar{x}$ , and

\nabla g(x)=\begin{cases}\frac{10}{3}x^{7/3}\cos\frac{1}{x}+x^{4/3}\sin\frac{1}{x}+4x^{3}&\mbox{ if }x>1,\\ \frac{10}{3}x^{7/3}\cos\frac{1}{x}+x^{4/3}\sin\frac{1}{x}+\frac{(2n+1)(2n^{2}+2n+1)}{n^{3}(n+1)^{3}}&\mbox{ if }x\in\big{(}\frac{1}{n+1},\frac{1}{n}\big{)},\ n=1,2,...\\ 0\quad\ &\mbox{ if }x=0,\\ -\nabla g(-x)\quad\ &\mbox{ if }\ x\in(-\infty,0)\setminus\big{\{}-\frac{1}{n}|\ n\in\mathbb{N}^{*}\big{\}}.\end{cases}

Put $U=(-1,1),$ $D=(-1,1)\setminus\{\frac{1}{n}|\ n\in\mathbb{Z}^{*}\},$ and $A=0.$ Then $\mu(U\backslash D)=0,$ $g$ is Lipschitz on $U$ with constant $\kappa=1,$ and differentiable at every point in $D,$ where $\mu$ is the Lebesgue measure on $\mathbb{R}.$ Furthermore, for each $x\in\big{(}\frac{1}{n+1},\frac{1}{n}\big{)}$ with $n\in\mathbb{N}^{*}$ we have

\begin{array}[]{rl}\left|\frac{\nabla g(x)-\nabla g(\bar{x})-A(x-\bar{x})}{|x-\bar{x}|}\right|&\leq\left|\frac{10}{3}x^{4/3}\cos\frac{1}{x}+x^{1/3}\sin\frac{1}{x}\right|+\frac{(2n+1)(2n^{2}+2n+1)}{n^{3}(n+1)^{3}|x|}\\ &\leq\frac{10}{3}x^{4/3}+x^{1/3}+\frac{(2n+1)(2n^{2}+2n+1)}{n^{3}(n+1)^{2}}\to 0\quad\mbox{as}\ n\to\infty.\end{array}

Combing this with $\nabla g(x)=-\nabla g(-x)$ for all $x\in(-\infty,0)\setminus\big{\{}-\frac{1}{n}|\ n\in\mathbb{N}^{*}\big{\}},$ we get

\lim\limits_{x\stackrel{{\scriptstyle D}}{{\rightarrow}}\bar{x}}\frac{\nabla g(x)-\nabla g(\bar{x})-A(x-\bar{x})}{|x-\bar{x}|}=0.

Hence, $g$ is twice differentiable at $\bar{x}$ in the extended sense. On the other hand, since $g$ is not differentiable at each point $\frac{1}{n}$ with $n\in\mathbb{Z}^{*},$ $g$ is not twice differentiable at $\bar{x}$ in the classical sense. Therefore, the extended twice differentiability does not imply the classical twice differentiability.

${\it Claim2:}$ $g$ is not prox-regular at $\bar{x}$ for $\bar{v}=0.$ Fix $r>0$ and put $u_{k}=\frac{1}{2k\pi},\ x_{k}=\frac{1}{\frac{\pi}{2}+{2k\pi}}$ for every $k\in\mathbb{N}^{*}.$ Then for each $k\in\mathbb{N}^{*}$ there exist $m_{k},n_{k}\in\mathbb{N}^{*}$ such that $u_{k}\in\big{(}\frac{1}{m_{k}+1},\frac{1}{m_{k}}\big{)}$ and $x_{k}\in\big{(}\frac{1}{n_{k}+1},\frac{1}{n_{k}}\big{)}.$ This implies that $2k\pi<m_{k}+1$ for all $k.$ So we have

\begin{array}[]{rl}&\langle\nabla g(u_{k})-\nabla g(x_{k}),u_{k}-x_{k}\rangle+r\left|u_{k}-x_{k}\right|^{2}\\ &=\Big{(}\frac{10}{3}\frac{1}{(2k\pi)^{7/3}}+\frac{(2m_{k}+1)(2m_{k}^{2}+2m_{k}+1)}{m^{3}_{k}(m_{k}+1)^{3}}-\frac{1}{\big{(}\frac{\pi}{2}+2k\pi\big{)}^{4/3}}-\frac{(2n_{k}+1)(2n_{k}^{2}+2n_{k}+1)}{n_{k}^{3}(n_{k}+1)^{3}}\Big{)}\big{(}\frac{1}{2k\pi}-\frac{1}{\frac{\pi}{2}+{2k\pi}}\big{)}\\ &\ \ \ \ +r\big{(}\frac{1}{2k\pi}-\frac{1}{\frac{\pi}{2}+{2k\pi}}\big{)}^{2}\\ &\leq\frac{\pi}{4k\pi\big{(}\frac{\pi}{2}+2k\pi\big{)}}\Big{(}\frac{10}{3}\frac{1}{(2k\pi)^{7/3}}+\frac{5}{(m_{k}+1)^{3}}-\frac{1}{\big{(}\frac{\pi}{2}+2k\pi\big{)}^{4/3}}+r\frac{\pi}{4k\pi\big{(}\frac{\pi}{2}+2k\pi\big{)}}\Big{)}\\ &\leq\frac{\pi}{4k\pi\big{(}\frac{\pi}{2}+2k\pi\big{)}}\Big{(}\frac{10}{3}\frac{1}{(2k\pi)^{7/3}}+\frac{5}{(2k\pi)^{3}}-\frac{1}{\big{(}\frac{\pi}{2}+2k\pi\big{)}^{4/3}}+r\frac{\pi}{4k\pi\big{(}\frac{\pi}{2}+2k\pi\big{)}}\Big{)}<0,\\ \end{array}

for all $k$ large enough. Note that $\lim\limits_{k\to\infty}\big{(}u_{k},\nabla g(u_{k})\big{)}=\lim\limits_{k\to\infty}\big{(}x_{k},\nabla g(x_{k})\big{)}=(0,0).$ Therefore, by Lemma 1, $f$ is not prox-regular at $\bar{x}=0$ for $\bar{v}=0.$

Example 2

(Twice differentiability does not imply prox-regularity). Consider the function $f:\mathbb{R}\to\mathbb{R}$ defined by

f(x):=\int_{0}^{x}g(t)dt\quad\mbox{where}\quad g(x)=\begin{cases}x^{2}\mbox{sin}\frac{1}{x^{2}}\quad\ &\mbox{ if }x\not=0,\\ 0\quad\ &\mbox{ if }x=0.\end{cases}

We see that $\nabla f(x)=g(x)$ for all $x\in\mathbb{R},$ and $f$ is twice differentiable at every point in $\mathbb{R}$ with

\nabla^{2}f(x)=\nabla g(x)=\begin{cases}2x\sin\frac{1}{x^{2}}-\frac{2}{x}\cos\frac{1}{x^{2}}\quad\ &\mbox{ if }x\not=0,\\ 0\quad\ &\mbox{ if }x=0.\end{cases}

We next prove that $f$ is not prox-regular at $\bar{x}:=0$ for $\nabla f(\bar{x})=0.$ Arguing by contradiction, suppose that $f$ is prox-regular at $0$ for $0.$ Then, by Lemma 1, there exist $r,\epsilon>0$ such that

\langle g(u)-g(x),u-x\rangle=\langle\nabla f(u)-\nabla f(x),u-x\rangle\geq-r|u-x|^{2},

(9)

for every $u,x\in\mathbb{B}_{\epsilon}(\bar{x}).$ Thus, for each $k\in\mathbb{N}^{*}$ sufficiently large, choosing $u_{k}=\frac{1}{\sqrt{2k\pi}}$ and $x_{k}=\frac{1}{\sqrt{\frac{\pi}{2}+{2k\pi}}},$ we have

\begin{array}[]{rl}&\langle g(u_{k})-g(x_{k}),u_{k}-x_{k}\rangle+r\left|u_{k}-x_{k}\right|^{2}\\ &=-\frac{1}{\frac{\pi}{2}+{2k\pi}}\Big{(}\frac{1}{\sqrt{2k\pi}}-\frac{1}{\sqrt{\frac{\pi}{2}+{2k\pi}}}\Big{)}+r\Big{(}\frac{1}{\sqrt{2k\pi}}-\frac{1}{\sqrt{\frac{\pi}{2}+{2k\pi}}}\Big{)}^{2}\\ &=\Big{(}\frac{1}{\sqrt{2k\pi}}-\frac{1}{\sqrt{\frac{\pi}{2}+{2k\pi}}}\Big{)}\Bigg{(}-\frac{1}{\frac{\pi}{2}+{2k\pi}}+r\Big{(}\frac{1}{\sqrt{2k\pi}}-\frac{1}{\sqrt{\frac{\pi}{2}+{2k\pi}}}\Big{)}\Bigg{)}\\ &=\Big{(}\frac{1}{\sqrt{2k\pi}}-\frac{1}{\sqrt{\frac{\pi}{2}+{2k\pi}}}\Big{)}\Bigg{(}-\frac{1}{\frac{\pi}{2}+{2k\pi}}+r\frac{\sqrt{\frac{\pi}{2}+{2k\pi}}-\sqrt{2k\pi}}{\sqrt{2k\pi}\sqrt{\frac{\pi}{2}+{2k\pi}}}\Bigg{)}\\ &=\Big{(}\frac{1}{\sqrt{2k\pi}}-\frac{1}{\sqrt{\frac{\pi}{2}+{2k\pi}}}\Big{)}\Bigg{(}-\frac{1}{\frac{\pi}{2}+{2k\pi}}+\frac{\pi r}{2\sqrt{2k\pi}\sqrt{\frac{\pi}{2}+{2k\pi}}\Big{(}\sqrt{\frac{\pi}{2}+{2k\pi}}+\sqrt{2k\pi}\Big{)}}\Bigg{)}<0.\end{array}

This contradicts (9) since $\lim\limits_{k\to\infty}u_{k}=\lim\limits_{k\to\infty}u_{k}=0=\bar{x}$ . Therefore, $f$ is not prox-regular at $\bar{x}=0$ for $\bar{v}=0.$

The following lemma collects some properties of extended twice differentiable functions that will be used in the sequel.

Lemma 2

((RW98, , Theorem 13.2)). Let $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ be twice differentiable at a point $\bar{x}$ in the extended sense. Then $\partial f(\bar{x})=\{\nabla f(\bar{x})\}$ and there exists a neigborhood $U$ of $\bar{x}$ such that

\emptyset\not=\partial f(x)\subset\nabla f(\bar{x})+\nabla^{2}f(x-\bar{x})+o(\|x-\bar{x}\|)\mathbb{B},

(10)

for every $x\in U.$ Furthermore, $f$ is strictly differentiable at $\bar{x},$ and

f(x)=f(\bar{x})+\langle\nabla f(\bar{x}),x-\bar{x}\rangle+\frac{1}{2}\langle x-\bar{x},\nabla^{2}f(\bar{x})(x-\bar{x})\rangle+o(\|x-\bar{x}\|^{2}).

(11)

Here $o(t)$ stands for some function of $t$ with $\lim\limits_{t\to 0}\frac{o(t)}{t}=0.$

Naturally, we say a mapping $F:\mathbb{R}^{n}\to\mathbb{R}^{m},$ $x\mapsto\big{(}F_{1}(x),F_{2}(x),...,F_{m}(x)\big{)}$ is twice differentiable at $\bar{x}$ in the extended sense if $F_{k}$ is twice differentiable at $\bar{x}$ in the extended sense for every $k=1,2,...,m.$ In the sequel, for such a mapping $F,$ the symbol $\nabla^{2}F(\bar{x})(w,v)$ stands for $\big{(}\langle\nabla^{2}F_{1}(\bar{x})w,v\rangle,\langle\nabla^{2}F_{2}(\bar{x})w,v\rangle,...,\langle\nabla^{2}F_{m}(\bar{x})w,v\rangle\big{)}$ for all $v,w\in\mathbb{R}^{n}.$

The following theorem provides sum rules of equality form for gradient graphical derivative, second subderivative and parabolic subderivative.

Theorem 3.1

Suppose that $\varphi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is twice differentiable at $\bar{x}$ in the extended sense, $\psi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is proper lower semicontinuous around $\bar{x}$ , and $\bar{v}\in\partial(\varphi+\psi)(\bar{x})$ . Then one has

D\partial(\varphi+\psi)(\bar{x}|\bar{v})(w)=\nabla^{2}\varphi(\bar{x})(w)+D\partial\psi\big{(}\bar{x}|\bar{v}-\nabla\varphi(\bar{x})\big{)}(w),

(12)

d^{2}(\varphi+\psi)\big{(}\bar{x}|\bar{v}\big{)}(w)=\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\big{\rangle}+d^{2}\psi\big{(}\bar{x}|\bar{v}-\nabla\varphi(\bar{x})\big{)}(w),

(13)

and

d^{2}(\varphi+\psi)(\bar{x})(w|z)=\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\big{\rangle}+\nabla\varphi(\bar{x})z+d^{2}\psi(\bar{x})(w|z),

(14)

for every $w\in\mathbb{R}^{n}$ and $z\in\mathbb{R}^{n}.$

Proof. We first prove (12). To this end, take any $w\in\mathbb{R}^{n}$ and $z\in D\partial(\varphi+\psi)(\bar{x}|\bar{v})(w).$ Then there exist sequences $t_{k}\downarrow 0$ and $(w_{k},z_{k})\to(w,z)$ such that

\bar{v}+t_{k}z_{k}\in\partial(\varphi+\psi)(\bar{x}+t_{k}w_{k})\quad\mbox{for all}\ k\in\mathbb{N}^{*}.

Since $\varphi$ is twice differentiable at $\bar{x}$ in the extended sense, it is Lipschitz continuous around $\bar{x},$ and by the sum rule of subdifferential (M18, , Theorem 2.19) and (10), we get

\begin{array}[]{rl}\partial(\varphi+\psi)(\bar{x}+t_{k}w_{k})&\subset\partial\varphi(\bar{x}+t_{k}w_{k})+\partial\psi(\bar{x}+t_{k}w_{k})\\ &\subset\nabla\varphi(\bar{x})+t_{k}\nabla^{2}\varphi(\bar{x})(w_{k})+o(\left\|t_{k}w_{k}\right\|)\mathbb{B}+\partial\psi(\bar{x}+t_{k}w_{k}),\end{array}

for all $k\in\mathbb{N}^{*}$ sufficiently large. Thus, for such numbers $k$ , it holds that

\big{(}\bar{v}-\nabla\varphi(\bar{x})\big{)}+t_{k}\Big{(}z_{k}-\nabla^{2}\varphi(\bar{x})(w_{k})+\frac{o(\left\|t_{k}w_{k}\right\|)}{t_{k}}\Big{)}\in\partial\psi(\bar{x}+t_{k}w_{k}),

or equivalently,

\big{(}\bar{x},\bar{v}-\nabla\varphi(\bar{x})\big{)}+t_{k}\Big{(}w_{k},z_{k}-\nabla^{2}\varphi(\bar{x})(w_{k})+\frac{o(\left\|t_{k}w_{k}\right\|)}{t_{k}}\Big{)}\in\mbox{gph}\partial\psi.

On the other hand,

\Big{(}w_{k},z_{k}-\nabla^{2}\varphi(\bar{x})(w_{k})+\frac{o(\left\|t_{k}w_{k}\right\|)}{t_{k}}\Big{)}\to\big{(}w,z-\nabla^{2}\varphi(\bar{x})(w)\big{)}\quad\mbox{as}\ k\to\infty.

Therefore,

\big{(}w,z-\nabla^{2}\varphi(\bar{x})(w)\big{)}\in T_{\mbox{gph}\partial\psi}\big{(}\bar{x},\bar{v}-\nabla\varphi(\bar{x})\big{)}.

In other words,

z-\nabla^{2}\varphi(\bar{x})(w)\in D\partial\psi(\bar{x}|\bar{v}-\nabla\varphi(\bar{x})(w).

This shows that

D\partial(\varphi+\psi)(\bar{x}|\bar{v})(w)\subset\nabla^{2}\varphi(\bar{x})(w)+D\partial\psi\big{(}\bar{x}|\bar{v}-\nabla\varphi(\bar{x})\big{)}(w).

(15)

Conversely, by using (15) and noting that $-\varphi$ is also twice differentiable at $\bar{x}$ in the extended sense with $\nabla^{2}(-\varphi)(\bar{x})=-\nabla^{2}\varphi(\bar{x}),$ we have

\begin{array}[]{rl}D\partial\psi(\bar{x}|\bar{v}-\nabla\varphi(\bar{x}))(w)&=D\partial\big{(}\varphi+\psi+(-\varphi)\big{)}(\bar{x}|\bar{v}-\nabla\varphi(\bar{x}))(w)\\ &\subset D\partial\big{(}\varphi+\psi\big{)}(\bar{x}|\bar{v})(w)+\nabla^{2}(-\varphi)(\bar{x})(w)\\ &=D\partial\big{(}\varphi+\psi\big{)}(\bar{x}|\bar{v})(w)-\nabla^{2}\varphi(\bar{x})(w).\end{array}

This infers that

\nabla^{2}\varphi(\bar{x})(w)+D\partial\psi(\bar{x}|\bar{v}-\nabla\varphi(\bar{x}))(w)\subset D\partial(\varphi+\psi)(\bar{x}|\bar{v})(w).

(16)

From (15) and (16) it follows that

D\partial(\varphi+\psi)(\bar{x}|\bar{v})(w)=\nabla^{2}\varphi(\bar{x})(w)+D\partial\psi(\bar{x}|\bar{v}-\nabla\varphi(\bar{x}))(w)\ \mbox{for every}\ w\in\mathbb{R}^{n}.

We next justify the validity of (13). Take any $w\in\mathbb{R}^{n}.$ Since $\varphi$ is twice differentiable at $\bar{x}$ in the extended sense, by (11), we see that

\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\big{\rangle}=\lim\limits_{w^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}w}\Delta^{2}_{t}\varphi\big{(}\bar{x}|\nabla\varphi(\bar{x})\big{)}(w^{\prime}).

Therefore,

\begin{array}[]{rl}d^{2}(\varphi+\psi)(\bar{x}|\bar{v})(w)&=\liminf\limits_{w^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}w}\Delta^{2}_{t}(\varphi+\psi)(\bar{x}|\bar{v})(w^{\prime})\\ &=\liminf\limits_{w^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}w}\Big{[}\Delta^{2}_{t}\big{(}\bar{x}|\nabla\varphi(\bar{x})\big{)}(w^{\prime})+\Delta^{2}_{t}\psi\big{(}\bar{x}|\bar{v}-\nabla\varphi(\bar{x})\big{)}(w^{\prime})\Big{]}\\ &=\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\big{\rangle}+\liminf\limits_{w^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}w}\Delta^{2}_{t}\psi\big{(}\bar{x}|\bar{v}-\nabla\varphi(\bar{x})\big{)}(w^{\prime})\\ &=\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\rangle+d^{2}\psi\big{(}\bar{x}|\bar{v}-\nabla\varphi(\bar{x})\big{)}(w).\end{array}

Finally, we show that (14) holds. The differentiability of $\varphi$ at $\bar{x}$ gives us that

\begin{array}[]{rl}d(\varphi+\psi)(\bar{x})(w)&=\liminf\limits_{w^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}w}\frac{(\varphi+\psi)(\bar{x}+tw^{\prime})-(\varphi+\psi)(\bar{x})}{t}\\ &=\liminf\limits_{w^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}w}\left[\frac{\varphi(\bar{x}+tw^{\prime})-\varphi(\bar{x})}{t}+\frac{\psi(\bar{x}+tw^{\prime})-\psi(\bar{x})}{t}\right]\\ &=\nabla\varphi(\bar{x})w+\liminf\limits_{w^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}w}\frac{\psi(\bar{x}+tw^{\prime})-\psi(\bar{x})}{t}\\ &=\nabla\varphi(\bar{x})w+d\psi(\bar{x})(w)\quad\forall w\in\mathbb{R}^{n}.\end{array}

Since $\varphi$ is twice differentiable at $\bar{x}$ in the extended sense, by (11), we get

\lim\limits_{z^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}z}\frac{\varphi(\bar{x}+tw+\frac{1}{2}t^{2}z^{\prime})-\varphi(\bar{x})-t\nabla\varphi(\bar{x})w}{\frac{1}{2}t^{2}}=\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\big{\rangle}+\nabla\varphi(\bar{x})z\quad\forall w\in\mathbb{R}^{n},z\in\mathbb{R}^{n}.

Therefore,

\begin{array}[]{rl}&d^{2}(\varphi+\psi)\big{(}\bar{x}\big{)}(w|z)=\liminf\limits_{z^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}z}\frac{(\varphi+\psi)(\bar{x}+tw+\frac{1}{2}t^{2}z^{\prime})-(\varphi+\psi)(\bar{x})-td(\varphi+\psi)(\bar{x})(w)}{\frac{1}{2}t^{2}}\\ &=\liminf\limits_{z^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}z}\Big{[}\frac{\varphi(\bar{x}+tw+\frac{1}{2}t^{2}z^{\prime})-\varphi(\bar{x})-t\nabla\varphi(\bar{x})w}{\frac{1}{2}t^{2}}+\frac{\psi(\bar{x}+tw+\frac{1}{2}t^{2}z^{\prime})-\psi(\bar{x})-td\psi(\bar{x})(w)}{\frac{1}{2}t^{2}}\Big{]}\\ &=\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\big{\rangle}+\nabla\varphi(\bar{x})z+\liminf\limits_{z^{\prime}\stackrel{{\scriptstyle t\downarrow 0}}{{\rightarrow}}z}\frac{\psi(\bar{x}+tw+\frac{1}{2}t^{2}z^{\prime})-\psi(\bar{x})-td\psi(\bar{x})(w)}{\frac{1}{2}t^{2}}\\ &=\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\big{\rangle}+\nabla\varphi(\bar{x})z+d^{2}\psi(\bar{x})(w|z)\quad\forall w\in\mathbb{R}^{n},z\in\mathbb{R}^{n}.\end{array}

This finishes the proof. $\hfill\Box$

Let $\psi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ be finite at $\bar{x}\in\mathbb{R}^{n}.$ Assume that there exists a neighborhood $\mathcal{O}$ of $\bar{x}$ on which $\psi$ can be represented as

\psi(x)=g\circ F(x)\quad\mbox{for all}\ x\in\mathcal{O},

(17)

where $F:\mathbb{R}^{n}\to\mathbb{R}^{m}$ is twice differentiable at $\bar{x}$ in the extended sense, and $g:\mathbb{R}^{m}\to\overline{\mathbb{R}}$ is proper lower semicontinuous, convex, and Lipschitz continuous around $F(\bar{x})$ relative to its domain with constant $\ell\in\mathbb{R}_{+},$ that is, there exists a neighborhood $\mathcal{V}$ of $F(\bar{x})$ such that $|g(y_{1})-g(y_{2})|\leq\ell\|y_{1}-y_{2}\|$ for all $y_{1},y_{2}\in\mbox{\rm dom}\,g\cap\mathcal{V}.$

We see that

(\mbox{\rm dom}\,\psi)\cap\mathcal{O}=\{x\in\mathcal{O}\ |\ F(x)\in\mbox{\rm dom}\,g\}.

(18)

Following (MMS20, , Definition 3.2), the composition $\psi=g\circ F$ is said to satisfy the metric subregularity qualification condition (MSQC) at $\bar{x}\in\mbox{\rm dom}\,\psi$ if there exist a constant $\kappa\in\mathbb{R}_{+}$ and a neighborhood $U$ of $\bar{x}$ such that

d(x,\mbox{\rm dom}\,\psi)\leq\kappa d\big{(}F(x),\mbox{\rm dom}\,g\big{)}\quad\mbox{for all}\ x\in U.

(19)

Proposition 1

Let $\psi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ be a function that is represented as (18) with the composition $g\circ F$ satisfying MSQC at $\bar{x}.$ Then we have

d\psi(\bar{x})(w)=dg\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w\big{)}\ \mbox{for all}\ w\in\mathbb{R}^{n},\ \partial_{p}\psi(\bar{x})=\partial\psi(\bar{x})=\nabla F(\bar{x})^{*}\partial g\big{(}F(\bar{x})\big{)},

and $T_{\mbox{\rm dom}\,\psi}(\bar{x})=\big{\{}w\in\mathbb{R}^{n}\ |\ \nabla F(\bar{x})w\in T_{\mbox{\rm dom}\,g}\big{(}F(\bar{x})\big{)}\big{\}}.$

If assume further that $w\in T_{\mbox{\rm dom}\,\psi}(\bar{x})$ and $g$ is parabolically epi-differentiable at $F(\bar{x})$ for $\nabla F(\bar{x})w$ , then the following assertions hold:

$(i)$ $z\in T^{2}_{\mbox{\rm dom}\,\psi}(\bar{x},w)\Leftrightarrow\nabla F(\bar{x})z+\nabla^{2}F(\bar{x})(w,w)\in T^{2}_{\mbox{\rm dom}\,g}\big{(}F(\bar{x}),\nabla F(\bar{x})w\big{)},$ and $\mbox{\rm dom}\,\psi$ is parabolically derivable at $\bar{x}$ for $w;$

$(ii)$ $d^{2}\psi(\bar{x})(w|z)=d^{2}g\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w|\nabla F(\bar{x})z+\nabla^{2}F(\bar{x})(w,w)$ for all $z\in\mathbb{R}^{n}$ ;

$(iii)$ $\mbox{\rm dom}\,d^{2}\psi(\bar{x})(w|\cdot)=T^{2}_{\mbox{\rm dom}\,\psi}(\bar{x},w)$ ;

$(iv)$ $\psi$ is parabolically epi-differentiable at $\bar{x}$ for $w.$

Proof. Since $F:\mathbb{R}^{n}\to\mathbb{R}^{m}$ is twice differentiable at $\bar{x}$ in the extended sense, by Lemma 2, we get

F(x)=F(\bar{x})+\langle\nabla F(\bar{x}),x-\bar{x}\rangle+\frac{1}{2}\nabla^{2}F(\bar{x})(x-\bar{x},x-\bar{x})+o(\|x-\bar{x}\|^{2}),

(20)

and $f$ is strictly differentiable at $\bar{x}.$ The latter along with the composition $g\circ F$ satisfying MSQC at $\bar{x}$ implies by (MMS20, , Theorem 3.4) that

d\psi(\bar{x})(w)=dg\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w\big{)}\ \mbox{for all}\ w\in\mathbb{R}^{n},

(21)

and by (MMS20, , Theorem 3.6) that

\partial_{p}\psi(\bar{x})\subset\partial\psi(\bar{x})=\nabla F(\bar{x})^{*}\partial g\big{(}F(\bar{x})\big{)}.

(22)

We next prove that $\nabla F(\bar{x})^{*}\partial g\big{(}F(\bar{x})\big{)}\subset\partial_{p}\psi(\bar{x}).$ To this end, take any $y\in\partial g\big{(}F(\bar{x})\big{)}.$ Since $g$ is convex, we have $y\in\partial_{p}g\big{(}F(\bar{x})\big{)}.$ Hence,

\begin{array}[]{rl}&\liminf\limits_{x\to\bar{x}}\frac{\psi(x)-\psi(\bar{x})-\big{\langle}\nabla F(\bar{x})^{*}y,x-\bar{x}\big{\rangle}}{\|x-\bar{x}\|^{2}}\\ &=\liminf\limits_{x\to\bar{x}}\frac{g\big{(}F(\bar{x})+\nabla F(\bar{x})(x-\bar{x})+\frac{1}{2}\nabla^{2}F(\bar{x})(x-\bar{x},x-\bar{x})+o(\|x-\bar{x}\|^{2})\big{)}-g\big{(}F(\bar{x})\big{)}-\big{\langle}y,\nabla F(\bar{x})(x-\bar{x})\big{\rangle}}{\|x-\bar{x}\|^{2}}\\ &=\liminf\limits_{x\to\bar{x}}\frac{g\big{(}F(\bar{x})+\Delta(x)\big{)}-g\big{(}F(\bar{x})\big{)}-\big{\langle}y,\Delta(x)\big{\rangle}+\big{\langle}y,\frac{1}{2}\nabla^{2}F(\bar{x})(x-\bar{x},x-\bar{x})+o(\|x-\bar{x}\|^{2})\big{\rangle}}{\|x-\bar{x}\|^{2}}\\ &\geq\liminf\limits_{x\to\bar{x}}\frac{g\big{(}F(\bar{x})+\Delta(x)\big{)}-g\big{(}F(\bar{x})\big{)}-\big{\langle}y,\Delta(x)\big{\rangle}}{\|x-\bar{x}\|^{2}}-\frac{1}{2}\|y\|\cdot\|\nabla^{2}F(\bar{x})\|>-\infty,\end{array}

where $\Delta(x):=\nabla F(\bar{x})(x-\bar{x})+\frac{1}{2}\nabla^{2}F(\bar{x})(x-\bar{x},x-\bar{x})+o(\|x-\bar{x}\|^{2})\rightarrow 0$ as $x\to\bar{x}.$ This shows that $\nabla F(\bar{x})^{*}y\in\partial_{p}\psi(\bar{x})$ , and thus

\nabla F(\bar{x})^{*}\partial g\big{(}F(\bar{x})\big{)}\subset\partial_{p}\psi(\bar{x}).

(23)

From (22) and (23) it follows that

\partial_{p}\psi(\bar{x})=\partial\psi(\bar{x})=\nabla F(\bar{x})^{*}\partial g\big{(}F(\bar{x})\big{)}.

(24)

Furthermore, by (21) and (MS20, , Proposition 2.2), we get

\begin{array}[]{rl}T_{\mbox{\rm dom}\,\psi}(\bar{x})&=\mbox{\rm dom}\,d\psi(\bar{x})\\ &=\big{\{}w\in\mathbb{R}^{n}\ |\ \nabla F(\bar{x})w\in\mbox{\rm dom}\,dg\big{(}F(\bar{x})\big{)}\big{\}}\\ &=\big{\{}w\in\mathbb{R}^{n}\ |\ \nabla F(\bar{x})w\in T_{\mbox{\rm dom}\,g}\big{(}F(\bar{x})\big{)}\big{\}}.\end{array}

Let us now suppose further that $w\in T_{\mbox{\rm dom}\,\psi}(\bar{x})$ and $g$ is parabolically epi-differentiable at $F(\bar{x})$ for $\nabla F(\bar{x})w.$ Since $g$ is Lipschitz continuous around $F(\bar{x})$ relative to its domain, and $\nabla F(\bar{x})w\in T_{\mbox{\rm dom}\,g}\big{(}F(\bar{x})\big{)},$ by (MS20, , Proposition 4.1), $\mbox{\rm dom}\,g$ is parabolically derivable at $F(\bar{x})$ for $\nabla F(\bar{x})w$ . Hence, the proofs of $(i)$ and $(ii)-(iv)$ can be, respectively, done as the ones of (MMS21, , Theorem 4.5) and (MS20, , Theorem 4.4), where $F$ was assumed to be twice differentiable at $\bar{x},$ but they actually needed the quadratic expansion of (20) and the strict differentiability of $F$ at $\bar{x},$ which are valid under the twice differentiability in the extended sense. $\hfill\Box$

In order to prove the next proposition we need the following lemma whose proof is the one of (MS20, , Proposition 4.6). For the sake of completeness we provide the proof with more details.

Lemma 3

Suppose $f:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is a proper lower semicontinuous convex function with $f(\bar{x})\in\mathbb{R}$ , $\bar{v}\in\partial f(\bar{x}),$ $w\in K_{f}(\bar{x},\bar{v}),$ and $f$ is parabolically epi-differentiable at $\bar{x}$ for $w.$ Then $d^{2}f(\bar{x})(w|\cdot)$ is proper lower semicontinuous and convex. Furthermore, $d^{2}f(\bar{x})(w|\cdot)^{*}(v)=\infty$ whenever $v\in\mathbb{R}^{n}\backslash\mathcal{A}(\bar{x},w),$ and $d^{2}f(\bar{x})(w|\cdot)^{*}(v)=-d^{2}f(\bar{x},v)(w)$ if $v\in\mathcal{A}(\bar{x},w)$ and $f$ is parabolically regular at $\bar{x}$ for $v,$ where $\mathcal{A}(\bar{x},w):=\{v\in\partial f(\bar{x})\ |\ df(\bar{x})(w)=\langle v,w\rangle\}$ and $d^{2}f(\bar{x})(w|\cdot)^{*}$ is the Fenchel conjugate of $d^{2}f(\bar{x})(w|\cdot).$

Proof. Since $f$ is a lower semicontinuous function, $f(\bar{x})\in\mathbb{R}$ and $df(\bar{x})(w)=\langle\bar{v},w\rangle\in\mathbb{R},$ by (RW98, , Proposition 13.64), $d^{2}f(\bar{x})(w|\cdot)$ is lower semicontinuous and

d^{2}f(\bar{x})(w|z)-\langle\bar{v},z\rangle\geq d^{2}f(\bar{x},\bar{v})(w)\quad\forall z\in\mathbb{R}^{n}.

(25)

Noting that $f$ is convex and $\bar{v}\in\partial f(\bar{x})$ , we have

d^{2}f(\bar{x},\bar{v})(w)=\liminf\limits_{w^{\prime}\to w}\frac{f(\bar{x}+tw^{\prime})-f(\bar{x})-t\langle\bar{v},w^{\prime}\rangle}{\frac{1}{2}t^{2}}\geq 0.

(26)

Thus $d^{2}f(\bar{x})(w|z)>-\infty$ for all $z\in\mathbb{R}^{n}.$ Combining this with $\mbox{\rm dom}\,d^{2}f(\bar{x})(w|\cdot)\not=\emptyset$ (due to the parabolic epi-differentiability of $f$ at $\bar{x}$ for $w$ ), we see that $d^{2}f(\bar{x})(w|\cdot)$ is a proper function. By (RW98, , Example 13.62),

\mbox{\rm epi}\,d^{2}f(\bar{x})(w|\cdot)=T^{2}_{\mbox{\rm epi}\,f}\Big{(}\big{(}\bar{x},f(\bar{x})\big{)},\big{(}w,df(\bar{x})(w)\big{)}\Big{)},

and since $f$ is parabolically epi-differentiable at $\bar{x}$ for $w,$ ${\mbox{\rm epi}\,f}$ is parabolically derivable at $\big{(}\bar{x},f(\bar{x})\big{)}$ for $\big{(}w,df(\bar{x})(w)\big{)}.$ This implies that $d^{2}f(\bar{x})(w|\cdot)$ is convex since $f$ is convex.

Take any $v\in\mathbb{R}^{n}.$ Let us consider the following two cases.

Case 1. $v\in\mathcal{A}(\bar{x},w).$ Then $w\in K_{f}(\bar{x},v)$ and by (MS20, , Proposition 3.6), we have

-d^{2}f(\bar{x},v)(w)=-\inf\limits_{z\in\mathbb{R}^{n}}\{d^{2}f(\bar{x})(w|z)-\langle v,z\rangle\}=d^{2}f(\bar{x})(w|\cdot)^{*}(v),

due to the parabolic regularity of $f$ at $\bar{x}$ for $v.$

Case 2. $v\in\mathbb{R}^{n}\backslash\mathcal{A}(\bar{x},w).$ Then either $v\not\in\partial f(\bar{x})$ or $df(\bar{x})(w)\not=\langle v,w\rangle.$ Put

\upsilon_{t}(z):=\frac{f(\bar{x}+tw+\frac{1}{2}t^{2}z)-f(\bar{x})-tdf(\bar{x})(w)}{\frac{1}{2}t^{2}}\quad\forall z\in\mathbb{R}^{n},t>0.

We have

\upsilon_{t}^{*}(v)=\frac{f(\bar{x})+f^{*}(v)-\langle v,\bar{x}\rangle}{\frac{1}{2}t^{2}}+\frac{df(\bar{x})(w)-\langle v,w\rangle}{\frac{1}{2}t}\quad\forall v\in\mathbb{R}^{n},t>0.

Since $f$ is parabolically epi-differentiable at $\bar{x}$ for $w,$ by (RW98, , Example 13.59), $\mbox{\rm epi}\,\upsilon_{t}$ converges to $\mbox{\rm epi}\,d^{2}f(\bar{x})(w|\cdot)$ as $t\downarrow 0.$ Noting that $\upsilon_{t}(\cdot)$ and $d^{2}f(\bar{x})(w|\cdot)$ are proper lower semicontinuous and convex functions, by (RW98, , Theorem 11.34), the latter implies that $\mbox{\rm epi}\,\upsilon_{t}^{*}$ converges to $\mbox{\rm epi}\,d^{2}f(\bar{x})(w|\cdot)^{*}$ as $t\downarrow 0.$ So, for any sequence $t_{k}\downarrow 0,$ by (RW98, , Proposition 7.2), there exists a sequence $v_{k}\to v$ such that

d^{2}f(\bar{x})(w|\cdot)^{*}(v)=\lim\limits_{k\to\infty}\upsilon_{t_{k}}^{*}(v_{k}).

If $v\not\in\partial f(\bar{x})$ then $f(\bar{x})+f^{*}(v)-\langle v,\bar{x}\rangle>0.$ Thus, by lower semicontinuity of $f^{*}$ , we see that

\begin{array}[]{rl}d^{2}f(\bar{x})(w|\cdot)^{*}(v)&=\lim\limits_{k\to\infty}\upsilon_{t_{k}}^{*}(v_{k})\\ &=\lim\limits_{k\to\infty}\frac{2}{t_{k}}\left(\frac{f(\bar{x})+f^{*}(v_{k})-\langle v_{k},\bar{x}\rangle}{t_{k}}+df(\bar{x})(w)-\langle v_{k},w\rangle\right)\\ &=\infty.\end{array}

If $df(\bar{x})(w)\not=\langle v,w\rangle$ then, by (26) and (RW98, , Proposition 13.5), $\langle v,w\rangle<df(\bar{x})(w).$ On the other hand, we have

f(\bar{x})+f^{*}(v_{k})-\langle v_{k},\bar{x}\rangle=f(\bar{x})+\sup\limits_{x\in\mathbb{R}^{n}}[\langle v_{k},x\rangle-f(x)]-\langle v_{k},\bar{x}\rangle\geq 0\quad\forall k.

Therefore,

\begin{array}[]{rl}d^{2}f(\bar{x})(w|\cdot)^{*}(v)&=\lim\limits_{k\to\infty}\upsilon_{t_{k}}^{*}(v_{k})\\ &=\lim\limits_{k\to\infty}\left(\frac{f(\bar{x})+f^{*}(v_{k})-\langle v_{k},\bar{x}\rangle}{\frac{1}{2}t_{k}^{2}}+\frac{df(\bar{x})(w)-\langle v_{k},w\rangle}{\frac{1}{2}t_{k}}\right)\\ &\geq\lim\limits_{k\to\infty}\frac{df(\bar{x})(w)-\langle v_{k},w\rangle}{\frac{1}{2}t_{k}}=\infty.\end{array}

So, we arrive at the desired conclusion. $\hfill\Box$

Following Mohammadi and Sarabi MS20 , we say that function $\psi(x):=g\circ F$ with $(\bar{x},\bar{v})\in\mbox{\rm gph}\,\partial\psi$ satisfies the basic assumptions at $(\bar{x},\bar{v})$ if the following conditions hold:

(H1)

the metric subregularity qualification condition (19) is valid at $\bar{x}$ ;
(H2)

for each $y\in\Lambda(\bar{x},\bar{v}),$ $g$ is parabolically epi-differentiable at $F(\bar{x})$ for every $u\in K_{g}\big{(}F(\bar{x}),y\big{)};$
(H3)

$g$ is parabolically regular at $F(\bar{x})$ for every $y\in\Lambda(\bar{x},\bar{v}).$

Here

\Lambda(\bar{x},\bar{v}):=\big{\{}y\in\partial g\big{(}F(\bar{x})\big{)}\ |\ \nabla F(\bar{x})^{*}y=\bar{v}\big{\}},

and

K_{g}\big{(}F(\bar{x}),y\big{)}:=\big{\{}w\in\mathbb{R}^{m}\ |\ dg\big{(}F(\bar{x})\big{)}(w)=\langle\bar{v},w\rangle\big{\}}

are the set of Lagrangian multipliers associated with $(\bar{x},\bar{v}),$ and the critical cone of $g$ at $(F(\bar{x}),y\big{)},$ respectively.

Let us consider the following optimization problem:

\min\limits_{x\in\mathbb{R}^{n}}-\langle z,\bar{v}\rangle+d^{2}g\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w|\nabla F(\bar{x})z+\nabla^{2}F(\bar{x})(w,w)\big{)}.

(27)

Proposition 2

Let $\psi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ be a function that is represented as (17) with the composition $\psi=g\circ F$ satisfying the basic assumptions $(H1)$ - $(H3)$ at $(\bar{x},\bar{v}).$ Then the following assertions hold:

$(i)$ For each $w\in K_{\psi}(\bar{x},\bar{v}),$ the dual problem of (27) is

\max\limits_{y\in\Lambda(\bar{x},\bar{v})}\left\langle y,\nabla^{2}F(\bar{x})(w,w)\right\rangle+d^{2}g\big{(}F(\bar{x}),y\big{)}\big{(}\nabla F(\bar{x})\big{)};

(28)

the optimal values of the primal and dual optimization problems (27) and (28) are equal and finite. Furthermore, $\Lambda(\bar{x},\bar{v},w)\cap\tau\mathbb{B}\not=\emptyset,$ where $\Lambda(\bar{x},\bar{v},w)$ is the optimal solution set of (28) and

\tau:=\kappa\ell\|\nabla F(\bar{x})\|+\kappa\|\bar{v}\|+\ell

(29)

with $\ell$ and $\kappa$ given in (17) and (19), respectively.

$(ii)$ $\psi$ is parabolically regular at $\bar{x}$ for $\bar{v},$ and

\begin{array}[]{rl}d^{2}\psi(\bar{x},\bar{v})(w)&=\max\limits_{y\in\Lambda(\bar{x},\bar{v})}\left\{\big{\langle}y,\nabla^{2}F(\bar{x})(w,w)\big{\rangle}+d^{2}g\big{(}F(\bar{x}),y\big{)}\big{(}\nabla F(\bar{x})w\big{)}\right\}\\ &=\max\limits_{y\in\Lambda(\bar{x},\bar{v})\cap(\tau\mathbb{B})}\left\{\big{\langle}y,\nabla^{2}F(\bar{x})(w,w)\big{\rangle}+d^{2}g\big{(}F(\bar{x}),y\big{)}\big{(}\nabla F(\bar{x})w\big{)}\right\},\end{array}

(30)

for every $w\in\mathbb{R}^{n},$ where $\tau$ is given by (29).

$(iii)$ $\psi$ is twice epi-differentiable at $\bar{x}$ for $\bar{v}.$

Proof. Take any $w\in K_{\psi}(\bar{x},\bar{v})$ and $y\in\Lambda(\bar{x},\bar{v}).$ Then $d\psi(\bar{x})(w)=\langle\bar{v},w\rangle,$ $y\in\partial g\big{(}F(\bar{x})\big{)}$ and $\nabla F(\bar{x})^{*}y=\bar{v}.$ So, by Proposition 1,

dg\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w\big{)}=\langle\bar{v},w\rangle=\langle\nabla F(\bar{x})^{*}y,w\rangle=\langle y,\nabla F(\bar{x})w\rangle.

This means $\nabla F(\bar{x})w\in K_{g}\big{(}F(\bar{x}),y\big{)}.$ By $(H2),$ $g$ is parabolically epi-differentiable at $F(\bar{x})$ for $\nabla F(\bar{x})w.$ Thus, by Lemma 3, the function $d^{2}g\big{(}F(\bar{x})\big{)}(\nabla F(\bar{x})w|\cdot)$ is a proper lower semicontinuous convex function. Hence, from (RW98, , Example 11.41) it follows that the Fenchel dual problem of (27) is

\max\limits_{\nabla F(\bar{x})^{*}y=\bar{v}}\left\langle y,\nabla^{2}F(\bar{x})(w,w)\right\rangle-d^{2}g\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w|\cdot)^{*}(y).

(31)

Pick any $y\in\mathbb{R}^{m}$ with $\nabla F(\bar{x})^{*}y=\bar{v}.$ If $y\not\in\partial g\big{(}F(\bar{x})\big{)}$ then, by Lemma 3,

d^{2}g\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w|\cdot)^{*}(y)=\infty.

(32)

Otherwise, we get $y\in\Lambda(\bar{x},\bar{v}).$ Then, by $(H3),$ $g$ is parabolically regular at $F(\bar{x})$ for $y.$ Note that $y\in\partial g\big{(}F(\bar{x})\big{)}$ and $dg\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w\big{)}=\langle y,\nabla F(\bar{x})w\rangle.$ So, by Lemma 3, we see that

d^{2}g\big{(}F(\bar{x})\big{)}\big{(}\nabla F(\bar{x})w|\cdot)^{*}(y)=-d^{2}g\big{(}F(\bar{x}),y)(w).

(33)

From (32) and (33) it follows that problem (31) can be written as problem (28). The rest of the proof $(i)$ runs as the one of (MS20, , Theorem 5.2), and the proof of $(ii)$ is similar to the proof of (MS20, , Theorem 5.4). So, they are omitted. Finally, we see that, by $(ii),$ $\psi$ is parabolically regular at $\bar{x}$ for $\bar{v}\in\partial\psi(\bar{x})=\partial_{p}\psi(\bar{x}),$ and, by (MS20, , Theorem 4.4), $\psi$ is parabolically epi-differentiable at $\bar{x}$ for every $w\in K_{\psi}(\bar{x},\bar{v}).$ Therefore, by (MS20, , Theorem 3.8), $\psi$ is twice epi-differentiable at $\bar{x}$ for $\bar{v}.$ $\hfill\Box$

Remark 1

Under the assumption of Proposition 2, $g$ is parabolically regular at $F(\bar{x})$ only for all $y\in\partial g\big{(}F(\bar{x})\big{)}$ with $\nabla F(\bar{x})^{*}y=\bar{v}.$ Thus, we cannot apply (MS20, , Proposition 4.6) to transforming (31) into (28), since (MS20, , Proposition 4.6) requires the parabolic regularity of $g$ at $F(\bar{x})$ for every $y\in\partial g\big{(}F(\bar{x})\big{)}.$ That is why Lemma 3 is utilized. We note that the results in Propositions 1&2 were established in MS20 for the case where $F$ is twice differentiable in the classical sense.

4 Quadratic growth and strong metric subregularity of the subdifferential

Let $f\colon\mathbb{R}^{n}\to\overline{R}$ and $\bar{x}\in\mbox{\rm dom}\,f.$ We say that $\bar{x}$ is a strong local minimizer of $f$ with modulus $\kappa>0$ if there is a number $\gamma>0$ such that the following quadratic growth condition (QGC) is satisfied:

f(x)-f(\bar{x})\geq\frac{\kappa}{2}\|x-\bar{x}\|^{2}\quad\mbox{for all}\quad x\in\mathbb{B}_{\gamma}(\bar{x}).

(34)

The exact modulus for QGC of $f$ at $\bar{x}$ is given by

{\rm QG}\,(f;\bar{x}):=\sup\big{\{}\kappa>0\;|\;\bar{x}\mbox{ is a strong local minimizer of $f$ with modulus $\kappa$}\big{\}}.

Lemma 4

((DMN14, , Corollary 3.3)). Let $f\colon\mathbb{R}^{n}\to\bar{\mathbb{R}}$ be a proper lower semicontinuous function and let $\bar{x}\in\mbox{\rm dom}\,f$ with $0\in\partial f(\bar{x})$ . Suppose that the subgradient mapping $\partial f$ is strongly metrically subregular at $\bar{x}$ for $0$ with modulus $\kappa>0$ and there are real numbers $r\in(0,\kappa^{-1})$ and $\delta>0$ such that

\displaystyle f(x)\geq f(\bar{x})-\frac{r}{2}\|x-\bar{x}\|^{2}\quad\mbox{for all}\quad x\in\mathbb{B}_{\delta}(\bar{x}).

(35)

Then for any $\alpha\in(0,\kappa^{-1})$ , there exists a real number $\eta>0$ such that

\displaystyle f(x)\geq f(\bar{x})+\frac{\alpha}{2}\|x-\bar{x}\|^{2}\quad\mbox{for all}\quad x\in\mathbb{B}_{\eta}(\bar{x}).

(36)

Lemma 5

((CHNT21, , Lemma 3.6)). Let $h:\mathbb{R}^{n}\to\bar{\mathbb{R}}$ be a proper function. Suppose that $h$ is positively homogenenous of degree $2$ in the sense that $h(\lambda w)=\lambda^{2}h(w)$ for all $\lambda>0$ and $w\in\mbox{\rm dom}\,h$ . Then for any $w\in\mbox{\rm dom}\,h$ and $z\in\partial h(w)$ , we have $\langle z,w\rangle=2h(w)$ .

The following result provides some characterizations of the quadratic growth and the strong metric subregularity of the subdifferential.

Theorem 4.1

Let $f:\mathbb{R}^{n}\to\bar{\mathbb{R}}$ be the function defined by $f(x)=\varphi(x)+\psi(x)$ for every $x\in\mathbb{R}^{n},$ where $\varphi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is twice differentiable at $\bar{x}$ in the extended sense, $0\in\nabla\varphi(\bar{x})+\partial\psi(\bar{x}),$ and $\psi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is subdifferentially continuous, prox-regular, and twice epi-differentiable at $\bar{x}$ for $-\nabla\varphi(\bar{x})$ . Then the following assertions are equivalent:

$(i)$ The quadratic growth condition (34) is satisfied.

$(ii)$ The subgradient mapping $\partial f$ is strongly metrically subregular at $(\bar{x},0),$ and

\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+d^{2}\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)\geq 0\ \mbox{for all}\ w\in\mathbb{R}^{n}.

(37)

$(iii)$ The subgradient mapping $\partial f$ is strongly metrically subregular at $(\bar{x},0),$ and $\bar{x}$ is a local minimizer for $f.$

$(iv$ For all $w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}\backslash\{0\}$ and $z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w),$ we have

\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle>0.

(38)

$(v)$ There exists a real number $c>0$ such that

\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle\geq c\|w\|^{2},

(39)

for all $w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}$ and $z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w).$

$(vi)$ For every $w\in\mathbb{R}^{n}\backslash\{0\},$ we have

\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+d^{2}\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)>0.

(40)

If one of the above assertions holds then

{\rm QG}(f;\bar{x})=\inf\left\{\frac{\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle}{\|w\|^{2}}\,\Big{|}\begin{array}[]{rl}&w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)},\\ &z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)\end{array}\right\},

(41)

with the convention that $0/0=\infty$ .

Proof. Under our assumption, by Theorem 3.1, we have

D\partial(\varphi+\psi)(\bar{x}|0)(w)=\nabla^{2}\varphi(\bar{x})(w)+D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w),

(42)

and

d^{2}(\varphi+\psi)\big{(}\bar{x}|0\big{)}(w)=\big{\langle}w,\nabla^{2}\varphi(\bar{x})w\big{\rangle}+d^{2}\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w),

(43)

for every $w\in\mathbb{R}^{n}$ . By (43) and (RW98, , Theorem 13.24), we see that $(i)\Leftrightarrow(vi)$ and $(iii)\Rightarrow(ii).$

We next prove that $0\in\partial_{p}f(\bar{x}).$ Since $\psi$ is subdifferentially continuous and prox-regular at $\bar{x}$ for $-\nabla\varphi(\bar{x})$ , we get

\liminf\limits_{x\to\bar{x}}\frac{\psi(x)-\psi(\bar{x})+\langle\nabla\varphi(\bar{x}),x-\bar{x}\rangle}{\|x-\bar{x}\|^{2}}>-\infty.

On the other hand, from the extended twice differentiability of $\varphi$ at $\bar{x}$ , by (11), it follows that

\varphi(x)=\varphi(\bar{x})+\langle\nabla\varphi(\bar{x}),x-\bar{x}\rangle+\frac{1}{2}\langle x-\bar{x},\nabla^{2}\varphi(\bar{x})(x-\bar{x})\rangle+o(\|x-\bar{x}\|^{2}),

which gives us the following estimations

\begin{array}[]{rl}\liminf\limits_{x\to\bar{x}}\frac{\varphi(x)-\varphi(\bar{x})-\langle\nabla\varphi(\bar{x}),x-\bar{x}\rangle}{\|x-\bar{x}\|^{2}}&=\liminf\limits_{x\to\bar{x}}\frac{\frac{1}{2}\langle x-\bar{x},\nabla^{2}\varphi(\bar{x})(x-\bar{x})\rangle+o(\|x-\bar{x}\|^{2})}{\|x-\bar{x}\|^{2}}\\ &\geq-\frac{1}{2}\|\nabla^{2}\varphi(\bar{x})\|>-\infty.\end{array}

Therefore,

\begin{array}[]{rl}\liminf\limits_{x\to\bar{x}}\frac{f(x)-f(\bar{x})}{\|x-\bar{x}\|^{2}}&=\liminf\limits_{x\to\bar{x}}\left[\frac{\varphi(x)-\varphi(\bar{x})-\langle\nabla\varphi(\bar{x}),x-\bar{x}\rangle}{\|x-\bar{x}\|^{2}}+\frac{\psi(x)-\psi(\bar{x})+\langle\nabla\varphi(\bar{x}),x-\bar{x}\rangle}{\|x-\bar{x}\|^{2}}\right]\\ &=\liminf\limits_{x\to\bar{x}}\frac{\varphi(x)-\varphi(\bar{x})-\langle\nabla\varphi(\bar{x}),x-\bar{x}\rangle}{\|x-\bar{x}\|^{2}}+\liminf\limits_{x\to\bar{x}}\frac{\psi(x)-\psi(\bar{x})+\langle\nabla\varphi(\bar{x}),x-\bar{x}\rangle}{\|x-\bar{x}\|^{2}}\\ &>-\infty.\end{array}

This shows that $0\in\partial_{p}f(\bar{x}).$

Hence, by (42) and (CHNT21, , Theorem 3.2), implication $(v)\Rightarrow(iv)\Rightarrow(iii)$ holds, and

{\rm QG}(f;\bar{x})\geq\inf\left\{\frac{\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle}{\|w\|^{2}}\,\Big{|}\begin{array}[]{rl}&w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)},\\ &z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)\end{array}\right\}.

(44)

We now prove $(ii)\Rightarrow(i).$ Suppose $\partial f$ is strongly metrically subregular at $\bar{x}$ for $0$ with modulus $\kappa>0,$ and (37) holds. By (43), we get

d^{2}f\big{(}\bar{x}|0\big{)}(w)\geq 0\,\ \mbox{for all}\ w\in\mathbb{R}^{n}.

(45)

Fix an arbitrary $r\in(0,\kappa^{-1}).$ Then there exists a real number $\delta>0$ such that

\displaystyle f(x)\geq f(\bar{x})-\frac{r}{2}\|x-\bar{x}\|^{2}\quad\mbox{for all}\quad x\in\mathbb{B}_{\delta}(\bar{x}).

(46)

Indeed, suppose by contrary that this claim does not hold. Then, for each $k\in\mathbb{N},$ there exists $x_{k}\in\mathbb{B}_{1/k}(\bar{x})$ with

f(x_{k})<f(\bar{x})-\frac{r}{2}\|x_{k}-\bar{x}\|^{2}.

Put $t_{k}:=\|x_{k}-\bar{x}\|$ and $w_{k}:=t^{-1}_{k}(x_{k}-\bar{x})$ for $k\in\mathbb{N}.$ We see that $\_k\downarrow 0$ as $k\to\infty.$ Furthermore, passing to a subsequence if necessary, we may assume that $\{w_{k}\}$ converges to some $\bar{w}\in\mathbb{R}^{n}$ as $k\to\infty.$ So we have

\begin{array}[]{rl}d^{2}f\big{(}\bar{x}|0\big{)}(\bar{w})&=\liminf\limits_{\begin{subarray}{\quad}\,\ t\downarrow 0\\ w\longrightarrow\bar{w}\end{subarray}}\frac{f(\bar{x}+tw)-f(\bar{x})-\tau\langle 0,w\rangle}{\frac{1}{2}t^{2}}\\ &\leq\liminf\limits_{k\to\infty}\frac{f(\bar{x}+t_{k}w_{k})-f(\bar{x})}{\frac{1}{2}t_{k}^{2}}\\ &=\liminf\limits_{k\to\infty}\frac{f(x_{k})-f(\bar{x})}{\frac{1}{2}\|x_{k}-\bar{x}\|^{2}}\leq-\frac{r}{2}<0.\end{array}

This contradicts (45). Therefore, there exists a real number $\delta>0$ such that (46) holds. By Lemma 4, the quadratic growth condition (34) holds, and we have $(ii)\Rightarrow(i).$

Finally, we prove $(i)\Rightarrow(v)$ and

{\rm QG}(f;\bar{x})\leq\inf\left\{\frac{\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle}{\|w\|^{2}}\,\Big{|}\begin{array}[]{rl}&w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)},\\ &z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)\end{array}\right\}.

(47)

Suppose that $\bar{x}$ is a strong local minimizer with modulus $\kappa$ as in (34). We derive from (34) and (3) that

d^{2}f(\bar{x}|0)(w)\geq\kappa\|w\|^{2}\quad\mbox{for all}\ w\in\mathbb{R}^{n}.

(48)

Since $\psi$ is subdifferentially continuous, prox-regular, and twice epi-differentiable at $\bar{x}$ for $-\nabla\varphi(\bar{x})\in\partial\psi(\bar{x}),$ it follows from (4) that

D(\partial\psi)\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}=\partial h\quad\mbox{with}\quad h(\cdot):=\dfrac{1}{2}d^{2}\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(\cdot)

(49)

Note from (3) and (48) that $h$ is proper and positively homogenenous of degree $2$ . By Lemma 5, for any $z\in D(\partial\psi)\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)=\partial h(w),$ we obtain from (48) and (49) that

\langle z,w\rangle=2h(w)=d^{2}\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w).

(50)

Therefore, for every $w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}$ and $z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w),$ by (42), (43), (48), and (50), we get

\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle=\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+d^{2}\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)=d^{2}f(\bar{x}|0)(w)\geq\kappa\|w\|^{2},

which clearly verifies $(v)$ and

\kappa\leq\inf\left\{\frac{\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle}{\|w\|^{2}}\,\Big{|}\begin{array}[]{rl}&w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)},\\ &z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)\end{array}\right\}.

Since $\kappa$ is an arbitrary modulus of the strong local minimizer $\bar{x},$ the latter implies that (47) holds. So by (44) and (47) we get (41). $\hfill\Box$

Remark 2

By choosing $\varphi:=0$ , we can get (CHNT21, , Theorem 3.7) from Theorem 4.1. In the case where $\varphi$ is twice continuously differentiable and $\psi$ is twice epi-differentiable and convex, other characterizations of the quadratic growth as well as the strong metric subregularity of the subdifferential can be found in (OM21, , Theorem 7.8).

We next consider the composite optimization problem

\min\limits_{x\in\mathbb{R}^{n}}f(x):=\varphi(x)+g\big{(}F(x)\big{)},

(51)

where $\varphi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is twice differentiable at $\bar{x}$ in the extended sense, $F:\mathbb{R}^{n}\to\mathbb{R}^{m}$ is twice differentiable, and $g:\mathbb{R}^{m}\to\overline{\mathbb{R}}:=(-\infty,+\infty]$ is a proper lower semicontinuous convex function Lipschitz continuous around $F(\bar{x})$ relative to its domain with constaint $\ell\in\mathbb{R}_{+}.$

The Lagrangian associated with (51) is defined by

L(x,y)=\varphi(x)+\langle F(x),y\rangle-g^{*}(y),

where $g^{*}(y):=\sup\limits_{v\in\mathbb{R}^{m}}[\langle y,v\rangle-g(v)]$ is the Fenchel conjugate of $g$ (see MS20 ).

Corollary 1

Let $0\in\nabla\varphi(\bar{x})+\partial\psi(\bar{x}),$ where $\varphi:\mathbb{R}^{n}\to\overline{\mathbb{R}}$ is twice differentiable at $\bar{x}$ in the extended sense, and $\psi:=g\circ F$ with $F:\mathbb{R}^{n}\to\mathbb{R}^{m}$ being twice differentiable at $\bar{x}$ in the extended sense and $g:\mathbb{R}^{m}\to\overline{\mathbb{R}}$ being a proper lower semicontinuous convex function Lipschitz continuous around $F(\bar{x})$ relative to its domain. Assume that the basic assumptions $(H1)$ - $(H3)$ hold for $\psi$ at $(\bar{x},\bar{v})$ with $\bar{v}:=-\nabla\varphi(\bar{x}),$ and $\psi$ is prox-regular at $\bar{x}$ for $\bar{v}.$ Then, the following assertions are equivalent:

$(i)$ The quadratic growth condition (34) is satisfied.

$(ii)$ $\partial f$ is strongly metrically subregular at $(\bar{x},0),$ and

\max\limits_{y\in\Lambda(\bar{x},\bar{v})}\left\{\big{\langle}\nabla^{2}_{xx}L(\bar{x},\bar{y})w,w\big{\rangle}+d^{2}g\big{(}F(\bar{x}),y\big{)}\big{(}\nabla F(\bar{x})w\big{)}\right\}\geq 0

for all $w\in K_{\psi}(\bar{x},\bar{v});$

$(iii)$ $\partial f$ is strongly metrically subregular at $(\bar{x},0),$ and $\bar{x}$ is a local minimizer of $f.$

$(iv$ For all $w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}\backslash\{0\}$ and $z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w),$ we have

\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle>0.

$(v)$ There exists a real number $c>0$ such that

\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle\geq c\|w\|^{2},

(52)

for all $w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}$ and $z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w).$

$(vi)$ For every $w\in K_{\psi}(\bar{x},\bar{v})\backslash\{0\},$ we have

\max\limits_{y\in\Lambda(\bar{x},\bar{v})}\left\{\big{\langle}\nabla^{2}_{xx}L(\bar{x},y)w,w\big{\rangle}+d^{2}g\big{(}F(\bar{x}),y\big{)}\big{(}\nabla F(\bar{x})w\big{)}\right\}>0.

If one of the above assertions holds then

{\rm QG}(f;\bar{x})=\inf\left\{\frac{\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+\langle z,w\rangle}{\|w\|^{2}}\,\Big{|}\begin{array}[]{rl}&w\in{\rm dom}D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)},\\ &z\in D\partial\psi\big{(}\bar{x}|-\nabla\varphi(\bar{x})\big{)}(w)\end{array}\right\},

with the convention that $0/0=\infty$ .

Proof. Under the given assumption, $\psi$ is prox-regular and subdifferentially continuous at $\bar{x}$ for $\bar{v},$ and by Proposition 2, $\psi$ is twice epi-differentiable at $\bar{x}$ for $\bar{v}.$ Furthermore, since $g$ is Lipschitz continuous relative to its domain and $F$ is Lipschitz continuous around $\bar{x},$ the composition $\psi=g\circ F$ is subdifferentially continuous at $\bar{x}$ for $\bar{v}.$ On the other hand, by Proposition 2, we have

d^{2}\psi(\bar{x},\bar{v})(w)=\max\limits_{y\in\Lambda(\bar{x},\bar{v})}\left\{\big{\langle}\nabla^{2}F(\bar{x})(w,w)\big{\rangle}+d^{2}g\big{(}F(\bar{x}),y\big{)}\big{(}\nabla F(\bar{x})w\big{)}\right\}\quad\forall w\in\mathbb{R}^{n},

which gives us that

\langle\nabla^{2}\varphi(\bar{x})w,w\rangle+d^{2}\psi\big{(}\bar{x}|\bar{v}\big{)}(w)=\max\limits_{y\in\Lambda(\bar{x},\bar{v})}\left\{\big{\langle}\nabla^{2}_{xx}L(\bar{x},y)w,w\big{\rangle}+d^{2}g\big{(}F(\bar{x}),y\big{)}\big{(}\nabla F(\bar{x})w\big{)}\right\},

for every $w\in\mathbb{R}^{n}.$ Therefore, noting that $d^{2}\psi(\bar{x},\bar{v})$ is a proper lower semicontinuous function with $\mbox{\rm dom}\,d^{2}\psi(\bar{x},\bar{v})=K_{\psi}(\bar{x},\bar{v}),$ we get the desired conclusion by applying Theorem 4.1 to the function $f:=\varphi+\psi$ with $\psi:=g\circ F.$ $\hfill\Box$

Remark 3

Under $(H1)$ - $(H3)$ , Mohammadi and Sarabi (MS20, , Theorem 6.3) showed that $(iii)\Leftrightarrow(vi)$ when $\varphi$ and $F$ are twice continuously differentiable around $\bar{x}$ . Since the latter implies the prox-regularity of $\psi,$ Corollary 1 is an extension of (MS20, , Theorem 6.3).

Example 3

Consider the following optimization problem:

\min\limits_{x\in\mathbb{R}}\varphi(x)+\psi(x),

(53)

where $\varphi(x)=2x+g(x)$ with $g(x)$ being taken from Example 1, and $\psi(x):=\delta_{\mathbb{R}^{2}_{-}}\circ F(x)$ with $F(x)=\big{(}F_{1}(x),F_{2}(x)\big{)}$ , $F_{1}(x)=-x$ and $F_{2}(x)=-x^{3}.$ By Example 1, $\varphi$ is twice differentiable at $\bar{x}=0$ in the extended sense and not prox-regular at $\bar{x}=0$ for $\bar{v}=0.$ Put

\Gamma=\{x\in\mathbb{R}\,|\,F_{i}(x)\leq 0,\ i=1,2\}=\mathbb{R}_{+}\quad\mbox{and}\quad g(y):=\delta_{\mathbb{R}^{2}_{-}}(y).

Then $g$ satisfies $(H2)$ and $(H3).$ Furthermore, we see that

d(x,\mbox{\rm dom}\,\psi)=d(x,\Gamma)=\begin{cases}0\ &\mbox{if}\ x\geq 0,\\ \left|x\right|\ &\mbox{if}\ x<0,\end{cases}

and

d\big{(}F(x),\mbox{\rm dom}\,g\big{)}=d\big{(}F(x),\mathbb{R}^{2}_{-}\big{)}=\begin{cases}0\ &\mbox{if}\ x\geq 0,\\ \sqrt{x^{2}+x^{6}}\ &\mbox{if}\ x<0,\end{cases}

which infers that $d(x,\mbox{\rm dom}\,\psi)\leq d\big{(}F(x),\mbox{\rm dom}\,g\big{)}.$ This shows that $(H1)$ holds at $\bar{x}.$
We next prove that $\bar{x}$ is a strong local minimizer. Indeed, for all $x\in\Gamma\cap[-1,1]$ and $n\in\mathbb{N}^{*},$ we have

x+x^{10/3}\cos\frac{1}{x}\geq 0\ \mbox{and}\ \frac{(2n+1)(2n^{2}+2n+1)}{n^{3}(n+1)^{3}}x+\frac{1}{(n+1)^{3}}-\frac{1}{n^{3}}\geq x^{4}\geq 0.

Therefore, we get

\varphi(x)-\varphi(\bar{x})\geq x\geq x^{2}\ \mbox{for all}\ x\in\Gamma\cap[-1,1].

Thus, $\bar{x}$ is a strong local minimizer. By Corollary 1, the assertions $(ii)$ - $(vi)$ hold.

5 Conclusion

We have proved some characterizations of the quadratic growth and the strong metric subregularity of the subdifferential of a function that can be represented as the sum of a function twice differentiable in the extended sense and a subdifferentially continuous, prox-regular, twice epi-differentiable function. Especially, for such a function, we have shown that the quadratic growth, the strong metric subregularity of the subdifferential at a local minimizer, and the positive definiteness of the subgradient graphical derivative at a stationary point are equivalent. Our results are new even for the case where the twice differentiability in the extended sense is replaced by the twice differentiability in the classical sense. In this research direction, it seems to us that finding out to which extent the established results can be applied to the analysis of convergence of numerical algorithms is a very interesting issue BLN18 ; HMS21 ; OM21 , which requires further investigation. Moreover, in order to widen the range of applications of the obtained results, more researches on the class of functions that are twice differentiable in the extended sense are needed.

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