Geometry of a prox-regular set

Shuvomoy Das Gupta

July 30, 2021

In this blog, we provide a proof about certain geometric properties of a prox-regular set based on Proposition 3.1 of the paper "Local Differentiability of Distance Functions" by R. A. Poliquin, R. T. Rockafellar and L. Thibault (link to pdf here).



We consider nonempty, closed set X\mathcal{X} in a finite-dimensional vector space, which is prox-regular at a point xˉX\bar{x}\in\mathcal{X}, i.e., Euclidean projection onto the set from xˉ\bar{x} is single-valued in some neighborhood of that point. Throughout the blog, the underlying space is a finite-dimensional vector space over the reals. To state the result and to prove it, we introduce certain notation and notions.

Notation and notions

Notation.

An open ball with center cc and radius rr is denoted by

B(c;r)={xxc<r}, B(c;r)=\left\{ x\mid\|x-c\| < r\right\},

where the norm is is the 2 norm throughout the blog.

Indicator function.

The indicator function of X\mathcal{X} is denoted by ι,\iota, with

ι(x)={0,xX,xX.\iota(x)=\begin{cases} 0, & x\in\mathcal{X}\\ \infty, & x\notin\mathcal{X}. \end{cases}

Projection.

A Euclidean projection from a point xx onto this set X\mathcal{X} is denoted by Π(x)\Pi(x), and one arbitrary projection in Π(x)\Pi(x) is denoted by πx\pi_{x}. It is defined as:

πxΠ(x)=argminyXxy2.(Projx)\pi_{x}\in\Pi(x)=\mathop{\textrm{argmin}_{y\in\mathcal{X}}\|x-y\|^{2}.}\quad(\textrm{Projx})

​ The distance function at the point xx is denoted by d(x)=xπxd(x)=\|x-\pi_{x}\|, where the norm is the 2 norm.

The distance squared function is locally Lipschitz.

A function is locally Lipschitz, if at any point, there is a neighborhood around that point where the function is Lipschitz continuous.

For any closed, nonempty set its distance function is 1-Lipschitz continuous everywhere (Rockafellar and Wets (2009))[Example 9.6]. This implies that for any β\beta, d2+βx2d^{2}+\beta\|x\|^{2} is locally Lipschitz on any point xx. To see that, consider xBx\in\mathcal{B} where B\mathcal{B} is some closed ball around xx. Then, for any x,yB,x,y\in\mathcal{B},

d2(x)+βx2d2(y)βy2(d(x)d(y))(d(x)+d(y))+βx2y2undefined=(x+y)(xy)d(x)+d(y)undefinedM1d(x)d(y)undefinedxy+βx+yundefinedM2xyundefinedxy(M1+M2β)xy,(DistSqdLcLip) \begin{align*} & |d^{2}(x)+\beta\|x\|^{2}-d^{2}(y)-\beta\|y\|^{2}|\\ \leq & |\left(d(x)-d(y)\right)\left(d(x)+d(y)\right)|+|\beta||\underbrace{\|x\|^{2}-\|y\|^{2}}_{=(\|x\|+\|y\|)(\|x\|-\|y\|)}|\\ \leq & \underbrace{|d(x)+d(y)|}_{\leq M_{1}}\underbrace{|d(x)-d(y)|}_{\leq\|x-y\|}\\ & +|\beta|\underbrace{|\|x\|+\|y\||}_{\leq M_{2}}\underbrace{|\|x\|-\|y\||}_{\leq\|x-y\|}\\ \leq & (M_{1}+M_{2}\beta)\|x-y\|,\quad(\textrm{DistSqdLcLip}) \end{align*}

where in the second line we have used (i) the existence of constant M1,M2>0M_{1},M_{2}>0​ due to the points x,yBx,y\in\mathcal{B}, and (ii) the reverse triangle inequality.

Proximal normal cone.

The proximal normal cone of X\mathcal{X} at a point xXx\in\mathcal{X} is defined as follows (Rockafellar and Wets (2009))

vN(x)τ>0 xΠ(x+τv),(PrxNrmlCn) v\in N(x)\Leftrightarrow\exists_{\tau>0}\;x\in\Pi(x+\tau v),\quad(\textrm{PrxNrmlCn})

which also implies that

τundefined(0,τ) Π(x+τundefinedv)={x}. \forall_{\widetilde{\tau}\in(0,\tau)}\;\Pi(x+\widetilde{\tau}v)=\{x\}.

Frechet subdifferential and Clarke subdifferential.

For any function ff (not necessarily convex), its Frechet subdifferential f\partial f at a point xx is defined as follows (Correa et al. (1992))

vf(x)lim infy0f(x+y)f(x)vyy0.v\in\partial f(x)\Leftrightarrow\liminf_{y\to0}\frac{f(x+y)-f(x)-\left\langle v\mid y\right\rangle }{\|y\|}\geq0.

On the other hand, The Clarke subdifferential of a locally Lipschitz function ff​ is defined as follows:

uClarkef(x)d [lim supyx,t0f(y+td)f(y)t]ud. u\in\partial^{\textrm{Clarke}}f(x)\Leftrightarrow\forall_{d}\;\left[\limsup_{y\to x,t\downarrow0}\frac{f(y+td)-f(y)}{t}\right]\geq\left\langle u\mid d\right\rangle .

For a locally Lipschitz function the Clarke subdifferential is nonempty everywhere (Correa et al. (1992))[Property 2.2].

Finding Frechet subgradient of an infimal convolution function.

Define the infimal convolution function between two lower-semicontinuous functions ff and gg as follows (Correa et al. (1992))[Lemma 3.6]:

(fg)(x)=infy{f(y)+g(xy)}.\begin{aligned} (f\square g)(x) & =\inf_{y}\left\{ f(y)+g(x-y)\right\} .\end{aligned}

Also, define

yxargminy{f(y)+g(xy)},y_{x}^{\star}\in\mathop{\textrm{argmin}}_{y}\left\{ f(y)+g(x-y)\right\},

which can be empty. If yxy_{x}^{\star} exists, then

(fg)(x)f(yx)g(xyx).(FrechSubRule)\partial(f\square g)(x)\subseteq\partial f(y_{x}^{\star})\cap\partial g(x-y_{x}^{\star}).\quad(\textrm{FrechSubRule})

An interesting implication of (FrechSubRule).

Note that,

d2(x)=minyXxy2=miny{ι(y)+xy2}=(ι2)(x),d^{2}(x)=\min_{y\in\mathcal{X}}\|x-y\|^{2}=\min_{y}\left\{ \iota(y)+\|x-y\|^{2}\right\} =(\iota\square\|\cdot\|^{2})(x),

​ where by definition:

πx=argminyXxy2=argminy{ι(y)xy2},\pi_{x}=\mathop{\textrm{argmin}}_{y\in\mathcal{X}}\|x-y\|^{2}=\mathop{\textrm{argmin}}_{y}\left\{ \iota(y)\square\|x-y\|^{2}\right\},

Hence, recalling that x2=2x\partial\|x\|^{2}=2x, we can find a Frechet subgradient of d2d^{2} using (FrechSubRule):

d2(x)=(ι2)(x)ι(πx)[xπx2]undefined=2(xπx)=ι(πx){2(xπx)},\begin{aligned} \partial d^{2}(x) & =\partial(\iota\square\|\cdot\|^{2})(x)\\ \subseteq & \partial\iota(\pi_{x})\cap\underbrace{\partial\left[\|x-\pi_{x}\|^{2}\right]}_{=2(x-\pi_{x})}\\ & =\partial\iota(\pi_{x})\cap\{2(x-\pi_{x})\},\end{aligned}

hence if d2(x)\partial d^{2}(x)\neq\emptyset, then d2(x)={2(xπx)}\partial d^{2}(x)=\{2(x-\pi_{x})\}. In other words,

x:d2(x)d2(x)=2(xπx).(FrechSubDistSqd)\forall_{x:\partial d^{2}(x)\neq\emptyset}\quad\partial d^{2}(x)=2(x-\pi_{x}).\quad(\textrm{FrechSubDistSqd})

Proving local convexity via Frechet subdifferential.

If a locally Lipschitz function ff​ has its Frechet subdifferential f\partial f​ monotone on {(x,u)grafxA}\{(x,u) \in \mathbf{gra}\partial f \mid x\in A\}​, where AA is a convex set, then ff is convex on AA (Correa et al. (1992))[Theorem 3.8, Remark after Property 2.7]. In other words, for a locally Lipschitz function ff

(x,u),(y,v)graf:x,yA uvxy0f:convex on A(LocCvx),\forall_{(x,u),(y,v)\in\mathbf{gra}\partial f:x,y\in A}\;\left\langle u-v\mid x-y\right\rangle \geq0\Rightarrow f:\textrm{convex on }A\quad(\textrm{LocCvx}),

​ where graf={(x,u)uf(x)}.\mathbf{gra}\partial f=\left\{ (x,u)\mid u\in\partial f(x)\right\}.

Furthermore, if ff is locally Lipschitz, then monotonicity of f\partial f is equivalent to the monotonicity of Clarkef\partial^{\textrm{Clarke}}f, so proving either is fine to establish convexity (Correa et al. (1992))[Remark after Property 2.7].

Cocoercive operator.

An operator AA is β\beta-cocoercive on some convex set SS if

(x,u),(y,v)SgraAxyuvβuv2.(CocrcvOpt)\forall_{(x,u),(y,v)\in S\cap\mathbf{gra}A}\quad\left\langle x-y\mid u-v\right\rangle \geq\beta\|u-v\|^{2}.\quad(\textrm{CocrcvOpt})

A β\beta-cocoercive operator on SS is also 1β\frac{1}{\beta}-Lipschitz on SS.

Locally hypomonotone operator.

An operator AA is ρ\rho-hypomonotone on a convex set SS if

(x,u),(u,v)SgraAuvxy+ρxy20.\forall_{(x,u),(u,v)\in S\cap\mathbf{gra}A}\quad\left\langle u-v\mid x-y\right\rangle +\rho\|x-y\|^{2}\geq0.

Proximal normal cone of a prox-regular set is locally hypomonotone.

The following result is a restatement of Equation (3.1) of (Poliquin et al. (2000)). If X\mathcal{X} is prox-regular at xˉX\bar{x}\in\mathcal{X}, then there exists some ρ>0\rho>0 and R>0R>0 such that the proximal normal cone is a ρ\rho-hypomonotone operator on an RR-neighborhood of (xˉ,0)(\bar{x},0), defined by

VR(xˉ,0)={(p,u)pxˉR,u0R},(VR)\mathcal{V}_{R}(\bar{x},0)=\left\{ (p,u)\mid\|p-\bar{x}\|\leq R,\|u-0\|\leq R\right\},\quad \textrm{(VR)}

i.e.,

(p,u),(q,v)VR(xˉ,0)graNuvpq+ρpq20.(HypMntn)\begin{aligned} \forall_{(p,u),(q,v)\in\mathcal{V}_{R}(\bar{x},0)\cap\mathbf{gra}N} & \quad\left\langle u-v\mid p-q\right\rangle +\rho\|p-q\|^{2}\geq 0. \quad(\textrm{HypMntn})\end{aligned}

Now we are in a position to state the main result regarding geometry of a prox-regular set and prove it.

Geoemetry of a prox-regular set.

If X\mathcal{X} is set that is prox-regular at xˉX\bar{x}\in\mathcal{X}, then there exist some ρ>0,R>0\rho>0,R>0 such that for any λ\lambda satisfying λ(0,2)\lambda\in(0,2) and λρ\lambda\leq\rho we have the following properties :

(i) for any xB(xˉ;λR2ρ)x\in B(\bar{x};\frac{\lambda R}{2\rho})​, we have (πx,2ρλ(xπx))VR(xˉ,0)graN\left(\pi_{x},\frac{2\rho}{\lambda}(x-\pi_{x})\right)\in\mathcal{V}_{R}(\bar{x},0)\cap\mathbf{gra}N​, where VR(xˉ,0)\mathcal{V}_R(\bar x,0) is the same set defined in (VR).

(ii) the projection operator Π\Pi is 22λ\frac{2}{2-\lambda}-Lipschitz continuous on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right),

(iii) on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right)​, the function ϕλ=d2+λ2λ2\phi_{\lambda}=d^{2}+\frac{\lambda}{2-\lambda}\|\cdot\|^{2} is convex and differentiable, with the derivative given by ϕλ(x)=2(xπx)+222λx.\nabla\phi_{\lambda}(x)=2(x-\pi_{x})+2\frac{2}{2-\lambda}x.

Proof.

Proof to (i).

First, we show that for for any xB(xˉ,λR2ρ)x\in B\left(\bar{x},\frac{\lambda R}{2\rho}\right), πxxˉR\|\pi_{x}-\bar{x}\|\leq R. For xB(xˉ,λR2ρ),x\in B\left(\bar{x},\frac{\lambda R}{2\rho}\right), we have

πxxˉ=πxx+xxˉa)πxxundefined=d(x)xˉx<(λR/2ρ)+xxˉundefined<(λR/2ρ)<λRρR,\begin{aligned} & \|\pi_{x}-\bar{x}\|\\ = & \|\pi_{x}-x+x-\bar{x}\|\\ \overset{a)}{\leq} & \underbrace{\|\pi_{x}-x\|}_{=d(x)\leq\|\bar{x}-x\|<(\lambda R/2\rho)}+\underbrace{\|x-\bar{x}\|}_{<(\lambda R/2\rho)}\\ < & \frac{\lambda R}{\rho}\\ \leq & R,\end{aligned}

where in the last line we have used λρ\lambda\leq\rho.

Set τλ2ρ>0\tau\coloneqq\frac{\lambda}{2\rho}>0​, then

Π(πx+τ{2ρλ(xπx)})=Π(x)πx,\Pi\left(\pi_{x}+\tau\left\{ \frac{2\rho}{\lambda}(x-\pi_{x})\right\} \right)=\Pi\left(x\right)\ni\pi_{x},

​ where the last inclusion follows from definition of projection. Thus 2ρλ(xπx)N(πx)\frac{2\rho}{\lambda}(x-\pi_{x})\in N(\pi_{x})​ by (PrxNrmlCn). Also, from d(x)xˉx<(λR/2ρ)d(x)\leq\|\bar{x}-x\|<(\lambda R/2\rho)​, we have

2ρλ(xπx)<R,\begin{aligned} \|\frac{2\rho}{\lambda}(x-\pi_{x})\|< & R,\end{aligned}

​​

thus completing proof to (i).

Proof to (ii).

From (i), we have for any x,yB(xˉ,λR2ρ)x,y\in B\left(\bar{x},\frac{\lambda R}{2\rho}\right), we have (πx,2ρλ(xπx)),(πy,2ρλ(yπy))VR(xˉ,0)graN\left(\pi_{x},\frac{2\rho}{\lambda}(x-\pi_{x})\right),\left(\pi_{y},\frac{2\rho}{\lambda}(y-\pi_{y})\right)\in\mathcal{V}_{R}(\bar{x},0)\cap\mathbf{gra}N, and using (HypMntn), we have

02ρλ(xπx)2ρλ(yπy)πxπy+ρπxπy2=2ρλ(xy)2ρλ(πxπy)πxπy+ρπxπy2=2ρλxyπxπy2ρλπxπy2+ρπxπy2undefined=2ρλ(1λ2)πxπy2=2ρλxyπxπy2ρλ(1λ2)πxπy2(1λ2)πxπy2xyπxπy.\begin{aligned} 0 & \leq\left\langle \frac{2\rho}{\lambda}(x-\pi_{x})-\frac{2\rho}{\lambda}(y-\pi_{y})\mid\pi_{x}-\pi_{y}\right\rangle +\rho\|\pi_{x}-\pi_{y}\|^{2}\\ & =\left\langle \frac{2\rho}{\lambda}(x-y)-\frac{2\rho}{\lambda}(\pi_{x}-\pi_{y})\mid\pi_{x}-\pi_{y}\right\rangle +\rho\|\pi_{x}-\pi_{y}\|^{2}\\ & =\frac{2\rho}{\lambda}\left\langle x-y\mid\pi_{x}-\pi_{y}\right\rangle \underbrace{-\frac{2\rho}{\lambda}\|\pi_{x}-\pi_{y}\|^{2}+\rho\|\pi_{x}-\pi_{y}\|^{2}}_{=-\frac{2\rho}{\lambda}\left(1-\frac{\lambda}{2}\right)\|\pi_{x}-\pi_{y}\|^{2}}\\ & =\frac{2\rho}{\lambda}\left\langle x-y\mid\pi_{x}-\pi_{y}\right\rangle -\frac{2\rho}{\lambda}\left(1-\frac{\lambda}{2}\right)\|\pi_{x}-\pi_{y}\|^{2}\\ \Leftrightarrow & \left(1-\frac{\lambda}{2}\right)\|\pi_{x}-\pi_{y}\|^{2}\leq\left\langle x-y\mid\pi_{x}-\pi_{y}\right\rangle .\end{aligned}

So we have shown that

(x,πx),(y,πy)graΠB(xˉ,λR2ρ)xyπxπy(1λ2)πxπy2,\forall_{(x,\pi_{x}),(y,\pi_{y})\in\mathbf{gra}\Pi\cap B\left(\bar{x},\frac{\lambda R}{2\rho}\right)}\quad\left\langle x-y\mid\pi_{x}-\pi_{y}\right\rangle \geq\left(1-\frac{\lambda}{2}\right)\|\pi_{x}-\pi_{y}\|^{2},

hence the projection operator Π\Pi is [(2λ)/2]\left[(2-\lambda)/2\right]-cocoercive on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right) from (CocrcvOpt). Because a β\beta-cocoercive operator on SS is also 1β\frac{1}{\beta}-Lipschitz on SS, we have the projection operator Π\Pi being [2/(2λ)]\left[2/(2-\lambda)\right]-Lipschitz continuous on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right), i.e.,

(x,πx),(y,πy)graΠB(xˉ,λR2ρ)πxπy22λxy.(LipCont)\forall_{(x,\pi_{x}),(y,\pi_{y})\in\mathbf{gra}\Pi\cap B\left(\bar{x},\frac{\lambda R}{2\rho}\right)}\quad\|\pi_{x}-\pi_{y}\|\leq\frac{2}{2-\lambda}\|x-y\|.\quad(\textrm{LipCont})

This proves (ii).

Proof to (iii).

Take a point xB(xˉ,λR2ρ).x\in B\left(\bar{x},\frac{\lambda R}{2\rho}\right).Due to (ii), the projection operator Π(x)\Pi(x) is single-valued on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right).

From (DistSqdLcLip) ϕλ\phi_{\lambda} is locally Lipschitz, so we will employ (LocCvx) to prove its convexity on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right). First, we will show that on the set

S={(x,u)(x,u)graϕλ,xB(xˉ,λR2ρ)}S=\left\{ (x,u)\mid(x,u)\in\mathbf{gra}\partial\phi_{\lambda},x\in B\left(\bar{x},\frac{\lambda R}{2\rho}\right)\right\}

the operator ϕλ\partial\phi_{\lambda}​ is monotone, which will help in proving that ϕλ\phi_{\lambda}​ is convex and differentiable on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right)​.

Recall from (FrechSubDistSqd) that

x:d2(x)d2(x)=2(xπx).\forall_{x:\partial d^{2}(x)\neq\emptyset}\quad\partial d^{2}(x)=2(x-\pi_{x}).

​​ Consider two points (x,u),(y,v)S(x,u),(y,v)\in S​​. Without loss of generality, we can assume that d2(x),d2(y)\partial d^{2}(x),\partial d^{2}(y)​​ are nonempty, because for the empty case (iii) is vacuously true. Then from (FrechSubDistSqd) we have d2(x)=2(xπx)\partial d^{2}(x)=2(x-\pi_{x})​​ and d2(y)=2(yπy)\partial d^{2}(y)=2(y-\pi_{y})​​. On these points,

ϕλ(x)=d2(x)+2λ2λ(x), and ϕλ(y)=d2(y)+2λ2λ(y).\partial\phi_{\lambda}(x)=\partial d^{2}(x)+2\frac{\lambda}{2-\lambda}(x),\textrm{ and }\partial\phi_{\lambda}(y)=\partial d^{2}(y)+2\frac{\lambda}{2-\lambda}(y).

​​ We want to show that for any such (x,u),(y,v)S(x,u),(y,v)\in S​​, we have

0ϕλ(x)ϕλ(y)xy=d2(x)+2λ2λ(x)d2(y)2λ2λ(y)xy=d2(x)d2(y)xy+2λ2λxy2.(GoalA)\begin{aligned} 0 & \leq\left\langle \partial\phi_{\lambda}(x)-\partial\phi_{\lambda}(y)\mid x-y\right\rangle \\ & =\left\langle \partial d^{2}(x)+2\frac{\lambda}{2-\lambda}(x)-\partial d^{2}(y)-2\frac{\lambda}{2-\lambda}(y)\mid x-y\right\rangle \\ & =\left\langle \partial d^{2}(x)-\partial d^{2}(y)\mid x-y\right\rangle +2\frac{\lambda}{2-\lambda}\|x-y\|^{2}.\quad(\textrm{GoalA})\end{aligned}

​​ To prove (GoalA), first we note that

d2(x)d2(y)xy=2(xπx)2(yπy)xy=2(xy)(πxπy)xy=2xy22πxπyxy.(EqPart1)\begin{aligned} \left\langle \partial d^{2}(x)-\partial d^{2}(y)\mid x-y\right\rangle & =\left\langle 2(x-\pi_{x})-2(y-\pi_{y})\mid x-y\right\rangle \\ & =2\left\langle (x-y)-(\pi_{x}-\pi_{y})\mid x-y\right\rangle \\ & =2\|x-y\|^{2}-2\left\langle \pi_{x}-\pi_{y}\mid x-y\right\rangle .\quad(\textrm{EqPart1})\end{aligned}

​​ Now, using Cauchy–Schwarz inequality, we have

πxπyxyπxπyundefined22λxyxy22λxy22πxπyxy42λxy2,\begin{aligned} \left\langle \pi_{x}-\pi_{y}\mid x-y\right\rangle & \leq\underbrace{\|\pi_{x}-\pi_{y}\|}_{\leq\frac{2}{2-\lambda}\|x-y\|}\|x-y\|\\ & \leq\frac{2}{2-\lambda}\|x-y\|^{2}\\ \Rightarrow-2\left\langle \pi_{x}-\pi_{y}\mid x-y\right\rangle & \geq-\frac{4}{2-\lambda}\|x-y\|^{2},\end{aligned}

​​ and putting this in (EqPart1), we have

d2(x)d2(y)xy2xy242λxy2(242λ)xy2=2λ2λxy2d2(x)d2(y)xy+2λ2λxy20,\begin{aligned} \left\langle \partial d^{2}(x)-\partial d^{2}(y)\mid x-y\right\rangle & \geq2\|x-y\|^{2}-\frac{4}{2-\lambda}\|x-y\|^{2}\\ & \geq\left(2-\frac{4}{2-\lambda}\right)\|x-y\|^{2}\\ & =\frac{-2\lambda}{2-\lambda}\|x-y\|^{2}\\ \Rightarrow\left\langle \partial d^{2}(x)-\partial d^{2}(y)\mid x-y\right\rangle +2\frac{\lambda}{2-\lambda}\|x-y\|^{2} & \geq0,\end{aligned}

​​ thus reaching (GoalA). So, we have shown that on SS​​, ϕλ\partial\phi_{\lambda}​​ is monotone on SS​​. As ϕλ\phi_{\lambda}​​ is locally Lipschitz, it means that Clarkef\partial^{\textrm{Clarke}}f​​ is monotone on SS​​, and due to (LocCvx), we have ϕλ\phi_{\lambda}​​ convex on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right)​​. This further implies that, for any xx​​ in B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right)​​, ϕλ(x)=Clarkeϕλ(x),\partial\phi_{\lambda}(x)=\partial^{\textrm{Clarke}}\phi_{\lambda}(x),​​ and due to the locally Lipschitz nature of ϕλ,\phi_{\lambda},​​ it has nonempty Clarke subdifferential everywhere (Correa et al. (1992))[Property 2.2]. So, all points in B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right)​​ is Frechet subdifferentiable, i.e., for any xx​​ in B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right)​​, we have ϕλ(x)=Clarkeϕλ(x)\partial\phi_{\lambda}(x)=\partial^{\textrm{Clarke}}\phi_{\lambda}(x)\neq\emptyset​​, and as we have shown before, on those it is in fact differentiable with the gradient

ϕλ(x)=2(xπx)+222λx.\partial\phi_{\lambda}(x)=2(x-\pi_{x})+2\frac{2}{2-\lambda}x.

Thus, on B(xˉ,λR2ρ)B\left(\bar{x},\frac{\lambda R}{2\rho}\right)​, the function ϕλ\phi_{\lambda}​ is convex and differentiable.

References

  • Rockafellar, R. Tyrrell, and Roger J-B. Wets. Variational analysis. Vol. 317. Springer Science & Business Media, 2009. [pdf]

  • Correa, Rafael, Alejandro Jofre, and Lionel Thibault. "Characterization of lower semicontinuous convex functions." Proceedings of the American Mathematical Society (1992): 67-72. [pdf]

  • Poliquin, René, R. Rockafellar, and Lionel Thibault. "Local differentiability of distance functions." Transactions of the American mathematical Society 352.11 (2000): 5231-5249. [pdf]