On partial smoothness, tilt stability and the ${\mathcal {VU}}$-decomposition

Abstract

Under the assumption of prox-regularity and the presence of a tilt stable local minimum we are able to show that a ${\mathcal {VU}}$ like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function.

Appendix A

The prove Proposition 15 we need the following results regarding the variation limits of rank-1 supports.

Proposition 56

([13], Corollary 3.3) Let $\{\mathcal {A}(v)\}_{v\in W}$ be a family of non-empty rank-1 representers (i.e. $\mathcal {A}(v)\subseteq \mathcal {S}\left( n\right)$ and $-\mathcal {P}\left( n\right) \subseteq 0^{+}\mathcal {A}(v)$ for all v) and W a neighbourhood of w. Suppose that $\limsup _{v\rightarrow w}\mathcal {A}(v)=\mathcal {A}(w)$. Then

$$\begin{aligned} \limsup _{v\rightarrow w}\inf _{u\rightarrow h}q\left( \mathcal {A}(v)\right) (u)=q\left( \mathcal {A}(w)\right) (h) \end{aligned}$$

(41)

Recall that $(x^{\prime },z^{\prime })\rightarrow _{S_{p}(f)}(\bar{x},z)$ means $x^{\prime }\rightarrow ^{f}\bar{x}$, $\ z^{\prime }\in \partial _{p}f(x^{\prime })$ and $z^{\prime }\rightarrow z$.

Corollary 57

Let $f:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}_{\infty }$ be proper and lower semicontinuous with $h\in b^{1}(\underline{\partial }^{2}f(\bar{x},\bar{z} )).$ Then

$$\begin{aligned} q\left( \underline{\partial }^{2}f(\bar{x},\bar{z})\right) \left( h\right) =\limsup _{(x^{\prime },z^{\prime })\rightarrow _{S_{p}(f)}(\bar{x},\bar{z} )}\inf _{u\rightarrow h}q\left( \partial ^{2,-}f(x^{\prime },z^{\prime })\right) (u). \end{aligned}$$

(42)

Proof

Use Proposition 56 and Remark 11. $\square$

Denote the infimal convolution of f by $f_{\lambda }(x):=\inf _{u\in {\mathbb {R}}^{n}}\left( f(u)+\frac{1}{2\lambda }\Vert x-u\Vert ^{2}\right)$. Recall that $f_{\lambda }\left( x\right) -\frac{1}{2\lambda }\left\| x\right\| ^{2}=-\left( f\ +\frac{\lambda }{2}\Vert \cdot \Vert ^{2}\right) ^{*}(\lambda x)$ and this $f_{\lambda }$ is always para-concave. Recall that in [13, Lemma 2.1], it is observed that f is locally $C^{1,1}$ iff f is simultaneously a locally para-convex and para-concave function. Recall [38, Proposition 4.15] that states that the limit infimum of a collection of convex sets is also convex and that the upper epi-limit of a family of functions has an epi-graph that is the limit infimum of the family of epi-graphs. Consequently the epi-limit supremum of a family of convex functions give rise to convex function.

Proof

(of Proposition 15) Begin by assuming f is locally para-convex. Let $\frac{c}{2}>0$ be the modulus of para-convexity of f on $B_{\delta }( \bar{x})$, $x\in B_{\delta }(\bar{x})$ with $z\in \partial f\left( x\right)$ and $\partial ^{2,-}f\left( x,z\right) \ne \emptyset$. Let $C_{t}(x)=\{h\mid x+th\in B_{\delta }(\bar{x})\}$ then we have

$$\begin{aligned} h\mapsto \left( \frac{2}{t^{2}}\right) \left( f(x+th)-f(x)-t\langle z,h\rangle \right) +\frac{c}{t^{2}}\left( \Vert x+th\Vert ^{2}-\Vert x\Vert ^{2}-t\langle 2x,h\rangle \right) \end{aligned}$$

convex on $C_{t}(x)$ since $x\mapsto f(x)+\frac{c}{2}\Vert x\Vert ^{2}$ is convex on $B_{\delta }(\bar{x})$. Next note that for every $K>0$ there exists a $\bar{t}>0$ such that for $0<t<\bar{t}$ we have $B_{K}(0)\subseteq C_{t}(x)$. Once again restricting f to $B_{\delta }(\bar{x})$ we get a family

$$\begin{aligned} \{h\mapsto \Delta _{2}f(x,t,z,h)+\frac{c}{t^{2}}\left( \Vert x+th\Vert ^{2}-\Vert x\Vert ^{2}-t\langle 2x,h\rangle \right) \}_{t<\bar{t}} \end{aligned}$$

(43)

of convex functions with domains containing $C_{t}(x)\,$(for each t) and whose convexity (on their common domain of convexity) will be preserved under an upper epi-limit as $t\downarrow 0$. Thus, using the fact that $\frac{c}{t^{2}}\left( \Vert x+th\Vert ^{2}-\Vert x\Vert ^{2}-t\langle 2x,h\rangle \right)$ converges uniformly on bounded sets to $c\Vert h\Vert ^{2}$, we have the second order circ derivative (introduced in [23]) given by:

$$\begin{aligned} f^{\uparrow \uparrow }(x,z,h)+c\Vert h\Vert ^{2}&:=\limsup _{(x^{\prime },z^{\prime })\rightarrow _{S_{p}}(x,z),t\downarrow 0}\inf _{u^{\prime }\rightarrow h}(\Delta _{2}f(x^{\prime },t,z^{\prime },u^{\prime }) \\&\qquad +\frac{c}{t^{2}}\left( \Vert x+th\Vert ^{2}-\Vert x\Vert ^{2}-t\langle 2x,h\rangle \right) ) \end{aligned}$$

which is convex on $B_{K}(0)$, for every $K>0$, being obtained by taking an epi-limit supremum of a family of convex functions given in (43). We then have $h\mapsto f^{\uparrow \uparrow }(x,z,h)+c\Vert h\Vert ^{2}$ convex (with $f^{\uparrow \uparrow }(x,z,\cdot )$ having a modulus of para-convexity of c).

From [4], Proposition 4.1 particularized to $C^{1,1}$ functions f we have that there exists a $\eta \in [x,y]$ such that

$$\begin{aligned} f(y)\in f(x)+\langle \nabla f(x),y-x\rangle +\frac{1}{2}\langle \overline{D}^{2}f(\eta ),(y-x)(y-x)^{T}\rangle . \end{aligned}$$

(44)

Using (44), Proposition 7 and the variational result corollary 56, we have when the limit is finite (for $\bar{z}:=\nabla f(\bar{x})$)

$$\begin{aligned}&f^{\uparrow \uparrow }(\bar{x},\bar{z},h)\\&\quad :=\limsup _{(x^{\prime },z^{\prime })\rightarrow _{S_{p}}(\bar{x},\bar{z}),\;t\downarrow 0}\inf _{u^{\prime }\rightarrow h}\Delta _{2}f(x^{\prime },t,z^{\prime },u^{\prime }) \\&\quad \le \limsup _{x^{\prime }\rightarrow \bar{x},\;t\downarrow 0}\inf _{u^{\prime }\rightarrow h}\Delta _{2}f(x^{\prime },t,\nabla f(x^{\prime }),u^{\prime })\le \limsup _{\eta \rightarrow \bar{x}}\inf _{u^{\prime }\rightarrow h}q\left( \overline{D}^{2}f(\eta )\right) (u^{\prime }) \\&\quad \le q\left( \overline{D}^{2}f(\bar{x})-\mathcal {P}(n)\right) (h)\le q\left( \underline{\partial }^{2}f(\bar{x},\bar{z})\right) (h)\le f^{\uparrow \uparrow }(\bar{x},\bar{z},h), \end{aligned}$$

where the last inequality follows from [23, Proposition 6.5].

Now assuming f is quadratically minorised and is prox-regular at $\bar{x}\$ for $\bar{p}\in \partial f(\bar{x})$ with respect to $\varepsilon$ and r. Let $g(x):=f(x+\bar{x})-\langle \bar{z},x+\bar{x}\rangle$. Then $0\in \partial g(0)$ and we now consider the infimal convolution $g_{\lambda }(x)$ which is para-convex locally with a modulus $c:=\frac{\lambda r}{2(\lambda -r)}$, prox-regular at 0 (see [35, Theorem 5.2]). We may now use the first part of the proof to deduce that $g_{\lambda }^{\uparrow \uparrow }(0,0,\cdot )$ is para-convex with modulus $c=\frac{2\lambda r}{(\lambda -r)}$ and $g_{\lambda }^{\uparrow \uparrow }(0,0,h)=q\left( \underline{\partial }^{2}g_{\lambda }(0,0)\right) (h)$ since $g_{\lambda }$ is $C^{1,1}$ (being both para-convex and para-concave). Using Corollary 56 and [8, Proposition 4.8 part 2.] we obtain

$$\begin{aligned} \limsup _{\lambda \rightarrow \infty }\inf _{h^{\prime }\rightarrow h}g_{\lambda }^{\uparrow \uparrow }(0,0,h^{\prime })=q\left( \limsup _{\lambda \rightarrow \infty }\underline{\partial }^{2}g_{\lambda }(0,0)\right) (h)=q\left( \underline{\partial }^{2}g(0,0)\right) (h). \end{aligned}$$

Thus $q\left( \underline{\partial }^{2}g(0,0)\right) (h)+r\Vert h\Vert ^{2}=\limsup _{\lambda \rightarrow \infty }\inf _{h^{\prime }\rightarrow h}\left( g_{\lambda }^{\uparrow \uparrow }(0,0,h^{\prime })+\frac{\lambda r}{(\lambda -r)}\Vert h\Vert ^{2}\right)$ is convex, being the variational upper limit of convex functions. One can easily verify that $\underline{\partial }^{2}g(0,0)=\underline{\partial }^{2}f(\bar{x},\bar{z})$ and $g^{\uparrow \uparrow }(0,0,h)=f^{\uparrow \uparrow }(\bar{x},\bar{z},h)$. $\square$