Functions of a multivariate variable

Vector variable - Matrix variable - Perspective functions - Divergences

Vector variable

Description f(x)

(∀ x ∈ ℝN)

proxγf(x)

(∀ γ ∈ ℝ+)

Matlab Python Ref
Euclidean norm $\|x\|_2 = \displaystyle\sqrt{\sum_{n=1}^N x_n^2}$ $\left(1 - \dfrac{\gamma}{\max\big\{\|x\|_2,\gamma\big\}}\right)x$ [Function]
[Prox]
[Class] [Combettes et al., 2005]
Infinity norm $\|x\|_\infty = \displaystyle\max_{1\le n\le N} |x_n|$ $\Big(\operatorname{sign}(x_n)\min\{|x_n|,s\}\Big)_{1\le n\le N}$

where $s\in\mathbb{R}$ is such that $\displaystyle\sum_{n=1}^N \max\{0, |x_n| - s\} = \gamma$

[Function]
[Prox]
(coming soon)
Maximum $\displaystyle\max_{1\le n\le N} x_n$ $\Big(\min\{x_n,s\}\Big)_{1\le n\le N}$

where $s\in\mathbb{R}$ is such that $\displaystyle\sum_{n=1}^N \max\{0, x_n - s\} = \gamma$

[Function]
[Prox]
(coming soon)
Vapnik $\displaystyle\max\big\{\|x\|_2-\varepsilon,0\big\}$

(with $\varepsilon>0$)

$\left\{\begin{aligned} &x &&\textrm{if $\|x\|_2 - \varepsilon\le 0$}\\ &\frac{\varepsilon}{\|x\|_2}x &&\textrm{if $0<\|x\|_2-\varepsilon\le\gamma$}\\ &\Big(1-\frac{\gamma}{\|x\|_2}\Big) x &&\textrm{otherwise}\\ \end{aligned}\right.$ [Function]
[Prox]
(coming soon) [Combettes et al., 2011]
Euclidean norm + positivity $\|x\|_2 + \iota_{[0,+\infty)^N}(x)$ $\left(1 - \dfrac{\gamma}{\max\big\{\|\max(x,0)\|_2,\gamma\big\}}\right)\max(x,0)$ [Function]
[Prox]
(coming soon)
Barrier of affine constraints $\left\{\begin{aligned} &-\log(b-a^{\top} x) &&\textrm{if $a^{\top} x < b$}\\&+ \infty &&\textrm{otherwise }\end{aligned}\right.$

(with $b \in \mathbb{R}$ and $a \in \mathbb{R}^N \setminus \{0\}$)

$x+\frac{b-a^{\top}x-\sqrt{(b-a^{\top}x)^2+4\gamma\|a\|^2}}{2\|a\|^2} a$ [Function]
[Prox]
(coming soon) [Bertocchi et al., 2018]
Barrier of hyperslab constraints $\left\{\begin{aligned} &-\log(b_{M}a^{\top} x) -\log(a^{\top} x-b_{m})&& \textrm{if $b_{m} < a^{\top} x < b_{M}$}\\&+ \infty &&\textrm{otherwise }\end{aligned}\right.$

(with $b_m \in \mathbb{R}$, $b_M \in \mathbb{R}$, $b_m < b_M$, and $a \in \mathbb{R}^N \setminus \{0\}$)

$x+\frac{\kappa(x,\gamma)-a^{\top}x}{\|a\|^2} a$

with $\kappa(x,\gamma)$ unique solution in $]b_{m},b_{M}[$ of: $0=z^3 -(b_{m}+b_{M}+ a^{\top}x) z^2 +(b_{m}b_{M} +a^{\top} x(b_{m}+b_{M})-2\gamma \|a\|^2)z -b_{m}b_{M}a^{\top}x+\gamma (b_{m}+b_{M})||a||^2$

[Function]
[Prox]
(coming soon) [Bertocchi et al., 2018]
Barrier of $\ell_2$ ball constraint $\left\{\begin{aligned} &-\log(\alpha-\|x-c\|^2)&&\textrm{if $\|x-c\|^2< \alpha$}\\&+ \infty &&\textrm{otherwise }\end{aligned}\right.$

(with $\alpha > 0$ and $c \in \mathbb{R}^N$)

$c+\frac{\alpha -\kappa(x,\gamma)^2}{\alpha -\kappa(x,\gamma)^2+2\gamma} (x-c)$

with $\kappa(x,\gamma)$ unique solution in $[0,\sqrt{\alpha}[$ of: $0=z^3 -\|x-c\| z^2 -(\alpha+2\gamma)z +\alpha \|x-c\|$

[Function]
[Prox]
(coming soon) [Bertocchi et al., 2018]

Matrix variable

Description f(X) = φ(s)

(∀ X = U Diag(s) V∈ ℝM × N)

proxγf(X)

(∀ γ ∈ ℝ+)

Matlab Python Ref
Permutation invariant $\varphi(s)$ $U\operatorname{Diag}\Big(\operatorname{prox}_{\gamma\varphi}(s)\Big)V^\top$ [Function]
[Prox]
(coming soon) [Lewis, 1996]
Nuclear norm $\displaystyle\|X\|_N = \|s\|_1$ $U\operatorname{Diag}\Big(\operatorname{prox}_{\gamma\|\cdot\|_1}(s)\Big)V^\top$ [Function]
[Prox]
(coming soon) [Lewis, 1996]
Frobenius norm $\displaystyle\|X\|_F = \|s\|_2$ $\operatorname{prox}_{\gamma\|\operatorname{Vec}(\cdot)\|_2}(X)$ [Function]
[Prox]
(coming soon) [Lewis, 1996]
Spectral norm $\displaystyle\|X\|_S = \|s\|_\infty$ $U\operatorname{Diag}\Big(\operatorname{prox}_{\gamma\|\cdot\|_\infty}(s)\Big)V^\top$ [Function]
[Prox]
(coming soon) [Lewis, 1996]
Nuclear norm + Ridge $\frac{1}{2}\|X\|_F ^2 +\mu \|X\|_N = \frac{1}{2} \|s\|_2^2 + \mu \| s \|_1$ with $\mu > 0$

$U\operatorname{Diag}(p)V^\top$

with $p =soft_{\frac{\mu \gamma}{\gamma+1}}(\frac{s}{1+\gamma})$
[Function]
[Prox]
(coming soon) [ Benfenati et al., 2018]
Frobenius norm + Ridge $\frac{1}{2}\|X\|_F ^2 +\mu \|X\|_F = \frac{1}{2} \|s\|_2^2 + \mu \| s \|_2$ with $\mu > 0$

$U\operatorname{Diag}(p)V^\top$

with $p =\left\{ \begin{aligned} &(1-\frac{\gamma \mu}{\|s\|})\frac{s}{1+\gamma} &&\textrm{if $\|s\| > \gamma\mu$}\\ &0 &&\textrm{otherwise} \end{aligned} \right.$
[Function]
[Prox]
(coming soon) [ Benfenati et al., 2018]
Squared Frobenius norm + Ridge $\frac{1}{2}\|X\|_F ^2 +\mu \|X\|_F^2 = \frac{1}{2} \|s\|_2^2 + \mu \| s \|_2^2$ with $\mu > 0$

$U\operatorname{Diag}(p)V^\top$

with $p =\frac{s}{1+\gamma(1+2\mu)}$
[Function]
[Prox]
(coming soon) [ Benfenati et al., 2018]
Schatten 3-penalty + Ridge $\frac{1}{2}\|X\|_F ^2 +\mu \mathcal{R}_3^3(X) = \frac{1}{2} \|s\|_2^2 + \mu \| s \|_3^3$ with $\mu > 0$

$U\operatorname{Diag}(p)V^\top$

with $p =(6\gamma\mu)^{-1}\Bigg(sign(s)\sqrt{(\gamma+1)^2+12|s|\gamma\mu}-\gamma-1\Bigg)$
[Function]
[Prox]
(coming soon) [ Benfenati et al., 2018]
Schatten 4-penalty + Ridge $\frac{1}{2}\|X\|_F ^2 +\mu \mathcal{R}_4^4(X) = \frac{1}{2} \|s\|_2^2 + \mu \| s \|_4^4$ with $\mu > 0$

$U\operatorname{Diag}(p)V^\top$

with $p =(8\gamma\mu)^{-\frac{1}{3}}\Bigg(\sqrt[3]{s+\sqrt{s^2+\zeta}}-\sqrt[3]{\sqrt{s^2+\zeta}-s}\Bigg)$

And $\zeta=\frac{(\gamma+1)^3}{27\gamma\mu}$

[Function]
[Prox]
(coming soon) [ Benfenati et al., 2018]
Schatten 4/3-penalty + Ridge $\frac{1}{2}\|X\|_F ^2 +\mu \mathcal{R}_{4/3}^{4/3}(X) = \frac{1}{2} \|s\|_2^2 + \mu \| s \|_{4/3}^{4/3}$ with $\mu > 0$

$U\operatorname{Diag}(p)V^\top$

with $p = \frac{1}{1+\gamma}\Bigg(s+\frac{4\gamma\mu}{3\sqrt[3]{2(1+\gamma)}}\Bigg(\sqrt[3]{\sqrt{s^2+\zeta}-s}-\sqrt[3]{s+\sqrt{s^2+\zeta}}\Bigg)\Bigg)$

And $\zeta=\frac{256(\gamma\mu)^3}{729(1+\gamma)}$

[Function]
[Prox]
(coming soon) [ Benfenati et al., 2018]
Schatten 3/2-penalty + Ridge $\frac{1}{2}\|X\|_F ^2 +\mu \mathcal{R}_{3/2}^{3/2}(X) = \frac{1}{2} \|s\|_2^2 + \mu \| s \|_{3/2}^{3/2}$ with $\mu > 0$

$U\operatorname{Diag}(p)V^\top$

with $p =\frac{1}{1+\gamma}\Bigg(s+\frac{9\gamma^2\mu^2}{8(1+\gamma)}sign(s)\Bigg(1-\sqrt{1+\frac{16(1+\gamma)}{9\gamma^2\mu^2}|s|}\Bigg)\Bigg)$
[Function]
[Prox]
(coming soon) [Benfenati et al., 2018]
logdet

$f(X) =\left\{ \begin{aligned} &-\log(\det(X)) &&\textrm{if $X\succ 0$}\\ &+\infty &&\textrm{otherwise} \end{aligned} \right.$

$U\operatorname{Diag}(p)V^\top$

with $p =\frac{1}{2} \Bigg( s+\sqrt{s^2+4\gamma}\Bigg)$
[Function]
[Prox]
(coming soon) [Benfenati et al., 2018]
Nuclear norm + logdet

$f(X) +\mu \|X\|_N $

with $f(X) =\left\{ \begin{aligned} &-\log(\det(X)) &&\textrm{if $X\succ 0$}\\ &+\infty &&\textrm{otherwise} \end{aligned} \right.$

$U\operatorname{Diag}(p)V^\top$

with $p =\frac{1}{2} \Bigg( s-\gamma\mu+\sqrt{(s-\gamma\mu)^2+4\gamma}\Bigg)$
[Function]
[Prox]
(coming soon) [Benfenati et al., 2018]
Squared Frobenius norm + logdet

$f(X) +\mu \|X\|_F^2$

with $f(X) =\left\{ \begin{aligned} &-\log(\det(X)) &&\textrm{if $X\succ 0$}\\ &+\infty &&\textrm{otherwise} \end{aligned} \right.$

$U\operatorname{Diag}(p)V^\top$

with $p =\frac{1}{2(2\mu\gamma+1)} \Bigg( s+\sqrt{s^2+4\gamma(2\gamma\mu+1)}\Bigg)$
[Function]
[Prox]
(coming soon) [Benfenati et al., 2018]
Schatten p-penalty + logdet

$f(X) +\mu \mathcal{R}_p^p(X) $

with $p\le1$ and $f(X) =\left\{ \begin{aligned} &-\log(\det(X)) &&\textrm{if $X\succ 0$}\\ &+\infty &&\textrm{otherwise} \end{aligned} \right.$

$U\operatorname{Diag}(d)V^\top$

with $d =(d_i)_{1\le i\le n}$ and $(\vee i\in {1,....n}) \quad d_i >0 $ and $\gamma\mu p d_i^p + d_i^2-\lambda_i d_i = \gamma$
[Function]
[Prox]
(coming soon) [Benfenati et al., 2018]
Inverse Schatten p-penalty + logdet

$f(X) +\mu \mathcal{R}_p^p(X^{-1}) $

with $p\le1$ and $f(X) =\left\{ \begin{aligned} &-\log(\det(X)) &&\textrm{if $X\succ 0$}\\ &+\infty &&\textrm{otherwise} \end{aligned} \right.$

$U\operatorname{Diag}(d)V^\top$

with $d =(d_i)_{1\le i\le n}$ and $(\vee i\in {1,....n}) \quad d_i >0 $ and $ d_i^{p+2} -\lambda_i d_i^{p+1} - \gamma d_i^p= \mu\gamma p $
[Function]
[Prox]
(coming soon) [Benfenati et al., 2018]

Perspective of convex functions

Description f(x,ξ)

(∀(x,ξ) ∈ ℝN×ℝ)

proxγf(x,ξ)

(∀ γ ∈ ℝ+)

Matlab Python Ref
Square $\begin{cases} \dfrac{\|x\|_2^2}{\xi} & \textrm{if $\xi > 0$}\\ 0 & \textrm{if $x = 0$ and $\xi=0$}\\ +\infty & \textrm{otherwise}.\end{cases}$ $\left\{\begin{aligned} &(0,0) &&\textrm{if $\|x\|_2^2\le -4\gamma\xi$}\\ &(0,\xi) &&\textrm{if $x=0$ and $\xi>0$}\\ &\Big(x-\dfrac{\gamma tx}{\|x\|_2}, \xi+\dfrac{\gamma t^2}{4}\Big) &&\textrm{otherwise} \end{aligned}\right.$

with $t\ge0$ such that

$\small \gamma t^3 + 4(\xi+2\gamma) t - 8\|x\|_2=0$
[Function]
[Prox]
(coming soon) [Combettes et al., 2017]
Squared root $\begin{cases} -\sqrt{\xi^2-\|x\|_2^2} & \textrm{if $\xi > 0$ and $\|x\|_2^2\le\xi$}\\ 0 & \textrm{if $x = 0$ and $\xi=0$}\\ +\infty & \textrm{otherwise}.\end{cases}$ $\left\{\begin{aligned} &\Big(x-\dfrac{\gamma tx}{\|x\|_2}, \xi+\gamma\sqrt{1+ t^2}\Big) &&\textrm{if $\xi+\sqrt{\gamma^2+\|x\|_2^2}>0$}\\ &(0,0) &&\textrm{otherwise} \end{aligned}\right.$

with $t\ge0$ such that

$\small 2\gamma t + \dfrac{\xi t}{\sqrt{1+t^2}} - \|x\|_2^2 = 0$

(assuming that $x/\|x\|_2=0$ if $x=0$)
[Function]
[Prox]
(coming soon) [Combettes et al., 2017]
Huber $\begin{cases} \rho|x|-\dfrac{\xi\rho^2}{2} & \textrm{if $\xi > 0$ and $|x|>\xi\rho$}\\ \dfrac{|x|^2}{2\xi} & \textrm{if $\xi > 0$ and $|x|\le\xi\rho$}\\ \rho|y| & \textrm{if $x = 0$}\\ +\infty & \textrm{if $x < 0$}\end{cases}$

(with $N=1$ and $\rho>0$)

$\left\{\begin{aligned} &(0,0) &&\textrm{if $2\gamma\xi+|x|^2\le0$ and $|y|\le\gamma\rho$}\\ &\Big(x-\gamma\rho\operatorname{sign}(x),0\Big) &&\textrm{if $\xi\le -\gamma\rho^2/2$ and $|x|>\gamma\rho$}\\ &\Big(x-\gamma\rho\operatorname{sign}(x),\xi+\gamma\rho^2/2\Big) &&\textrm{if $\xi>-\gamma\rho^2/2$ and $|x|>\rho\xi+\gamma\rho(1+\rho^2/2)$}\\ &\operatorname{prox}_{\gamma|\cdot|^2/(\cdot)}(x,\xi) &&\textrm{if $\xi>-\gamma\rho^2/2$ and $|x|\le\rho\xi+\gamma\rho(1+\rho^2/2)$} \end{aligned}\right.$ [Function]
[Prox]
(coming soon) [Combettes et al., 2017]
Vapnik $\begin{cases} \displaystyle\inf_{y\in\mathbb{R}} \; |y| + \iota_{[-\varepsilon\xi,\varepsilon\xi]}(x-y) & \textrm{if $\xi \ge 0$}\\ +\infty & \textrm{if $\xi < 0$}\end{cases}$

(with $N=1$ and $\rho>0$)

$\left\{\begin{aligned} &(0,0) &&\textrm{if $\xi+\varepsilon|x|\le0$ and $|x|\le\gamma$}\\ &\Big(x-\gamma\operatorname{sign}(x),0\Big) &&\textrm{if $\xi\le-\gamma\varepsilon$ and $|x|>\gamma$}\\ &\Big(x-\gamma\operatorname{sign}(x),\xi+\gamma\varepsilon\Big) &&\textrm{if $\xi>-\gamma\varepsilon$ and $|x|>\varepsilon\xi+\gamma(1+\varepsilon^2)$}\\ &\Big(\varepsilon(\xi+\varepsilon|x|)\operatorname{sign}(x)/(1+\varepsilon^2),(\xi+\varepsilon|x|)/(1+\varepsilon^2)\Big) &&\textrm{if $|x|>-\xi/\varepsilon$ and $\varepsilon\xi\le|x|\le\varepsilon\xi+\gamma(1+\varepsilon^2)$}\\ &(y,\xi) &&\textrm{if $\xi\ge0$ and $|x|\le\varepsilon\xi$} \end{aligned}\right.$ [Function]
[Prox]
(coming soon) [Combettes et al., 2017]

φ-divergences

Description f(x,y)

(∀(x,y) ∈ ℝ2)

proxγf(x,y)

(∀ γ ∈ ℝ+)

Matlab Python Ref
Absolute
difference
$\left\{\begin{aligned} &|x-y| && \mbox{if $x\ge0$ and $y\ge0$}\\ &+\infty && \mbox{otherwise}\end{aligned}\right.$ $\small\left\{\begin{aligned} &\frac12\big(x+y+p,x+y-p\big) && \mbox{if $|p| < x+y$}\\ &(x-\gamma,0) && \mbox{if $x > \gamma$ and $y \le -\gamma$}\\ &(0,y-\gamma) && \mbox{if $y > \gamma$ and $x \le -\gamma$}\\ &(0,0) && \mbox{otherwise} \end{aligned}\right.$

where

$\small p=\operatorname{sign}(x-y)\max\big\{|x-y|-2\gamma,0\big\}$
[Function]
[Prox]
(coming soon) [El Gheche et al., 2018]
Hellinger $\left\{\begin{aligned}&(\sqrt{x}-\sqrt{y})^2 && \mbox{if $x\ge0$ and $y\ge0$}\\ &+\infty && \mbox{otherwise.}\end{aligned}\right.$ $\small\left\{\begin{aligned} &\Big(x + \gamma (p-1),y+\gamma \big(p^{-1}-1\big)\Big) &&\mbox{if $x \ge \gamma$ or $\left(1-\dfrac{x}{\gamma}\right) \left(1-\dfrac{y}{\gamma}\right) < 1$}\\[1em] &(0,0) && \mbox{otherwise} \end{aligned}\right.$

where

$\small\; p>\max\left\{1-\dfrac{x}{\gamma},0\right\}\;$

is such that

$\small p^{4}+ \left(\dfrac{x}{\gamma}-1\right)p^3 + \left(1-\dfrac{y}{\gamma}\right)p-1= 0$
[Function]
[Prox]
(coming soon) [El Gheche et al., 2018]
Chi-square $ \left\{\begin{aligned} &\dfrac{(x-y)^2}{y} &&\textrm{if $x \ge 0$ and $y>0$}\\ &0 &&\textrm{if $x = y = 0$}\\ &+\infty &&\textrm{otherwise}\end{aligned}\right.$ $\small \left\{\begin{aligned} &\Big(x + 2 \gamma(1-p),y+\gamma (p^2-1)\Big) &&\mbox{if $x > -2\gamma$ and $y > - \left(x+\frac{1}{4\gamma}x^2\right)$}\\[1em] &\big(0,\max\{y-\gamma,0\}\big) && \mbox{otherwise} \end{aligned}\right.$

where

$\small\; p \in \left]0,1+\dfrac{x}{2\gamma}\right[ \;$

is such that

$\small p^3+\left(1+\dfrac{y}{\gamma}\right)p - \dfrac{x}{\gamma} -2 = 0$
[Function]
[Prox]
(coming soon) [El Gheche et al., 2018]
Kullback-Leibler $ \left\{\begin{aligned} &x \log\Big(\dfrac{x}{y}\Big) && \mbox{if $x>0$ and $y>0$}\\ &0 &&\mbox{if $x=0$ and $y \ge 0$}\\ &+\infty && \mbox{otherwise}\end{aligned}\right.$ $\small \left\{\begin{aligned} &\Big(x + \gamma \log(p) -\gamma,y+\gamma p^{-1}\Big) && \mbox{if $\exp\left(\dfrac{x}{\gamma}-1\right) > -\dfrac{y}{\gamma}$}\\[1em] &(0,0) && \mbox{otherwise} \end{aligned}\right.$

where

$\small\; p>\exp\left(-\dfrac{x}{\gamma}-1\right) \;$

is such that

$\small\;p \log(p) + \left(\dfrac{x}{\gamma}-1\right)p - p^{-1} - \dfrac{y}{\gamma} = 0$
[Function]
[Prox]
(coming soon) [El Gheche et al., 2018]
Jeffrey $\small \left\{\begin{aligned}&(x-y) \Big(\log(x)-\log(y)\Big) && \mbox{if $x>0$ and $y>0$}\\ &0 && \mbox{if $x=y=0$}\\ &+\infty && \mbox{otherwise.}\end{aligned}\right.$ $\small \begin{cases} \Big(x + \gamma \big(\log(p) + p-1),y-\gamma \big(\log(p) -p^{-1}+1)\Big) &\mbox{if $W(e^{1-\gamma^{-1}x})W(e^{1 - \gamma^{-1}y}) < 1$}\\[1em] (0,0) & \mbox{otherwise} \end{cases}$

where

$\small\; p\in\left[W(e^{1-\gamma^{-1}x}),\Big(W(e^{1 - \gamma^{-1}y})\Big)^{-1}\right] \;$

is such that

$\small (p+1) \log(p) - p^{-1} + p^2 + \Big(\dfrac{x}{\gamma}- 1\Big) p + 1 - \dfrac{y}{\gamma} = 0$
[Function]
[Prox]
(coming soon) [El Gheche et al., 2018]
Rényi $ \left\{\begin{aligned} &\dfrac{x^\alpha}{y^{\alpha-1}} &&\textrm{if $x \ge 0$ and $y>0$}\\ &0 &&\textrm{if $x = y = 0$}\\ &+\infty &&\textrm{otherwise}\end{aligned}\right.$ $\small \left\{\begin{aligned} &\left(x - \frac{\gamma \alpha}{ p^{\alpha-1}},y+ \frac{\gamma (\alpha-1)}{p^{\alpha}}\right) &&\mbox{if $x>0$ and $\dfrac{\gamma^{\frac{1}{\alpha-1}} \, y}{1-\alpha} < \left(\dfrac{x}{\alpha}\right)^{\frac{\alpha}{\alpha-1}}$ }\\[1em] &\big(0,\max\{y,0\}\big) && \mbox{otherwise} \end{aligned}\right.$

where

$\small\; p > \Big(\dfrac{\alpha \gamma}{x}\Big)^{\frac{1}{\alpha-1}} \;$

is such that

$\small\dfrac{x}{\gamma}\;p^{\alpha+1} - \dfrac{y}{\gamma}\,p^{\alpha} - \alpha p^{2} + 1-\alpha = 0$
[Function]
[Prox]
(coming soon) [El Gheche et al., 2018]
$I_\alpha$ $ \left\{\begin{aligned} &-\sqrt[\alpha]{x \, y^{\alpha-1}} &&\textrm{if $x \ge 0$ and $y\ge0$}\\ &+\infty &&\textrm{otherwise}\end{aligned}\right.$ $\small \left\{\begin{aligned} &\left(x +\frac{\gamma}{\alpha}\big(p^{\alpha-1}-1\big),y + \frac{\gamma(\alpha-1)}{\alpha} \big(p^{-1}-1\big)\right) &&\mbox{if $\alpha x \ge \gamma\;$ or $\;1-\frac{\alpha y}{\gamma(\alpha - 1)}<\left(\frac{\gamma}{\gamma-\alpha x}\right)^{\frac{1}{\alpha-1}}$}\\[1em] &(0,0) && \mbox{otherwise} \end{aligned}\right.$

where

$\small\; p > \left(\max\Big\{1- \dfrac{\alpha x}{\gamma},0\Big\}\right)^{\frac{1}{\alpha-1}}\;$

is such that

$\small p^{2\alpha} + \left(\dfrac{\alpha x}{\gamma}-1\right) p^{\alpha+1} + \left(\alpha-1-\dfrac{\alpha y}{\gamma}\right) p -\alpha+1=0$
[Function]
[Prox]
(coming soon) [El Gheche et al., 2018]