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Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator $$ \begin{aligned} &\Gamma_0\rig …

Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with met …

Fix a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and let $F:L^2_{\mathbb{P}}(\mathcal{F})\rightarrow (-\infty,\infty]$ be bounded-blow, convex, lower semi-continuous, and not identically infi …

Setup Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by $$ f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\c …

Let $X$ be a Hilbert space, and suppose that $f:X^2\rightarrow \mathbb{R}$ is a Lipschitz, supercoercive, convex function such that (for every $y \in X$) the set $$ \operatorname*{argmin}_{x\in X} f(x …

Let $F$ be a lower semicontinuous functional on a Banach space $X$, wrt its strong topology. Is there a known form for the relaxation (lower semicontinuous envelope) of $F$ with respect to the weak t …

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$, $$ f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\}, $$ is $C^1$ if …

Suppose I have an optimization problem of the form $$ \inf_{\{x \in \mathbb{R}^d: g(x)=0\}} f(x), $$ for some convex function $f$ and non-convex l.s.c. function $g$. Can we reinterpret the Lagrang …

If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then $$ \operatorname{argmin}_{x \in X} f(x) = \operatorname{argmin}_{x \in X} g\circ f(x) . $$ X and Y …

In this post one says that a functional $F:H\rightarrow [0,\infty]$ on an infinite-dimensional Hilbert space $H$ is (strongly) coercive if there exists a constant $k>0$ such that $$ F(x)\geq k\|x\|_H^ …

Suppose that I know $f_n\rightarrow f$ and $g_n\rightarrow g$ are both continuous maps from a Complete Riemmanian Manifold $X$ to $\mathbb{R}$ which converge pointwise almost everywhere. Then is it t …

Most (stochastic) "gradient descent" type algorithms (such as Nesterov-accelerated gradient-descent or ADAM) seem to be well-defined only for functions which are either: lower semi-continuous + conve …

Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex de …

Let $f:H\to \mathbb{R}$ be a strictly convex Fréchet differentiable, coercive function on a separable Hilbert space $H$ and let $C_1,C_2\subseteq H$ be closed and convex. I want to optimize $$ \tag{(A …

Is there a characterization of all continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying: $f(0)=0$ $f$ is monotonically increasing $f$ is concave My intuition is that $f$ should admit …