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$\newcommand\R{\Bbb R}\newcommand\la{\lambda}$Remark 1: Let \begin{equation*} g(s):=q^T(P+sI)^{-2}q \end{equation*} for real $s$ such that $P+sI$ is invertible. The function $g$ is nonnegative an …

$\newcommand\Om\Omega\newcommand\R{\Bbb R}$The answer is no. Indeed, for real $k>0$, let $$G_k:=\{(x,y)\in\Bbb R^2\colon x>1,|y|<k\sqrt x\}. $$ For $(x,y)\in G_2$, let $$f_0(x,y):=y^2/x-1.$$ For all $ …

$\newcommand\opt{\operatorname{opt}}\newcommand\de{\delta}\newcommand\si{\sigma}$The answer is no. Indeed, take any $h\in(0,1)$. Let $X=\{1,2,3,4\}$. Suppose that $$\si_0=\frac12\de_1+\frac12\de_4,\qu …

we could greedily solve the first two transport problems separately and then compose the plans, but I suspect this not optimal. Indeed, this is not optimal in general. Here is an example: $X=Y=Z=\{ …

$\newcommand\tr{\operatorname{tr}}$Let $$Q(B):=(Y-XB)^T(Y-XB).$$ Since the column spaces of the matrices $X^TX$ and $X^T$ are the same, there is a matrix $B_*$ such that $$X^TXB_*=X^T Y.$$ For each $z …

Your question is Find the minimum of $f(x,y)=(x-2)^2+y$ subject to $y-x^3\ge0$, $y+x^3\le0$ and $y\ge0$. The restrictions $y-x^3\ge0$, $y+x^3\le0$, and $y\ge0$ can be rewritten as $0\le y\le-x^3$ or …

For any of your diagonal matrices $D$, let $J:=J_D$ be the set such that $D_{i,j}=1(i=j\in J)$ for all $i,j$ in the set $[N]:=\{1,\dots,N\}$, where, for any matrix $M$, its $(i,j)$-entry is denoted by …

The answer is yes. Indeed, the cdf $F$ of the binomial distribution is log concave -- see e.g. Theorem 2 on p. 152, used with $\alpha=1$, $r=\infty$, and $q$ being the probability mass function of the …

Rewrite the constraint as $$x_n \le f_n(\rho_1,\dots,\rho_n):=\ln\Big(B\log_2\Big(1+\frac{e^{\rho_n} g_n^2}{\sum_{i=1}^{n-1} e^{\rho_i} g_i^2+\sigma^2}\Big)\Big).$$ The problem is then to prove the co …

In part (P3) of Definition 2.1 in the paper you linked, it is also required that $g$ be $\partial$-differentiable at $x$ (meaning that both $\partial g(x)$ and $\partial(-g)(x)$ are nonempty), which i …

$\newcommand{\n}{\lfloor{r/c}\rfloor}$We have \begin{equation*} s_{r,c}=\sqrt{\n c^2+(r-\n c)^2}. \end{equation*} Indeed, by continuity and interchangeability of the coordinates, \begin{equation*} …

A counterexample is given by $n=2$, $[a_1,b_1]=[-2,1]$, $[a_2,b_2]=[-1,2]$. (Make a picture.) Even if the $n$-box $S$ is required to be a subset of $[0,\infty)^n$, the answer will still be no. E.g., …

$\newcommand\conv{\operatorname{conv}}\newcommand\ext{\operatorname{ext}}\newcommand\p{\partial}$The answer is yes. Indeed, let $K:=\conv P$ (the convex hull of $P$), let $\p K$ be the boundary of $K$ …

$\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}\newcommand{\al}{\alpha}\newcommand{\be}{\beta}\newcommand{\tal}{\tilde\al}\newcommand{\ga}{\gamma}\newcommand{\K}{\mathfrak K}$The answer is yes -- …

$\newcommand\R{\mathbb R}$The answer is no. E.g., let $A=B=\R$, $B_a=[1-a^2,2]$ for $a\in A$, and $\ell(b)=b^3-3b$ for $b\in B$. Then $\ell(b)$ and $B_a$ are perfectly smooth, but $L$ is not different …