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$\newcommand\diag{\operatorname{diag}}$Write $X^{-1}$ as $Y_1-Y_2$, where $Y_1$ and $Y_2$ are positive-semidefinite symmetric matrices. Detail: This can be done for any nonsingular symmetric matrix $ …

This inequality is not true. E.g., $$g\Big(\frac1{10},\frac1{10},\frac\pi4\Big)=-1783.256\ldots\not\ge1.$$ The OP has changed the problem, thus invalidating the previous answer. After the change, the …

$\newcommand\al\alpha$Let $$u:=\sum_{1\le i<j\le6}c_{ij}^2\quad\text{and}\quad v:=\sum_{1\le i<j<k\le6}c_{ij}c_{ik}c_{jk},$$ where $c_{ij}:=\cos\al_{ij}$. This answer is somewhat similar to the previ …

Write $a,b,c$ instead of $\alpha,\beta,\gamma$. Let $p(t)$ be the polynomial in question. Let $$u:=\cos^2a+\cos^2b+\cos^2c,\quad v:=\cos a\,\cos b\,\cos c,$$ $$t_*:=\frac{1}{30} \left(15-\sqrt{15} \sq …

$\newcommand\tr{\operatorname{tr}}$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 …

Let $c:=\Phi>0$ and $a:=t_{\max}>0$. For any given value (say $m$) of $ET$, the variance $VT$ of $T$ is maximized when $T$ takes only the endpoint values, $0$ and $a$, so that $P(T=a)=m/a=1-P(T=0)$, a …

$\newcommand{\n}{\lfloor{r/c}\rfloor}$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*} …

Section 1.3 of Chapter Nonsmooth Optimization by V. F. Demyanov in the book Nonlinear Optimization discusses in some detail the Dini, Hadamard, Shor, Clarke, and Michel–Penot notions of the subdiffere …

$\newcommand\R{\mathbb R}$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 …

Not in general. E.g., let $n=3$, $A(x_1,x_2,x_3):=1-x_1 x_2+2 x_3 x_2+x_1 x_3$, and $B(x_1,x_2,x_3):=2-x_1 x_2+x_3 x_2+2 x_1 x_3$. Then for $x_3\in(0,2\sqrt2)$ $$G(x_3)=\max_{t\in\mathbb R}\frac{A(t, …

We only need to consider $n\ge5$. Let us move each of the $n$ "equidistant points" on the equator slightly towards one of the two poles of the globe, so that after such a movement the points on the sp …

The minimum here can be found exactly, in a finite number of steps. Indeed, the target function is concave. So, its minimum on the (compact) (transportation) polytope $P:=\Pi(\alpha,\beta)$ is attaine …

$\newcommand\R{\mathbb R}\newcommand\tz{\tilde z}\newcommand{\de}{\delta}$Yes, the minimizer $x_{A,b}$ of $\frac12 x^TAx + b^Tx$ subject to $Cx\le d$ is continuous with respect to $A$ and $b$ -- provi …

The derivative $f'_n$ of $f_n$ is a nonzero real-analytic (and even entire) function. Therefore, $f'_n$ can have only finitely many zeroes in any bounded interval. So, all the local maxima of $f_n$ ar …