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Let $d_1,d_2\in \mathbb{N}_+$, a stopping time $\tau$, and consider a system of BSDEs \begin{align} Y_{\tau}^1 & = \xi^1+\int_{t\wedge \tau}^{\tau}\, f_1(t,Y_t^1,Z_t^1,Y_t^2,Z_t^2)dt - \int_{t\...
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Let $T$ be some random variable on $[0,1]$, and define \begin{equation} \alpha(t) \triangleq \mathbb{E}[T \vert T\le t],\\ \beta(t) \triangleq \mathbb{E}[T \vert T>t], ~t\in[0,1]. \end{equation} ...
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A coherent risk measure named Entropic Value-at-Risk was introduced as follows: Let $(\Omega,\mathcal{F},\mathbf{P})$ be a probability space, $X$ be a random variable and $\beta$ be a positive ...
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I am working on a bilinear inverse problem arising in multi-channel signal processing. My problem background is to reconstruct a certain one-dimensional information $\mathbf{w} $ of an object from ...
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Consider a coin that comes up heads with probability $0 < p < \frac{1}{2}$. Fix some integer $N > 0$. We choose in advance a number of flips to run. Write $H, T$ for the total number of heads ...
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I am looking for comparision results for nonlinear integral Volterra equations with parameters. This was partially motivated by this paper. There, the author establishes, under mild hypothesis, the ...
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I want to establish some useful criteria for uniqueness of solutions to the following: $$Mx=b,\\ \text{subject to}\ ||x(2k-1:2k)||=1, k=1,2,\cdots,5,$$ where $M\in\mathbb{R}^{10\times10},\ x\in\mathbb{...
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We want to get from 0 to 1 on the real axis with a moving point $P(x(t))$, that moves only in the right direction, as soft as possible in a minimum time. We introduce the class $\mathcal{S}$ ...
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I'm studying the OptNet paper (Amos & Kolter, 2017), which integrates quadratic programs (QPs) into neural network layers and enables end-to-end learning through differentiable optimization. In ...
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Assume that matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}=\mathbf{A}\mathbf{B}$ are given. I aim to find matrices $\mathbf{E}_1$, $\mathbf{E}_2$, and $\mathbf{E}_3$ such that \begin{align} \...
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Consider measures $\mu$ and $\nu$ in $\mathbb{R}^d$ with equal mass and no atoms, supported on a compact set, and make additional reasonable assumptions as necessary. Consider the optimal transport ...
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Let $(W_t)_{t\ge 0}$ be a one-dimensional standard Brownian motion on its natural filtration $(\mathcal{F}_t)$. For fixed constants $$0 < \underline{\sigma} < \overline{\sigma} < \infty,$$ ...
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The question is related to this algorithm in linear optimization. In the algorithm, projective transformations are used as said by wikipedia: Since the actual algorithm is rather complicated, ...
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I posted this question a few days ago to https://or.stackexchange.com/questions/13173/optimization-over-loop-spaces but didn't receive any replies, so I thought I would try here (if this is improper ...
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I'm stumped by the following variational problem which came up in the course of my research. Let $X_1, X_2 \in \mathbb{R}^{m \times d}$ and $Y_1, Y_2 \in \mathbb{R}^{n \times d}$ be fixed matrices of ...
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I have functions $f : \Bbb{R}^n \to \Bbb{R}_{\geq 0}$ and $g : \Bbb{R}^n \to \Bbb{R}_{\leq 0}$ and would like to find a $\mathbf{x}_* \in \Bbb{R}^n$ where $g(\mathbf{x}_*)=0$ and $f \left( {\bf x}_* \...
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I am trying to wrap my head around some features of singular stochastic control, and one of the things that has been bothering me is that authors sometimes take the singular control to be left-...
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I’ve been studying the optimal transport problem and I understand that in one dimension it can be solved quite easily: because $\mathbb{R}$ is totally ordered, the cumulative distribution function ...
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Given vectors ${\bf p}_1, {\bf p}_2, \dots, {\bf p}_n \in {\Bbb R}^d$ and ${\bf q}_1, {\bf q}_2, \dots, {\bf q}_n \in {\Bbb R}^d$, define the $d \times n$ matrices $$ {\bf P} := \begin{bmatrix} ...
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In engineering we are mainly interested in linear time-invariant (LTI) systems which are bounded input, bounded output (BIBO). It's easy to prove that BIBO condition is equivalent to $$\int_{-\infty}^{...
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Let $f$ be an $\rho$-weakly convex function, $f:\mathbb{R}^d\to \mathbb{R}$. The Moreau envelope of $f$, $f_\lambda$ for a parameter $0<\lambda<1/\rho$ is defined as $$ f_{\lambda}(x) := \min_{y}...
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Problem Statement Given a positive integer $n \geq 3$, consider a permutation $\pi = (a_1, a_2, \ldots, a_n)$ of $\{1, 2, \dots, n\}$. For each $i$ ($1 \leq i \leq n-1$), define $d_i$ as the minimum ...
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whether we can find postive definite $\{A_k\}$ such that $\frac{\kappa(\lambda_{\max}((I+\alpha D_k)^{-1}(I-\alpha(A_k-D_k))))}{\kappa(A_k)}=\Theta(n)$. Here $A_k\in\mathbb{R}^{n\times n}$ is positive ...
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I thought of the following problem. Due to its simple and natural formulation, I guess that it must had been already studied, but I could not find any reference. I'm interested in knowing what is the ...
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Let $f : \mathbb{R}^n \to \mathbb{R}$ be a locally Lipschitz function, and let $\bar{x} \in \mathbb{R}^n$. Assume the directional derivative $$ f'(\bar{x}; v) := \lim_{t \downarrow 0} \frac{f(\bar{x} +...
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Let $a,\varphi\in \mathbb{R}^N$. Consider the trigonometric polynomial $$f(\phi;t):=\sum_{n=1}^Na_ne^{i \phi_n} e^{i nt}.$$ My question is: what can be said about the quantity $$\omega(a)=\inf_{\phi\...
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More precisely, assume that $f:X\times Y\to \mathbb{R} \cup \{+\infty \}$ is a function where $Y\subset X$ is compact, $\partial_{x}^{C}f\left( x_{0},y\right) $ is the Clarke subdifferential of at $...
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Let $n,N\in \mathbb{N}_+$. What is the minimal diameter $r\ge 0$, in $\ell^{\infty}$, of $[0,r]^n$ which there are $N$ points at a pairwise distance of $1$? Specifically, are there known estimates ...
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Consider the following cooperative game played on the circle $S^1$, which we identify with $[0, 1]$ with its endpoints identified. Let $0 < \alpha < 1$ be an irrational number. A total of $N \...
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I was wondering how we can solve this control problem with equality integral constraints: $$ \min _{y \in K} J(y)=\left\|y-y_d\right\|_{0, \Omega}^2+\|u\|_{0, \Omega}^2 $$ subject to $$ \left\{\begin{...
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I often notice tankers of the type illustrated in the figure below. The cross section is neither circular nor elliptical. Is it a "notable" geometric shape? Which function or property does ...
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Given $n < m < n^2$, fat rank-$m$ matrix ${\bf A} \in {\Bbb R}^{m \times n^2}$ (that has full row rank) and vector ${\bf y} \in {\Bbb R}^m$, $$\begin{align} \underset{{\bf X} \in {\Bbb R}^{n \...
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Let $A \in \mathbb{R}^m$ be some set with the property that it is easy (polynomial-time computation) to generate random elements $r \in A$. Is it then also easy to compute $$ P_A(x) := \arg \min_{y\in ...
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Problem Formulation Given real values $ z_{i,j}^{(k)} $ for indices $ i = 1,\ldots, n, \quad j = 1,\ldots, m, \quad k = 1,\ldots, p, $ our goal is to estimate the parameters $\alpha_{i}^{(k)}$, $\...
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Given an (real) algebraic variety $$ V := \left\{ x \in \mathbb{R}^{n} (\mathbb{C}^{n}) \mid h_{1}(x) = 0, \dots, h_{m}(x) = 0 \right\} $$ where $h_{1},\dots,h_{m} \in \mathbb{R}[x_{1},\dots,x_{n}]$, ...
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I have the following discrepancy that satisfies this type of inequality. I want to know if this can be related to some kind of weak convexity. If so, what is the name of this property? Additionally, ...
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Assume that I have a graph $G(V, E)$, where each vertex $v$ is assigned a non-negative weight $w(v) \geq 0$. I aim to find a partition of the graph into cliques $C_1, C_2, \ldots, C_k$ (Here, $k$ is ...
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Nate is given a chance to play a game - there are $n \geq 2$ turns. On each turn, he is given a number, uniformly drawn from $[0, 1]$ independent of other draws. On that turn, he may choose to either ...
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Suppose I have a linear program of the form $\max c^Tx$ such that $Ax \leq b$. I am curious as to the continuity of the solutions to such a linear program under perturbations of the entries of $A$. ...
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I am working on a quadratically constrained quadratic program (QCQP) of the form:$$ \min_{x} \quad \frac{1}{2} x^T P x + q^T x + r$$ $$ \text{subject to} \qquad x^{T}x \leq 1 $$ where $P \in S^{++}_{...
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Consider a rational map $A : S^1 \to U(n)$, i.e. a matrix of rational functions such that evaluation at any $z \in S^1 \subset \mathbf C$ is unitary. These objects show up in digital signal processing ...
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When reading the paper Control for Schrödinger operators on tori by N.Burq and M.Zworski (Math. Res. Lett., 19(2):309–324, 2012), an inequality confused me: Define the flat torus $\mathbb{T}^2=\mathbb{...
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Suppose I have a quadratic program (with positive semidefinite cost matrix) with affine (polytopic) constraints. It is known that the solution to this is piecewise affine, with the ``pieces'' defined ...
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I am faced with an information-theoretic upper bound, such as \begin{align} \sqrt{\alpha'}2^{I_\alpha(X;Y)}, \end{align} where $I_\alpha(X;Y)$ is the Rényi mutual information with parameter $\alpha>...
Math_Y's user avatar
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Let $\mathcal{A} = \{ A_1, \ldots, A_J, : A_j \in \mathbb{R}^{s\times s}\}$, and define the lower spectral radius (LSR) (aka joint subradius) by $LSR(\mathcal{A}) = \lim_{n\rightarrow\infty} \inf_{A_{...
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The problem Assume $p > 1$. Consider the function $$f(x,y) = x^p y^{1-p}, \qquad x,y > 0.$$ Note that $$ f'' = p(p-1)x^{p-2}y^{-1-p} \begin{bmatrix} y \\ & x \end{bmatrix} \begin{bmatrix} 1 &...
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I'm working with a high-dimensional covariance matrix and exploring Kronecker product approximations to make it computationally manageable. Here's the setup: I have a graph $G$ represented by a $D\...
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I have a continuously differentiable curve $r : \mathbb{R} \to \mathbb{R}^{n}$, $n \ge 2$, with the properties \begin{align} \lvert r'(t) \rvert = 1 \text{ for all } t \in \mathbb{R}, \\ \lvert r'(t) -...
node's user avatar
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I am working on a binary optimization problem. So far I have derived the following constraint functions. \begin{align} \begin{bmatrix} \left( P + \sum_{i=1}^n (\sum_{j=1}^n x_{i, j} \alpha_j) e_i e_i^...
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Nate has $n \geq 2$ coins $\{C_i\}_{0 \leq i \leq n-1}$ that each turn up heads with probability $\frac{i}{n-1}$ each, but he is not sure which ones are which. He has \$1 with which to bet with. On ...
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