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Let $(X,P)$ be a probability measure space. Let $K$ be a convex compact subset of $\mathbb R^d$ and let $F:X \to 2^{K}$ be a set-valued map. Assume that $F$ is: closed (i.e $F(x)$ is closed for ...
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I am trying to compute the nonnegative quantity $$ \underset{y\in\mathbb{S}^{d-1}}{\sup}\int_{0}^{t}(\Vert A(\tau)y\Vert_{1}- \Vert A(\tau)y\Vert_{q})d\tau, \quad 1 < q < \infty $$ where $\...
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Suppose I have the following optimization problem $$ \min\limits_{\mathbf{x},\mathbf{y}} f(\mathbf{x},\mathbf{y}) \tag{1} $$ It is already known that the target function $f$ is continuous and ...
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Inspired by some recent real-world situations, I thought of this problem: An adversary has selected a positive real number $p \ge 1$ not known to you. You have to pick numbers $x_1, x_2, \dots$ in ...
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Let $X \subseteq \mathbb{R}^n$ be compact and say the function $f \colon X \to \mathbb{R}$ is locally Lipschitz continuous. Say $\mathcal{X}$ is the set of all solutions of the following optimization ...
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Let $Y,Z$ be $n\times k$ matrices and assume all columns have been standardized such that their means are zero and variances 1. I seek an $n\times n$ permutation matrix $P$ such that $$\left\Vert Y^{T}...
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Chow's theorem gives a criterion for the reachable set of the points of a manifold $M$ to be full. Specifically, if we have a distribution $D$ whose iterated commutators span the tangent bundle then ...
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Let $X$ be an infinite-dimensional Banach space. I have the following optimization problem. $$\begin{array}{ll} \underset{x \in X}{\text{minimize}} & f(x)\\ \text{subject to} & g_1(x) \leq 0\\ ...
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Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
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Assuming that an area $A$ on the plane has a known density distribution function $\rho (x, y)\geqslant 0$, now the goal is to obtain $n$ points $p_1, p_2, ..., p_n$ on the area so that $\iint_{}^{} \...
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We consider the wave equation $$\left\{ \begin{array}{ll} u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\ u=0, \quad u \in \partial \Omega, t>0 \\ u(0,x)=u_{0}(x), \quad u_{t}(0,x)=u_{1}(x), x \...
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The gradient descent (GD) with constant stepsize $\alpha^{k}=\alpha$ takes the form $$x^{k+1} = x^{k} -\alpha\nabla f(x^{k}).$$ Then, by constructing a continuous-time version of GD iterates ...
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I have been reading the paper of Goerner and Ulancowicz - "Quantifying economic sustainability" in which it is suggested that there is a tradeoff between sustainability and efficiency. ...
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This question is motivated by the consideration of linear control systems in Brunovsky normal form. The idea is that you have $m$ smooth functions with unconstrained dynamics, and the control input ...
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Let $A$ be an $m\times n$ matrix with entries in $F$ ($F=\mathbb{R}$ or $F=\mathbb{C}$). It is well-known that $A$ has decompositions of the form $$\displaystyle A = \sum_{k=1}^r\lambda_k\hspace{2mm} ...
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Problem Description I want to hold meetings where some given number of people will participate. They have some vacant dates respectively but they don't have the same date on which all of them can ...
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Some years ago, there was that question on this forum:"How to quantify noncommutativity?". I am asking that question in a context, human movement, which implies kinematic chains (like in ...
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Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem? \begin{align} \min_{\mathbf{...
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Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
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I have a function $h(y,x_1,x_2,\ldots,x_n)$. It is known that the minimum value of $h$ for any $y$ is attained when $x_1 = x_n$ and $x_2 = x_3 = \cdots = x_{n-1}$. Now consider the following function \...
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I want to implement the following optimization problem from the following paper Randomized Gossip Algorithms, Page 10 Eq 53: \begin{align} \text{minimize} &\qquad s\\ \text{subject to} & \...
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Traditional infeasibility cut is derived under the assumption that the feasibility problem is LP instead of ILP and relies on the dual form of the LP. Is there a systematic way of adding valid cuts ...
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I am having a hard time determining the motion of $X$ in the ODE $X''+\nabla f(X)=0$ with initial conditions arbitrary $X(0)$ and zero velocity, i.e. $X'(0) = 0$. $X$ is in $\mathbb{R}^{n}$, and $f$ ...
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Consider the linear time invariant system: $$\tag{1}\label{eq1} \dot{x}(t) = Ax(t) + Bu(t), \ \ x(0)=x_0\in\mathbb{R}^n, $$ where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$. Let $p_M(s)...
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Let $f$ be a balanced Boolean function. Are there $g$ linear functions, with $$\frac1{2^n}\mathrm{card} \big(\big\{\mathrm{sign} (g (x)) = 2f (x) -1, x \in \{0,1\}^n\big\}\big) > 0.55\quad ?$$ $g ...
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Original question Given $f(x):\mathbb{R}^+\to\mathbb{R}^+$, which is monotonically decreasing and convex. Then define a function $g(x) = xf(x)$, I am wondering whether $g(x)$ is quasiconvex for $x>...
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In the context of LP-formulations for the Traveling Salesman Problem the MTZ constraints prevent subtours via $n$ (i.e. effectively $n-1$) additional variables $$u_1=1\\2\le u_2,\,\dots ,\,u_n\le n\\ ...
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What is the relationship between the controllability of the ODE $$\dot x(t) = v(x) + u(t)$$ using a control $u$ and the controllabilty of the transport equation $$\rho_t(t,x) + \mathrm{div}(v(x) \rho(...
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Is there a way to efficiently check if all matrices in the following set are Hurwitz stable (eigenvalues strictly in the left-hand plane)?$$\left\{ A \in \Bbb R^{n \times n} : \ell_{i,j}\leq A_{i,j} \...
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The following problem is motivated from Generalized Procrustes Analysis. I am wondering if it is possible to obtain a closed form minimizer (which may involve SVD or some other decomposition of a ...
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Given a p.s.d. matrix $A$, one may want to find: $$ \max_x x^t A x \mbox{ such that } x \mbox{ has entries }1 \mbox{ or } {-1}. $$ This hard problem has a well known relaxation based on the so called ...
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As we all know, a classic optimization problem can be represented in the following way: Given a function $f: A \to \mathbb{R}$, find an element $x_0 \in A$ such that $f(x_0) \le f(x)$ for all $x \in ...
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Given real symmetric matrix $\mathbf{R} \in \mathbb{S}^{n\times n}$ and matrices $\mathbf{X}_n, \mathbf{X}_{n-1} \in \mathbb{R}^{n \times m}$, $$\begin{array}{ll} \underset{\mathbf{A} \in \mathbb{R}^{...
Universal_Bulin's user avatar
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If I had to partition the unit square $[0,1]\times[0,1]$ into $k^2$ rectangles such that the sum of their diagonals is minimum possible, I would simply choose the $k \times k$ grid of squares. Now ...
bleh's user avatar
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Suppose we have $d$ cylindrical metal bars, with radius $l$, attached orthogonal to a support in random places: Now we have to attach bars with radius $k$ EVENLY SPACED, with distance $p$ between ...
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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 ...
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I came across the algorithmic perfomance analysis model of smoothed analysis. In all references that I read a Gaussian distribution was used for perturbation (e.g. Spielman and Teng 2004 for the ...
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This question is a continuation of Minimization of an entropy type functional Let $\mathcal P_c$ be the set of probability densities on $[0,1]$ with mean $c\in [0,1]$, i.e. $p\in \mathcal P_c$ iff $$\...
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Let $\mathcal P$ be the set of probability densities on $[0,1]$ with mean $1/2$, i.e. $p\in \mathcal P$ iff $$\int_0^1 p(x)dx=1,\quad \int_0^1 xp(x)dx=\frac{1}{2}\quad \mbox{and}\quad p(x)\ge 0, ~~\...
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As we all know, a classic optimization problem can be represented in the following way: Given: a function $f: A \rightarrow \mathbb{R}$ from some set $A$ to the real numbers Sought: an element $x_0 ∈ ...
Shaun Han's user avatar
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To recall Arrow's theorem: Suppose we have a finite set $X$ of voters and a finite set $Y$ of candidates. An election is a map $\phi: X \rightarrow T$ where $T$ is the space of total orderings of $Y$. ...
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Let us consider the integral operator $T:\mathbb{R}^{n\times d}\to [0,\infty)$ such that for all $K\in \mathbb{R}^{n\times d}$, $$ T(K)=\int_0^1 \operatorname{tr}(KK^\top \Sigma_t) \,d t, $$ where $\...
John's user avatar
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Let $A=[a_{ik}]$ be a matrix with the consecutive ones property in each column, i.e. each column consists of a single consecutive block of $1$'s (with zeros everywhere else). Is there anything at all ...
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We consider a function $f(s_{1:p}, a_{1:p})$, where $p>1$ is an integer, $s_{1:p}$ denotes $(s_1,\ldots,s_p)^\top \in R^p$, and $a_{1:p}$ denotes $(a_1,\ldots,a_p)^\top \in R^p$. Question: What is ...
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Consider the deterministic controlled system: $$\dot x(t) = Ax(t) + Bu(t), \ t \in [0, T]$$ $$x(0) = x_0$$ where $x: [0, T] \to \mathbb R^n$ is the controlled state process, $A \in \mathbb R^{n \times ...
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According to Wikipedia, a function $f: \mathbb{R}^n \to \mathbb{R} \cup \{-\infty, +\infty\}$ is called coercive if, $$f(x) \to +\infty \text{ as } \|x\| \to +\infty$$ and it is super-coercive if $$\...
Norman's user avatar
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Suppose $A$ is a diagonal matrix with eigenvalues $1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n}$ and $x$ is drawn from standard Gaussian in $n$ dimensions. Define $z_n$ as follows $$z_n=E_{x\sim \...
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I’m looking for a nice readable introductory text to stochastic control theory. Background wise, I know some general stochastic analysis and deterministic optimal control theory. Some criterion I’m ...
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Nate River
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When solving the LQR problem to find the optimal feedback matrix $K^*$, i.e. solving \begin{align*} \min_K &\int_0^\infty \left(x^TQx + u^TRu\right)dt,\\ \text{s.t. }&\dot{x} = Ax+Bu,\\ &u=...
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Given matrices $A, B \in \mathbb{R}^{n\times n}$, I would like to solve the following optimization problem, $$\begin{array}{ll} \underset{v \in \mathbb{R}^n}{\text{maximize}} & \|Av\|_2+\|Bv\|_2\\ ...
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