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Operations research, linear programming, control theory, systems theory, optimal control, game theory

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13 views

A parametrization of a class of stable matrices

Let $A\in\mathbb{R}^{n\times n}$ be a non-derogatory matrix with real and strictly negative eigenvalues. Furthermore, for simplicity, suppose that $\mathrm{tr}(A)=-1$. Q. I'm wondering whether it ...
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1answer
105 views

Linear matrix inequality

I have the following linear matrix inequality: $$F^T P + PF < 0,$$ where $P$ is a positive definite matrix and $F$ is a matrix with appropriate dimension. Let $Q$ be a positive definite matrix ...
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0answers
63 views

An inequality concerning the solution of a Lyapunov equation

Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
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0answers
46 views

Using mollifiers (or some other idea) to solve constrained minimax problem

Sorry in advance if this sounds like a more SE question. Consider a continuously parametrized family of $L$-Lipschitz continuous $f_\theta: X \rightarrow \mathbb R_+$ on a metric space $X=(X,d)$. Let ...
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0answers
32 views

Minimum Preserving Transformations

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 ...
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2answers
349 views

A trace-constrained maximization problem in the cone of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...
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0answers
92 views

Bounds of Procrustes problem

We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following: \begin{align} f(A,B)...
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1answer
102 views

How to solve this system of Matrix equations? (Coupled riccati equations?)

I am trying to solve for K in the following problem: $ 3I = A_1 + A_2 + A_3$ $ A_1 K A_1 = K_1 $ $ A_2 K A_2 = K_2 $ $ A_3 K A_3 = K_3 $ Where $I$ is the identity, $K, K_1, K_2, K_3, A_1, A_2, ...
3
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2answers
224 views

A structural optimization problem

I have $n$ objects $O_i$, each of them having $3$ values, $O_i = (A_i, B_i, C_i)$. I am trying to group them into $k$ groups $P_u$ such as $P_u =(A_u, B_u, C_u)$ such that $$\text{minimize} \quad \...
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0answers
43 views

Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element

In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks. Consider the system consisting of the controlled object and regulator. The ...
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1answer
91 views

Maximizing a function that is sum of gaussians

Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function \begin{align} \mathcal{K}(\mathbf{x},\mathbf{y})= \alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...
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0answers
32 views

Geometric control condition

In control theory of hyperbolic equations, we talk always about the G.C.C(Geometric control condition), it is a constraint on the domain to ensure the exact controllability of hyperbolic systems but I ...
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0answers
11 views

how to impose a terminal condition in a minimisation problem?

Consider the problem of minimising $$ J(u(.)) \triangleq \int_0^T l(x(t),u(t)) d t + \phi(x(T)), \: T> 0 $$ over a space of controls $\mathcal{U}$ with the constraint $$ \dot x(t) = f(x(t),u(t)), ...
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0answers
60 views

Minimizing Frobenius norm with sparsity constraints

I am trying to solve the following minimization problem \begin{equation*} \begin{aligned} & \underset{X,Y \in \mathbb{R}^{n\times k}}{\text{minimize}} & & \| X Y^\top A - B \|_{\text F}^2 ...
2
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1answer
108 views

Optimal-score partitions

The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...
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0answers
25 views

Boundary controllability of the heat equation and observation

I'm studying the boundary controllability of the heat equation \begin{array}{c} y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\ y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\ u(...
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2answers
87 views

References for control-affine theory

In Control theory from the geometric viewpoint by Agrachev and Sachkov, the authors mention the concept of control-affine (affine in control $u_i$) systems: $$\dot{x} = (f_0 + \sum_{i=1}^{m} u_i f_i)x ...
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0answers
24 views

Should penalty and Lyapunov drift be normalised in drift-plus-penalty optimisation?

I want to use drift-plus-penalty method to optimize control decisions in the system that I am considering in my research. In particular, I want to minimize penalty function (which in my case is ...
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0answers
33 views

Lower semicontinuity of proximal gradient descent sequence

I'm using an alternate optimization scheme to minimize a function $F(a,b,c) = G(a,b,c) + H(c)$ where $G$ is continuous and differentiable, $H$ is not differentiable but lower semi-continuous and ...
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0answers
46 views

An “almost separable” optimization problem on a graph

I am trying to solve an unconstrained optimization problem with the following properties: I am given a graph $G=(V,E)$ with functions $f_{ij}:\mathbb{R}^2\to \mathbb{R}$ for each edge $(i,j)$. I am ...
3
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1answer
87 views

Shortest Manhattan-norm paths among disjoint rectangles

I am looking for the fastest possible algorithm for solving the following problem: I am given a collection of disjoint axis-aligned rectangles in the plane, and I need to pre-process these rectangles ...
2
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0answers
122 views

The asymptotic stability of a control system

Let a control system be described by the following nonlinear ordinary differential equations: $(1)\quad \frac{dx(t)}{dt}=f_0(x(t))+\sum_{j=1}^m f_j(x(t))u_j,\;\;x(t)\in D\subset\mathbb R^n,\;\;u\in\...
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0answers
69 views

Observability inequality for the heat equation

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions: \begin{array}{c} \varphi _{t}+\...
3
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1answer
255 views

Minimize the variance of a Boltzmann distribution

N.B.: Sorry for cross-posting from https://stats.stackexchange.com/posts/347804/edit (I realized it was the wrong venue for the question, but couldn't find an easy way to transfer the question here). ...
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0answers
223 views

How to promote a blog?

Math behind might be interesting. Quite recent bloggingg activity might have interesting math model. The point is that bloggers compete for subscribers and at the same time cooperate gaining ...
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0answers
27 views

Best possible convex lower bound for an optimization problem

Suppose we have a primal problem $$ p^{*}=\min_x f(x), \\\text{s.t.}~~ h_i(x) \leq 0, $$ where $f(.)$ and $h_i(.)$ are possibly non-convex. Then its Lagrangian is $$\mathcal{L}(x,z_i)= f(x) + \...
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1answer
128 views

Optimization of a continuous function

This is more like an optimization problem but any solution is appreciated. I have a data set with input specifying power(demand) to be generated for a particular time period(TP). Input: Time --- ...
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0answers
44 views

Is this set of polynomial constraints convex? Can I optimise it?

Basically I want to optimise a problem with a constraint on the (variable dependant) roots of a polynomial, which I would like to be assigned depending on other constraints. I reduced the problem to ...
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0answers
35 views

Model Predictive Control discrete theory for Continuous Non linear System

I'm studying Model Predictive Control, and basically, the most solid theory is developed for Discrete-Time Systems. For Continuous Non-Linear System is advised to linearize the system at a point of ...
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1answer
71 views

Improved estimates of $n$ quantities via $n$ measurements

$\newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\epsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\si}{\sigma} \newcommand{\Si}{\...
25
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1answer
557 views

$\mathbb{E}[X^4]=1$, $X,Y$ iid, what's the best upper bound of $\mathbb{E}[(X-Y)^4]$?

Let $X,Y$ be i.i.d. random variables, $\mathbb{E}[X^4]=1$, what's the best upper bound for $\mathbb{E}[(X-Y)^4]$ ? A trivial upper bound is $16$, since $(X-Y)^4 \leq 8 (X^4+Y^4)$ then take ...
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0answers
44 views

On the defect of a flow network

This problem in graph theory was actually motivated by some problems in Theory of Fractals. To formulate the problem I need to recall some definitions related to flow network. A flow network is a ...
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0answers
43 views

Optimization with bounds on the control and its derivative

I would like to understand the following optimization problem. Let $F(t,x)$ be a continuous function defined on $[0,1]\times [0,1]$, which is increasing in $t$ and convex in $x$ (I have in mind $F(t,x)...
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0answers
38 views

Gamma convergence for control problems with changing domains

$\Gamma$-convergence is a notion of convergence for functionals which has the nice property that if $x_\varepsilon$ are minimisers for a family of functionals $\{F_\varepsilon, \varepsilon > 0\}$ ...
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0answers
34 views

Constrained Optimization Over Two Functions

I don't work in optimization; however, recently I have found myself needing to optimize a variety of functionals. I'm wondering if there is a way to efficiently approximate a solution to this ...
2
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1answer
89 views

Hu-Gomory trees and Optimum Communication tree

It is known that that can be several trees in a graph that follow the conditions of "Cut-tree" (also called Hu-Gomory tree). For example (https://stackoverflow.com/questions/25297470/igraphs-gomory-...
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0answers
59 views

Showing a modified system of quadratic equations is stable

I have and $n$ dimensional dynamical system, given by $\dot{x} = M D(x) P x - \frac{c}{2}x$ $P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...
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0answers
30 views

Smoothness of an optimal control problem for a point process

Let $\theta \in \{0,1\}$ be an unknown state of the world. Let $P_0 := Prob(\theta = 1)$ at time $0$. Let $G_t$ and $B_t$ be two Poisson processes with stochastic intensity $\lambda_g e_t \theta$ ...
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0answers
51 views

Suggestions to solve an optimization problem that involves quadratic forms

I am in a crucial part of my research, I have arrived at an optimization problem that I can not solve, I need to solve it to be able to perform simulations and thus complete my research, due to this ...
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0answers
45 views

$H_\infty$-norm of a time delay system

I have a linear dynamical stable delayed system as follows, $\dot{x}=Ax(t-\tau)+Bu$, where $A_{n \times n}$ is a stable matrix with all its eigenvalues located on the open left half of complex plane. ...
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0answers
39 views

$H_2$-norm of a time delay system

I have a linear dynamical stable delayed system as follows, $\dot{x}=Ax(t−τ)+Bu$, where $A_{n \times n}$ is a stable matrix with all its eigenvalues located on the open left half of complex plane. I ...
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5answers
258 views

Nearest matrix orthogonally similar to a given matrix

Given $A,B\in\Bbb R^{n\times n}$ is there technique find $$\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_F\mbox{ or }\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_2$$ within additive approximation error in $\...
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0answers
96 views

Optimal control of SDEs

I've set up a system of stochastic differential equations that I'd like to control. I'm new to optimal control theory and SDEs (and, admittedly, weak on PDEs), so I'm not certain if I've set this up ...
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2answers
336 views

Proving that a matrix is positive semidefinite

Let matrices $A, B$ be positive semidefinite. Can we prove that $A(I+BA)^{-1}$ is positive semidefinite?
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1answer
89 views

existence of solution to a martingale optimal transport type problem

I encounter the following problem during the course of my research: Given a random variable $Y=(Y_1,Y_2)$ with values in $\mathbb R^2$ and the cost function $c(x,y)=(x_1-y_1)(x_2-y_2)$ where $x=(x_1,...
2
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1answer
106 views

Quadratic matrix equation for $X\in \mathbb{C}^{ n\times p}$ with Hermitian parameters

Let $A\in \mathbb{C}^{ n\times n}$ and $B \in \mathbb{C}^{p \times p}$ be Hermitian matrices with $p < n$. Find matrix $X$ such that $X^*AX=B.$ Solution in the case of positive definite $A$ and $...
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1answer
140 views

Supremum norm of certain quantity

Is there any easy way of finding supremum of the quantity $$\sum_{i,j=1}^n|z_i-z_j|,$$ where $|z_i|=1$ for $1\leq i\leq n$ ? We are considering complex variables of course.
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2answers
203 views

A (reverse)-Minkowski type inequality for symmetric sums

Let $(u_1, u_2, u_3, u_4)$ and $(v_1, v_2, v_3, v_4)$ be vectors in $\mathbb R_+^4$. Is the following inequality true? \begin{align*} \left(\sum_{{[4] \choose 3}} \sqrt{u_i u_j u_k}\right)^{2/3} + \...
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0answers
39 views

Rank Optimization over semi-definite constrains

Let $X$ and $Y$ be finite dimensional Hilbert spaces $\mathbb{C}^n$ and $\mathbb{C}^m$, $L(X)$ and $L(Y)$ be the sets of linear operators of $X$ and $Y$, $\text{Herm}(X)$ and $\text{Herm}(Y)$ be the ...
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0answers
28 views

Best Optimization Algorithm: SPSA vs RL and in RL

If I understand correctly, Simultaneous Perturbation Stochastic Approximation is an optimization method whose input parameter is basically just an initial guess given that you can find obtain a "...