Operations research, linear programming, control theory, systems theory, optimal control, game theory
1
vote
0answers
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
9
votes
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 ...
1
vote
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)...
1
vote
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
votes
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
$$\text{minimize} \quad \...
0
votes
0answers
43 views
Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element
In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks.
Consider the system consisting of the controlled object and regulator. The ...
0
votes
1answer
91 views
Maximizing a function that is sum of gaussians
Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function
\begin{align}
\mathcal{K}(\mathbf{x},\mathbf{y})=
\alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...
1
vote
0answers
32 views
Geometric control condition
In control theory of hyperbolic equations, we talk always about the G.C.C(Geometric control condition), it is a constraint on the domain to ensure the exact controllability of hyperbolic systems but I ...
0
votes
0answers
11 views
how to impose a terminal condition in a minimisation problem?
Consider the problem of minimising
$$
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)), ...
0
votes
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
votes
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-...
0
votes
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(...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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
votes
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
votes
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\...
2
votes
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
votes
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).
...
2
votes
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 ...
0
votes
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) + \...
1
vote
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 --- ...
0
votes
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 ...
0
votes
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 ...
4
votes
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
votes
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 ...
1
vote
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 ...
2
votes
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)...
1
vote
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 $\{F_\varepsilon, \varepsilon > 0\}$ ...
1
vote
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
votes
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-...
1
vote
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 ...
1
vote
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 $\lambda_g e_t \theta$ ...
0
votes
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 ...
0
votes
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. ...
0
votes
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 ...
6
votes
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 $\...
2
votes
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 ...
3
votes
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?
1
vote
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
votes
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 $...
5
votes
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.
9
votes
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} + \...
1
vote
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 ...
0
votes
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 "...
