Operations research, linear programming, control theory, systems theory, optimal control, game theory
1
vote
0answers
13 views
Pontryagin principle for optimal control problem with a terminal cost involving the control
Let $T >0$ and consider the problem of minimizing
$$
P(v(.)) \triangleq \int_0^T l(x_t,v_t) d t + h(x_T)
$$
over a broad class of control $v(.)$ where
\begin{equation}
\dot x_t = f(x_t,v_t),
\...
-2
votes
0answers
35 views
Optimal control of PDE in Matlab [on hold]
Is there a good book explaining how put in Matlab optimal control problems of a system with partial differential equations?
0
votes
0answers
24 views
The subdifferential of a function
The function is $f(\mathbf{x}) = \sqrt{1-\exp(-\phi \|\mathbf{x}\|_2^2)}$, where $\phi>0$ is a constant and $\mathbf{x}$ is a vector in $\mathbb{R}^p$. What is the subdifferential of $f(\cdot)$ at $...
2
votes
0answers
61 views
What is the most efficient path for a robot without turning radius?
I recently programmed a most efficient path for a robot going from $(x_1, y_1 , \theta_1)$ to $(x_2, y_2)$. The bot does a combination of turning and moving at any given point, based on the difference ...
0
votes
0answers
18 views
A question related to parametric linear programming
Consider the following parametric linear problem:
\begin{align}
\min z(t)=c^T x\\
Ax=b(t)\\
0\leq x\leq u.
\end{align}
We know $z(t)$ is a piecewise linear function. Let $x(t_1)$ and $x(t_2)$ be the ...
1
vote
1answer
98 views
Is it possible to “solve” iterative (convex/non-convex) optimization problems via learning (one-shot)?
I posted a following question in MSE, but I think it should be posted here in MO. Since I don't know how to transfer the post from MSE to MO, I have pasted the question below. Thank you in advance and ...
2
votes
0answers
68 views
A community effort: equilibrium in quitting games [on hold]
This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...
9
votes
2answers
253 views
Maximum of a quantity for two normal orthogonal vectors in $\mathbb{R}^n$
Let's define for every pair of vectors $u,v\in\mathbb{R}^n$, a quantity as follows:
$$f(u,v) = \sum_{1\leq i,j\leq n}|u_iu_j-v_iv_j|.$$
I want to find:
$$M(n)= \max \{f(u,v): u,v\in \mathbb{R}^n, |u|=...
4
votes
0answers
135 views
Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?
Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...
1
vote
1answer
108 views
How to solve this optimization problem efficiently?
Let, $D\in\mathbb{C}^{1\times M}$ is a row vector with $M$ elements
$V\in\mathbb{C}^{3^M\times M}$ is a given matrix
$T$ is a scalar (real and $>1$)
$\textbf{The problem at hand is as follows:}$
...
3
votes
1answer
60 views
Echange of Infimum Integral with Pointwise Infimum
Setup
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
3
votes
1answer
146 views
it's convex sequence inequality
A sequence $a_0,a_1,\dots,a_n$ of real numbers is called concave if $a_{0}=0$, and for each $0<i<n$, we have $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.
Find the largest $c(n)$ such that for every ...
0
votes
0answers
25 views
Kronecker sum matrix for a singular matrix pencil
The Kronecker sum of two matrices $A \in \mathbb{R^{n \times n}}$ and $B \in \mathbb{R^{m \times m}}$ is defined by the matrix.
$$A \oplus B = A \otimes I_m + I_n \otimes B \in \mathbb{R^{mn \...
15
votes
4answers
744 views
What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?
My question is to find the minimum of the following expression:
$$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$
over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...
0
votes
0answers
34 views
a game-theoretical / optimization problem
Does anyone know how to solve (or a good reference) for the following problem?
Let n and m be two positive integers. Let z \mapsto G^{i,j}(z) be a real polynomial, for each i=1,...,n and j=1,...,m. ...
0
votes
1answer
58 views
maximization of a log norm function
Considering the following optimization program:
$$
maximize \ \ \ \log \left( \|x\|_\infty \right)
$$
$$
subject \ to \ \ Ax\leq b, \ x \geq 0
$$
can we rewrite this program as a convex ...
3
votes
2answers
195 views
A (linear) optimization problem subject to (linear) matrix inequality constraints
Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
1
vote
1answer
69 views
LASSO problem but with a maximization instead of minimization
I have the following optimization problem (like the LASSO problem but with maximization instead of minimization):
$\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...
2
votes
0answers
168 views
A parametrization of stable matrices
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is ...
1
vote
1answer
123 views
Linear matrix inequality
I have the following linear matrix inequality:
$$F^T P + PF < 0,$$
where $P$ is a positive definite matrix and $F$ is a matrix with appropriate dimension.
Let $Q$ be a positive definite matrix ...
2
votes
0answers
79 views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
0
votes
0answers
49 views
Using mollifiers (or some other idea) to solve constrained minimax problem
Sorry in advance if this sounds like a more SE question.
Consider a continuously parametrized family of $L$-Lipschitz continuous $f_\theta: X \rightarrow \mathbb R_+$ on a metric space $X=(X,d)$. Let ...
1
vote
0answers
36 views
Minimum Preserving Transformations [closed]
If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then
$$
\operatorname{argmin}_{x \in X} f(x)
=
\operatorname{argmin}_{x \in X} g\circ f(x) .
$$
X and Y ...
9
votes
2answers
379 views
A trace-constrained maximization problem in the cone of positive definite matrices
Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...
1
vote
0answers
98 views
Bounds of Procrustes problem
We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following:
\begin{align}
f(A,B)...
1
vote
1answer
103 views
How to solve this system of Matrix equations? (Coupled riccati equations?)
I am trying to solve for K in the following problem:
$ 3I = A_1 + A_2 + A_3$
$ A_1 K A_1 = K_1 $
$ A_2 K A_2 = K_2 $
$ A_3 K A_3 = K_3 $
Where $I$ is the identity, $K, K_1, K_2, K_3, A_1, A_2, ...
3
votes
2answers
230 views
A structural optimization problem
I have $n$ objects $O_i$, each of them having $3$ values, $O_i = (A_i, B_i, C_i)$. I am trying to group them into $k$ groups $P_u$ such as $P_u =(A_u, B_u, C_u)$ such that
$$\text{minimize} \quad \...
0
votes
0answers
52 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 ...
1
vote
2answers
152 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\...
0
votes
1answer
37 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
$$
\dot x(t) = f(x(t),u(t)), ...
0
votes
0answers
76 views
Minimizing Frobenius norm with sparsity constraints
I am trying to solve the following minimization problem
\begin{equation*}
\begin{aligned}
& \underset{X,Y \in \mathbb{R}^{n\times k}}{\text{minimize}}
& & \| X Y^\top A - B \|_{\text F}^2 ...
2
votes
1answer
115 views
Optimal-score partitions
The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...
0
votes
1answer
51 views
Boundary controllability of the heat equation and observation
I'm studying the boundary controllability of the heat equation
\begin{array}{c}
y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\
y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\
u(...
2
votes
3answers
144 views
References for control-affine theory
In Control theory from the geometric viewpoint by Agrachev and Sachkov, the authors mention the concept of control-affine (affine in control $u_i$) systems:
$$\dot{x} = (f_0 + \sum_{i=1}^{m} u_i f_i)x ...
0
votes
0answers
30 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
36 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
48 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 ...
3
votes
1answer
166 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
74 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
297 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
226 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
28 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
132 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
36 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
73 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}{\...
27
votes
1answer
584 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)...
