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

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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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**4**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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**

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**0**answers

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

**1**answer

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

**2**answers

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**

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**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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)...