**6**

votes

**2**answers

360 views

### How to solve such an optimization problem

I encounter the following optimization problem, but I can't solve it.
Given $N$ variables satisfying $0 \leq x_1 \leq x_2 \leq x_3 \leq ... \leq x_N \leq 1$ and an integer $K$ no large than $N$, find ...

**1**

vote

**1**answer

154 views

### Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:
$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,
where
(1) $A \in R^{n \times n}$ is a given matrix whose element ...

**0**

votes

**1**answer

71 views

### A problem on about a matrix norm on $\mathfrak{su}(4)$

Given a fixed $B \in \mathfrak{su}(4)$ is it possible to solve for $F$:
$\sigma^{\text{max}}\left(\frac{A}{F(A)} + B \right) = 1$, $\forall A \in \mathfrak{su}(4)$. A theorem in ...

**2**

votes

**0**answers

20 views

### kalman filter with delayed (outdated) information

We have a linear system with observation as follows:
$x(t+1)=Ax(t)+Bu(t)+w(t)$
$y(t)=Cx(t)+z(t)$
for given information $y(0\sim t)$, we can construct the dynamics of $\hat{x}(t)$ using the Kalman ...

**2**

votes

**0**answers

55 views

### First Variation of Dyson Series/Magnus Expansion

Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...

**6**

votes

**1**answer

232 views

### Generalization of the equilateral triangle?

I consider points in the two-dimensional plane.
An equilateral triangle is a set of three points in the plane which are equidistant.
Suppose now I have $n$ points $x_1,...,x_n$. What is the ...

**27**

votes

**1**answer

1k views

### Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...

**0**

votes

**1**answer

144 views

### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...

**0**

votes

**1**answer

368 views

### eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...

**4**

votes

**1**answer

41 views

### What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system
$$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1)
$$
where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...

**2**

votes

**2**answers

381 views

### Why does not the Hamiltonian depend on the derivative of the state?

I am reading "Optimal Control Theory" from Kirk. When solving the optimal control problem, there is defined The Hamiltonian $H(x(t),u(t),p(t), t)$ as $g(x(t), u(t), t) + p [a(x(t), u(t), t]$ where ...

**0**

votes

**1**answer

66 views

### equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...

**0**

votes

**0**answers

17 views

### Zeros of non-lipschitz functions (when noisy estimates are available only)

Given noisy (martingale difference) of a Lipschitz continuous function $f$ it is known how to compute zeros of it. It is the stochastic approximation approach (by Borkar, Kushner and Yin etc.). Is ...

**1**

vote

**0**answers

49 views

### An optimization problem for a Markov Chain

Consider a Markov Chain $\{X_n\}$ whose transition probability depends on some parameter $\theta$ ($p_{ij}(\theta)$). Now I want to optimize the following quantity
$$\lambda(\theta) = ...

**4**

votes

**0**answers

68 views

### Sufficiency of stationary policy for negative stochastic dynamic programming

Consider a Markov Decision Process with Borel state space $X$ and Borel action space $U$, like the one defined in the book "Stochastic Optimal Control: Discrete-time case" by Bertsekas and Shreve. All ...

**2**

votes

**0**answers

27 views

### Linear control systems

Are there some algorithms to compute the error bound of the difference between the original system and the balance system using the Balance truncation method?

**0**

votes

**0**answers

45 views

### Representation of piecewise functions using variational inequalities

I need some help about how to represent piecewise smooth functions defined over N subintervals by using variational inequalities. From a paper I'm analyzing I get:
"If we can find a function ...

**1**

vote

**1**answer

91 views

### Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...

**1**

vote

**1**answer

78 views

### A certain instance of the Set Covering problem

Is there any useful structure associated with the following instance of the Set Covering problem?
Let $G$ be a weighted graph and let $\mathcal{P}$ denote the set of all shortest paths between all ...

**4**

votes

**4**answers

305 views

### Linear complementarity problem - Tridiagonal and Convex Case

I need an algorithm for the following LCP:
$\mathbf{Mz}+\mathbf{q} \ge \mathbf{0}$
$\mathbf{z} \ge \mathbf{0}$
$\mathbf{z}^{\mathrm{T}}(\mathbf{Mz}+\mathbf{q}) = 0$
Here, $\mathbf{M}$, is a ...

**0**

votes

**0**answers

36 views

### A Optimization problem using co-ordinates of joint numerical range.

Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as
\begin{align}
...

**2**

votes

**1**answer

141 views

### Optimization problem whose cardinality never exceeds 7 for some reason

I am working on a problem in which I have a collection of $n$ points, $x_1,\dots,x_n$, in the plane, as well as a positive definite matrix $\Sigma$ and another point $\mu$ in the plane. I am trying ...

**6**

votes

**2**answers

148 views

### Relativistic Control Theory

I am looking for literature that combines General relativity and control theory.
So far I found a video lecture on "Integrability meets Control Theory: Harmonic maps in GR", other than that not so ...

**0**

votes

**0**answers

31 views

### Stability condition for linear time varying Kalman filter

For the traditional time invariant system:
$x(k+1)=Ax(k)+w(k)$
$y(k)=Cx(k)+v(k)$
the stability of $\Delta(t)\triangleq x(t)-\hat{x}(t)$ is the observability of (A, C) pair.
What if now C is a ...

**6**

votes

**4**answers

559 views

### solving Lyapunov-like equation

The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...

**0**

votes

**2**answers

60 views

### Approximate solution to large mixed integer programming problem

What are the available approaches to find an approximate solution to a large mixed integer programming problem?
I ran my problem in the Gurobi MIP solver.
It can find a feasible solution in ...

**1**

vote

**0**answers

19 views

### Minimal rectangular confidence regions

For a given multivariate pdf $f$ (mainly the gaussian one) I'm looking to compute a minimal rectangular confidence region for a given level $\alpha$. For example, I would like to solve problems of the ...

**1**

vote

**1**answer

164 views

### System of quadratic complex equations

I want to solve this system of N non-linear equations without using a numerical method:
$x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$
With
$\left| ...

**3**

votes

**0**answers

99 views

### Rank 1 Approximation of Elementwise Inverse Matrix

I'm wondering whether there is a good way to solve the following optimisation problem.
Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that
$$ \| D_1 A ...

**14**

votes

**3**answers

603 views

### “Entropy” proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ ...

**1**

vote

**1**answer

35 views

### How to find all possible solution in the problem of simplex mesthod in maple? [closed]

I have an object to maximize and some constraints in Maple, but Maple just give me only one solution. How can I get more than one solution, i.e. I would like to know all possible solution for the ...

**0**

votes

**0**answers

19 views

### Examples of POMDPs where the actions impact the transitions of the underlying markov Chain

I am not sure if the following is a legitimate question for this board.
I am looking for examples of Partially observed Markov decision processes (preferably infinite horizon, Discrete time, Discrete ...

**5**

votes

**1**answer

261 views

### Geometric applications of Ekeland's variational principle

I'm looking for geometric applications of Ekeland's variational principle in order to see it at work in a context I'm familiar with. Let me recall the principle itself:
Definition. Let $(X,d)$ be a ...

**2**

votes

**1**answer

179 views

### Optimal Control

Consider the optimal control problem with an optimal trajectory $x^*(t)$ and an initial point $(x_1^0,x_2^0)$
\begin{equation}
\begin{split}
\dot{x}=A x + Bu
\\\
J=\int^\infty_0(x_2^2+\epsilon u^2)dt
...

**2**

votes

**3**answers

603 views

### An Optimization problem

The following unusual optimization problem came up and I don't know where to begin:
Maximize over the real variables $x_1, \dots, x_n$ the sum
$$
S = \sum_{r = 1}^n \frac{1}{x_1 + \dots + x_r}
$$
...

**8**

votes

**0**answers

198 views

### Hasse-Weil Bound and Chebyshev Inequality

I was reading about the Hasse-Weil bound for the number of points in on a curve over the finite field $\mathbb{F}_q$.
$$ \big| |C(\mathbb{F}_q)| - (q+1) \big| \leq 2g \sqrt{q} $$
However, this ...

**1**

vote

**0**answers

43 views

### Special case of Forced Harmonic Oscillator

$$ \frac{d^2l}{dt^2} + c \frac{dl}{dt} +k= F_0\sqrt{1-l^2}$$
Either a closed form or something like given initial conditions, the max value of $l$. For small $l$, this boils down to simple step ...

**3**

votes

**1**answer

108 views

### A problem related to routing in a graph

I have come across a new problem - I want to know whether this problem is similar to some existing problem or not.
The new problem is this. There is a tourist who has a having the following ...

**2**

votes

**0**answers

84 views

### Quickly checking an inequality on a convex region

I previously posted this question to math.sx at: http://math.stackexchange.com/questions/748015/quickly-checking-if-an-inequality-holds-on-a-convex-region but I'm thinking that MO may be more ...

**-2**

votes

**1**answer

699 views

### Determine noise distribution [closed]

I'm trying to solve the following least squares problem:
$\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$
where $Ax = b$ and $\tilde{b} = b + w$
Question:
How do I determine which probability ...

**2**

votes

**4**answers

354 views

### Efficient algorithm for finding the minima of a piecewise linear function

Consider real numbers $a_i$ and $b_i$ for $i=1\dots n$ and define a function by
$f(x) = \max_i ( a_i + b_i x )$
We desire to find $\min_x f(x)$. Obviously this occurs at an intersection of two ...

**0**

votes

**0**answers

61 views

### Weak convergence of 4-th degrees

Good day!
We have an equation $y'+Ay=Bu$ where $y=\{\theta,\varphi\}$, $A, B$ are nonlinear operators.
$u \in L^\infty(\Gamma)$, $\theta, \varphi \in W = \{y \in L^2(0,T;V) : y'\in L^2(0,T;V')\}$, ...

**3**

votes

**1**answer

67 views

### Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$.
I would like to know how many extrema $p$ has on the standard simplex
...

**4**

votes

**2**answers

134 views

### Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$:
$$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in ...

**3**

votes

**2**answers

204 views

### Moreau-Yosida regularization in Banach spaces

For a seminar I am working on a Moreau-Yosida regularization in Banach spaces.
The regularization is defined by
$$f_\lambda(x) := \inf \left \{ \frac{\|x-y\|^2}{2\lambda} +f(y) : y \in X \right \}, ...

**0**

votes

**0**answers

40 views

### Sequence of solutions of approximated problem converging to stationary solution of the original problem

Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...

**0**

votes

**0**answers

18 views

### Kalman Filter for coupled difference equations with stochastic volatilty

I am trying to estimate the following discrete-time system using the Kalman Filter:
y_t = a*y_t-1 + b*x_t-1 + c+ sigma_1(t)*Z_1,t
x_t = d*x_t-1 + e*y_t-1 + f+ sigma_2*exp(g*i_t-1)*Z_2,t
v_t = ...

**1**

vote

**0**answers

176 views

### An optimization problem on the sphere

Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer.
Let vector ...

**4**

votes

**0**answers

123 views

### Convexified threshold of a function

Upd. The question in a nutshell: find convex set on plane which is «closest» to a given non-convex set.
It is given integrable function $0\leq f(x,y)\leq 1$ with bounded support: $f(x,y)=0$ when ...

**5**

votes

**1**answer

124 views

### Are there any known results on numerical ranges of rank-one positive semi-definite matrices?

In my problem, I came across numerical ranges of rank-one positive semidefinite matrices. Through Toeplitz-Hausdorff theorem and some other extensions, I know if there are at most three matrices, then ...