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

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6
votes
2answers
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
1answer
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
1answer
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
0answers
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kalman filter with delayed (outdated) information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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
4answers
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
0answers
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
1answer
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
2answers
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
0answers
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
4answers
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
2answers
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
0answers
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
1answer
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
0answers
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
3answers
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
1answer
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
0answers
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
1answer
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
1answer
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
3answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
4answers
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
0answers
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
1answer
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
2answers
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
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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 ...