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

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2
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0answers
92 views

Copositivity under tensor products

Is there any symmetric real matrix $A$ such that $A^{\otimes n}$ is copositive for all positive integers $n$, but such that A is neither positive semidefinite nor has just non-negative entries? ...
0
votes
1answer
96 views

Sufficient and necessary condition for BIBO stability

I am looking for a reference for the proof of the next claim: "BIBO—bounded input bounded output—stability. We claim that a necessary and sufficient condition for a system described by a linear, ...
0
votes
0answers
57 views

numerical solver for stochastic optimal control problems

can any one recommend numerical solver (c/c++ library preferred) for stochastic optimal control problems? For deterministic optimal control I found something like that: ...
3
votes
0answers
92 views

Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...
0
votes
1answer
205 views

Linear programming with exponentially many constraints and variables [closed]

From class, I learnt that problems like traveller salesman have a Linear programming representation with exponentially many constraints. Using method of separation, this problem is solved rather ...
7
votes
3answers
368 views

The distribution of the shortest path through $n$ points

In the big picture, I'd like to know: if I sample $n$ points uniformly at random in the unit square, what is the probability that the shortest path that visits each one of them is very small? More ...
1
vote
2answers
150 views

Combinatorial optimization problem involving infinite spin system

In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity. ...
1
vote
0answers
125 views

lower bound for difference between max cut and min cut

Let $G=(V,E)$ be a graph on $n$ vertices with edge weights $w=(w_e)_{e\in E}$. Let $M^+$ and $M^-$ be the maximum and the minimum cut values, i.e. ...
0
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0answers
27 views

Optimization problem involving an entrywise function

Let $X$ a $n\times p$ real-valued matrix and $Y$ a $p\times q$ real-valued matrix. Let $\phi:\mathbb{R} \to \mathbb{R}$ a function. What is the appropriate way to deal with the following optimization ...
3
votes
1answer
117 views

A difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
2
votes
4answers
648 views

Algorithm to find the “optimal” path in a given graph

Assume that $G=(V,E)$ is an undirected connected graph and that $H: V \to \mathbb R$ is a function that assign at each vertex $v \in V$ its height $H(v)$. Think of the pair $(G,H)$ as an energy ...
0
votes
1answer
75 views

Generalized assignment problem with no integrality gap

Suppose I am solving the generalized assignment problem, so that I am given matrices $U$ and $W$ and a vector $c$ (all three of which have, say, positive entries), and I want to solve ...
0
votes
1answer
244 views

A problem 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 ...
3
votes
1answer
52 views

Kálmán 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 Kálmán ...
2
votes
0answers
152 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 ...
4
votes
1answer
80 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, ...
0
votes
1answer
94 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 $SU(n)$. ...
7
votes
2answers
447 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 ...
2
votes
0answers
35 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?
1
vote
2answers
243 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 ...
2
votes
2answers
298 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 ...
1
vote
1answer
100 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 ...
0
votes
0answers
44 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} ...
7
votes
2answers
180 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
2answers
162 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
26 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 ...
4
votes
1answer
185 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 ...
1
vote
1answer
170 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| ...
2
votes
1answer
151 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 ...
16
votes
3answers
991 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
50 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 ...
8
votes
0answers
266 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
48 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
117 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
103 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 ...
0
votes
0answers
66 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
77 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 ...
1
vote
0answers
181 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 ...
2
votes
2answers
109 views

Find the optimal set of subsets

Consider a set of $N$ individuals and let their distance be given by $R$, a $N\times N$ matrix. In that, $R(1,2)$ is the distance between individual 1 and 2. Now lets say that I want to separate the ...
5
votes
1answer
144 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 ...
2
votes
1answer
205 views

derivative of sum of singular values

can someone point me to the direction how to calculate the derivatives of a sum of singular values of a matrix? I am trying to minimize $$\min_A \parallel A \parallel_*+ \cdots $$ where $\parallel A ...
1
vote
1answer
1k views

Maximizing linear objective function with absolute values

This has be asked on other forums, though couldn't find authoritative answer. I have a linear program over the reals and don't want to introduce integer or binary variables. The objective function ...
2
votes
2answers
228 views

Finding the maximum of a multivariate polynomial of degree one

I need to find the global maximum of the function \begin{align} f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\ &+\ldots \\ &+ p_n ...
1
vote
0answers
264 views

Incoherence of the row/column span

Due to V.Chandrasekaran., et al‎ (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that: $$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$ where the lower bound is achieved (for ...
1
vote
0answers
138 views

Complexity of Nested Linear Optimization

My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint: ...
4
votes
2answers
231 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 ...
0
votes
1answer
57 views

generalization from linear programming solution [closed]

I have a series of similar linear programs that depend on an input vector $a\in A$ and whose solution is an output vector $b\in B$. I can solve them individually, but this is wasteful. I suspect that ...
0
votes
1answer
120 views

Eigenvalues of a given parametrized matrix.

Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as \begin{align} ...
0
votes
2answers
101 views

Union of linear inequalities cover whole space?

We have $n$ variables $a_0,a_1,\ldots,a_n$ such that $a_i\geq a_{i+1}$. There are $k$ sets of linear inequality constraints on the $a_i$. I need to check that any choice of $a_i$ satisfies at least ...
1
vote
0answers
145 views

How to solve such an optimization problem efficiently?

Given a symmetric positive semi-definite matrix $\mathbf Y$ and a convex set $\mathcal M$ which is a subset of all symmetric positive semi-definite matrices (consider a simple case of $\mathcal M$: a ...