**0**

votes

**1**answer

134 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}
\mathbf{C}=\left(\mathbf{I}*\frac{1}{\...

**0**

votes

**2**answers

104 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

**0**answers

151 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 ...

**2**

votes

**0**answers

147 views

### An optimization in Markov Chain

We are given two correlated random variables $V$ and $X$ supported over a finite alphabets $\mathcal{V}$ and $\mathcal{X}$. Suppose the marginal $P_V$ and conditional distribution $P_{X|V}$ are ...

**0**

votes

**1**answer

441 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 ...

**1**

vote

**0**answers

123 views

### Maximizing an integral over a convex region

Let $C$ denote a compact, convex region in the plane containing the origin with unit area, and let $f$ be a probability distribution on $C$. Let $f^\ast$ denote the distribution that maximizes the ...

**5**

votes

**2**answers

931 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 \}, ~...

**1**

vote

**0**answers

48 views

### Discrete Optimal Control and Monotone Policies

Let $x = (x_1,x_2) \in \mathbb{N}^2$ be the state, $u$ be the control, and the dynamics be given by $x^{(k+1)} = f(x^{(k)}, u^{(k)}, w^{(k)})$ where $w^{(k)}$ is an IID noise source. For some stage ...

**0**

votes

**1**answer

130 views

### Kalman filter with long term bias

I was reading about the Kalman filter and I do not understand how it should be used when our measurements have a long term offset like GPS location updates do.
As I understand, the Kalman filter ...

**2**

votes

**0**answers

36 views

### Continuity of minimizer of a function with respect to another variable

Suppose the real function $f(w,X)=wg(X)+h(X)$ ($g$ and $h$ are other functions) is differentiable with respect to scalar $w$ and vector $X \in \mathbf{R}^m$ everywhere and $f$ is bounded below. What ...

**2**

votes

**2**answers

366 views

### Why eigenvectors optimize this orthogonally constrained nonlinear minimization problem?

Given a $p \times p$ positive definite matrix $\Sigma$, why eigenvectors of $\Sigma$, stacked as columns of a matrix $R \equiv [r_1 \, r_2 \, \ldots \, r_p]$, optimize the following orthogonally ...

**6**

votes

**3**answers

761 views

### Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$:
$$
\mathrm{arg}\max_R \,\...

**1**

vote

**0**answers

104 views

### Forcing a set of complex points to be closed under conjugation

I am a PhD student and I have to address the optimization of a real scalar function of complex variables $f(z_1,z_2,\ldots,z_n)$. The function is real valued in my case because each complex $z_i$ ...

**3**

votes

**2**answers

214 views

### Generalized Moore Graphs

A generalized Moore graph - as defined by Cerf, Cowan, Mullin and Stanton - is one for which the girth G and diameter D satisfy G ≥ 2D - 1. These graphs retain many of the optimal properties of Moore ...

**4**

votes

**2**answers

561 views

### Simplified knapsack problem

There is a problem that I can not solve.
Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...

**3**

votes

**1**answer

526 views

### Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices

I have an engineering back ground. Due to work, I came across this problem
\begin{align}
&\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\
s.t.~&\left(\mathbf{A}_0+\sum_{i=1}^{K}y_i\mathbf{A}_i\...

**0**

votes

**1**answer

68 views

### About the suboptimality of linear estimators

Let $X$ be a random variable and $N$ a Gaussian noise independent from $X$. We observe $Y=X+N$ and want to estimate $X$ based on $Y$ to minimize the mean square error $mmse(X|Y):=E(\hat X(Y)-X)^2$.
...

**2**

votes

**0**answers

90 views

### Maximum Principle with Banach Control Space

This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in which,...

**0**

votes

**0**answers

208 views

### If $\phi_n$ is a sequence of mollifier converging to the identity, does $\inf f\ast \phi_n \to \inf f$?

Let $\phi_n$ be a sequence of mollifier converging to the identity
$$
\phi_n(x) \to \delta_{0}(x), \text{pointwise},
$$
with $\delta_{0}(\cdot)$ the delta function at zero, and $\phi_n \in C^\infty_{\...

**2**

votes

**1**answer

224 views

### More than controlability: Speed of controllability!

Consider the continuous linear time-invariant system
$$
\begin{array}{l}
\dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\
\mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t)
\end{array}...

**0**

votes

**1**answer

940 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. ...

**3**

votes

**0**answers

273 views

### An optimization problem over real symmetric matrices

Given an $n\times s$ matrix $P$ of positive real numbers and $T\geq n$, find (either by a formula or an algorithm) the real symmetric $n\times n$ Z-matrix $A$ which maximizes $\min\limits_{1\leq i<...

**2**

votes

**1**answer

216 views

### Upper bounds on the worst-case traveling salesman tours in the unit square

The paper [1] proves that, if we place $N$ points in the unit square, then the length $\ell$ of the euclidean TSP tour of those points must satisfy $$\ell \leq \sqrt{2N} + 7/4~~.$$ I'm wondering, can ...

**2**

votes

**1**answer

238 views

### optimization over positive semidefinite matrices

I wonder what is the most explicit characterization that can be given for the solution to the ($N$-dimensional) problem of maximizing the criterion
$$
-\textrm{trace}[AS^{-1}] - b^\top Sb
$$
over ...

**-2**

votes

**1**answer

261 views

### A kind of economic objective function in assignment

I recently thought about a concept that seems like it should come up in economics, but I don't know if there's a name for it and where people would have encountered it elsewhere: Suppose we have a ...

**2**

votes

**1**answer

58 views

### Stability of a stable systems with a converging input

Does the following hold?
Let $x=0$ be an equilibrium point for the system $\dot x(t)=f(x(t))$ and suppose the existence and uniqueness conditions of solutions on $[t_0, +\infty)$ are satisfied. If ...

**1**

vote

**1**answer

72 views

### On impulsive optimal control with functions of not bounded variation

I have the following optimal control problem
$$
J=\int_0^TF(t,y_1(t),y_2(t))dt \to \min,
$$
subject to
\begin{align}
&\dot y_1(t) = f(t,y_1(t),y_2(t)) + g(t)\nu(t),\\
&\dot y_2(t) = \nu(t),...

**3**

votes

**1**answer

198 views

### Intuition on a certain class of quadratic optimization problems

Let $\mathcal{X} = \{\mathbf{X}\in\mathbb{C}^{d\times d}:\|\mathbf{X}\|\leq 1\}$, where $\|\cdot\|$ is the Frobenius norm. Let $\mathbf{y}\in\mathbb{C}^{d\times 1}$. We are familiar with the following ...

**1**

vote

**1**answer

312 views

### Semi Definite Relaxation for a Quadratic Feasibility Problem using CVX

Consider the following Semi-Definite Feasibility problem
\begin{align}
\max_{\mathbf{Z}}~0 \\\
\mathrm{trace}(\mathbf{Z})\leq \rho \\\
\mathrm{trace}(\mathbf{S}_1\mathbf{Z}) \geq \alpha \\\
\mathrm{...

**3**

votes

**1**answer

179 views

### Explicit formula for an LMI solution

Suppose we have a linear matrix inequality (aka LMI aka spectahedron aka linear matrix pencil):
$$A_{0}+x_{1}A_{1}+x_{2}A_{2}+\ldots+x_{m}A_{m} \succeq 0.$$
(The notation $X \succeq Y$ means that $X-...

**3**

votes

**3**answers

743 views

### A NICE necessary and sufficient condition on positive semi-definiteness of a matrix with a special structure!

Let
$$
A =
\begin{pmatrix}
\sum_{j\ne 1}a_{1j} & -a_{12} & \cdots & -a_{1n}\\
-a_{21} & \sum_{j\ne 2}a_{2j} & \cdots & -a_{2n}\\
\vdots & \vdots & \ddots & \...

**4**

votes

**2**answers

221 views

### A certain type of constrained Rayleigh-Ritz ratio

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two hermitian matrices. Consider the problem
\begin{align}
\max_{\mathbf{u}^H\mathbf{u}=1}~\mathbf{u}^H\mathbf{A}_1\mathbf{u} \\\
\mathbf{u}^H\mathbf{A}_2\...

**0**

votes

**1**answer

384 views

### Anyone has Kushner's book “Introduction to stochastic control” 1971? I need a theorem from it

In a paper I'm reading, it refers to Theorem 8, Page 217 of the book
"Introduction to Stochastic Control" H. J. Kushner, New York: Holt, Reinhart, and Winston 1971. Unfortunately I don't have it and ...

**3**

votes

**0**answers

105 views

### Copositivity in matrix pencils

Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...

**4**

votes

**0**answers

139 views

### Are there some numerical test to check if a map is a contraction?

Let's say I have a multivariate function
$$
f:D \to D, D \subset \mathbb R ^n, D \text{ compact},
$$
for which there is no closed form.
That is the only way to evaluate the function is to do it ...

**2**

votes

**0**answers

412 views

### minimize a cost function with matrix traces

Hi, I have a cost function of the form
$$F(X) = \operatorname{tr}(X'AX)+\operatorname{tr}(X'B),\quad\textrm{ s.t. }X'X=I.$$
$X$ is a $m\times n$ matrix, ($m>n$), with orthonormal columns. $A$ is ...

**-1**

votes

**1**answer

141 views

### Regularized Gradient with respect to a matrix (with a specific structure)

Suppose we have a typical logdet function $\mathcal{L}$
$$
\mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q},
$$
where $\...

**3**

votes

**1**answer

360 views

### Delta-convex functions and inner products

A delta-convex (d.c.) function is one which can be written as the difference of two convex functions.
The space of d.c. functions includes all C2 functions, and is interesting because it allows many ...

**2**

votes

**0**answers

193 views

### Tools for “infinite-dimensional linear programming”

I was wondering, whether you could point me to some tools with which I could tackle the following "infinite-dimensional linear programming" problem:
Notation:
$a=1,2,\ldots, A$, $x\in\Omega:=\left\{...

**0**

votes

**1**answer

398 views

### solve non-convex quadratic constrained quadratic programming

$\min_{\beta}\beta^{T} A \beta$
$s.t. \ \beta^{T} C \beta=1\ and\ \beta\geqslant 0$
Here $A,C\in \mathbb{R}^{M\times M}$, $\beta \in \mathbb{R}^{M}$
I saw in one paper saying that it could be ...

**3**

votes

**1**answer

619 views

### How to maximize the determinant of a matrix of the form VDV^H

Hi,
I have a matrix of the form $A=VDV^H$,
where $V$ is a $M \times 2M$ complex matrix, $D$ is a $2M \times 2M$ diagonal real matrix, thus the dimension of $A$ is $M \times M$.
My problem is how ...

**1**

vote

**2**answers

225 views

### On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has Lipschitz constant $M$ and considers the scheme
$$ x(t+1) = x(...

**-1**

votes

**1**answer

167 views

### To what equal constant in the Gibbs lemma

The Gibbs lemma is broadly used in games theory and in mathematical economics (optimal distributions of resourses, Cournot competition e.t.c.). Here it is:
Lemma (Gibbs). $f_1,f_2,\ldots,f_n$ be ...

**2**

votes

**0**answers

572 views

### A detail in the proof of the Motzkin-Straus theorem

The Motzkin-Straus theorem says that the global optimum of the quadratic program
$$\max f(x)=\frac{1}{2} x^{t}Ax,\qquad \mbox{ subject to }\sum x_{i}=1 \mbox{ and } x_{i}\geq 0,$$ where $A$ is the ...

**10**

votes

**2**answers

516 views

### Optimal inspection path on a sphere

Suppose you would like to "inspect" every point of a unit-radius
sphere $S \subset \mathbb{R}^3$ by walking along a path $\gamma$
on $S$, but you can only see a distance $d$ from where you stand.
...

**0**

votes

**0**answers

311 views

### A tricky optimization problem over matrices

Hi
I have the following problem whose solution has lured me for some months now....
All matrices are complex $N\times N$.
Let $A$ be a positive definite matrix with all eigenvalues strictly smaller ...

**1**

vote

**0**answers

98 views

### null controllability of linear wave equation

Consider the linear wave equation :
$$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$
Are there sufficient conditions on the functions $k(x)$ and $h(t)$ for which $(z,z_t)$ vanish ...

**2**

votes

**0**answers

424 views

### Incremental minimum spanning tree

Given a connected graph $G=(V,E)$ with a weight function $w:E\to\mathbb{R}$ and a subset $E_0\subseteq E$ such that the subgraph $(V,E_0)$ is connected, I am looking for a sequence $E_0\subseteq E_1\...

**2**

votes

**0**answers

98 views

### A Conjecture related to minimization of product of determinants over permutations

Hi
I have the following problem (and a conjecture which holds in Matlab).
Given an $N\times N$ matrix $H$, represent it by its $2N\times 2N$ real-valued representation (which I will also denote by $H$...

**2**

votes

**0**answers

175 views

### Quadratic optimization with parameter in constraint

Disclaimer: I posted the same question on math.stackexchange. However, the FAQ suggests to post research-level questions in this forum.
Question: Given a function $q: \mathbb R^{N\times N}\mapsto \...