**5**

votes

**2**answers

864 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

127 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

350 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

663 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

212 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

542 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

452 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 \\\
...

**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

89 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 ...

**0**

votes

**0**answers

203 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 ...

**2**

votes

**1**answer

223 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)
...

**0**

votes

**1**answer

900 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

267 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 ...

**2**

votes

**1**answer

208 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

233 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

248 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

71 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) = ...

**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

301 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 \\\
...

**3**

votes

**1**answer

173 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 ...

**3**

votes

**3**answers

706 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

211 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} \\\
...

**0**

votes

**1**answer

364 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

104 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

402 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

347 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

190 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:
$a=1,2,\ldots, A$, ...

**0**

votes

**1**answer

369 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

577 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

223 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) = ...

**-1**

votes

**1**answer

165 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

539 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

497 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

300 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

398 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 ...

**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 ...

**2**

votes

**0**answers

169 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 ...

**1**

vote

**0**answers

59 views

### strong stability for the wave equation

Consider the $n-$dimensional wave equation
$$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$
where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...

**1**

vote

**0**answers

116 views

### Compactness of the set of solutions to an ODE

I am interested in optimal control problems with state constraints in the form of
ordinary differential equations given by
$\dot{x} = f(x,p), \quad t \in [0,T], \quad x(0) = x_{0}, \quad p \in K$,
...

**0**

votes

**1**answer

76 views

### Representation of all pass transfer functions/inner functions as Blaschke product.

What is the proof for: 'An all pass transfer function/inner function can be represented by a Blaschke Product' ?

**1**

vote

**1**answer

148 views

### Control a linear system to the kernal space of the output matrix.

Consider the following controllable and observable linear system
$$\dot x=Ax+Bu, y=Cx,$$
where $x\in \mathbb{R}^n, u\in \mathbb{R}^m, y\in \mathbb{R}^p$.
The observability of $(A,C)$ ...

**2**

votes

**0**answers

77 views

### Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious.
General gist of the problem
I have a variational problem on a ...

**2**

votes

**2**answers

180 views

### What is the dual of an semidefinitely representable (SDR) cone?

The Question
Let $V\simeq \mathbb{R}^r$ be an $r$-dimensional vector space with the usual Euclidean inner product.
Let $\mathcal K\subset V$ be a cone defined as
$$
\mathcal K=\Big\{x\in V\ ...