**0**

votes

**0**answers

100 views

### Modifying a QP to incorporate more constraints

Consider the following problem:
$$\min \sum_{i=1}^n (Y_i - Z^{(i)})^2 \\
\text{subjected to}~ \epsilon_k^{\top}(X_j-X_k) \leq Z^{(j)}-Z^{(k)} ~ \forall k,j = 1 \ldots n. $$
where $\epsilon_1, ...

**1**

vote

**0**answers

70 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

137 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 ...

**0**

votes

**0**answers

42 views

### fundamental optimal-trajectory result known?

It's well-known and obvious that if you have a spaceship
and your sole constraint is an upper bound
on magnitude of acceleration/deceleration,
the fastest way to get to a distant star (a fixed ...

**2**

votes

**1**answer

198 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

321 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

196 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

146 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

134 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

102 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

53 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

60 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

185 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

195 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

149 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

320 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

112 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

240 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 ...

**1**

vote

**0**answers

84 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

136 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

148 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

128 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

192 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

164 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

201 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 ...

**2**

votes

**1**answer

178 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

209 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

135 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 ...

**1**

vote

**0**answers

261 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 ...

**0**

votes

**0**answers

168 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

87 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

197 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

83 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

132 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

53 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

92 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

60 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

108 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

62 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 ...

**1**

vote

**2**answers

146 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 ...

**3**

votes

**0**answers

98 views

### Multiple Number Partitioning / “Multiprocessor Scheduling”

Consider the well-known Number Partitioning / "Multiprocessor Scheduling Problem": You want to partition a set $S$ of objects (jobs) $i$ with weights $w_i$ (process time) into subsets $S_k$ (e.g. ...

**2**

votes

**1**answer

81 views

### Maximal probability of “infinitely often” over MDP

Let us consider a Markov Decision Process (MDP) with a Borel state space $X$. Often, the optimization problems over MDP involve optimization of some objectives dependent on the reward function
$$
...

**3**

votes

**0**answers

51 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 ...

**0**

votes

**1**answer

82 views

### Avoiding epsilon in mixed integer linear and quadratically constrained programs

I would like to represent the following constraint as MILP constraint where $x \in [a, b]$ with fixed $a, b \in \mathbb{R}$ and $y \in \lbrace 0, 1 \rbrace$.
$(x = 0 \wedge y = 1) \vee (x \neq 0 ...

**3**

votes

**1**answer

439 views

### Symplectic block-diagonalization of a real symmetric Hamiltonian matrix

Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...

**1**

vote

**1**answer

352 views

### for what arguments the function reaches maximum?

Hi,
What is the maximum of the following function?:
$f(x_i,w_i)=\frac { \sum w_i}{ \sum \frac {w_i}{x_i} } - \frac{ 1 - \prod \left ( 1 - w_{i}\right )}{ 1 - \prod \left ( 1 - \frac{w_{i}}{ ...

**2**

votes

**1**answer

174 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
...

**3**

votes

**1**answer

189 views

### Riemannian metric on a space of “not-quite-smooth” (hyper)surfaces?

Apologies if the title's a bit vague; I'll try to explain everything below, and please let me know if I should clarify anything.
I've been looking for a while at variational problems on polytopes. ...

**2**

votes

**1**answer

134 views

### Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...

**7**

votes

**1**answer

299 views

### Probability density that minimizes the sample range

Let $\mathcal{F}$ denote the set of all "concave probability distributions" on the unit interval, that is, all functions $f:[0,1]\to \mathbb{R}$ such that $f$ is concave, $f(x)\geq 0$ for all $x\in ...