**0**

votes

**1**answer

72 views

### Finding the right σ-algebra. Question on uncertainty related to the secretary problem

Assume a number of iid. items is presented and the task was to stop under the objective of picking the best item.
In this setting it is relevant what is the distribution of the values of the ...

**2**

votes

**1**answer

32 views

### Edge-disjoint paths avoiding some subgraphs

Let $G$ be a directed graph on $n$ vertices. Let $H_1$, ..., $H_k$ be marked subgraphs of $G$. (Specifically, each $H_i$ consists of a subset of the vertices of $G$ and a subset of the edges of the ...

**8**

votes

**1**answer

263 views

### An inequality on the simplex involving $x^x$

Is anything known about the behavior of the function $$f(x)=\prod_{i=1}^n x_i^{x_i}$$ on the standard simplex, i.e. the set $\{x\in\mathbb{R}^n:\sum_{i=1}^n x_i=1, x_i\geq0\}$? I ask because I have ...

**1**

vote

**0**answers

55 views

### The state-transition-matrix of a physical system,

Here's a simple but potential research problem that I am learning about.
Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...

**1**

vote

**0**answers

32 views

### computational-expensive signal reconstruct - a combination problem [closed]

My problem is:
I have a time series signal (vibration signal), use BSS algorithm (Blind Source Separation, we can regard it as a black box), separate the source signal into 100 components.
Now I ...

**2**

votes

**1**answer

33 views

### Specific discrete system $x_n = A(n,u)\cdot x_{n-1}$ control papers

Basic discrete control theory mostly studies systems which can be represented as $x_n=A(n)x_{n-1}+B(n)u_n$.
I wonder if optimal control of specific discrete systems of the type $x_n = A(n,u)\cdot ...

**0**

votes

**0**answers

60 views

### Optimization with vectors

I am trying to solve the following optimization problem as a small part of a research project, and I do not know if there exists closed form solutions. My linear algebra is very rusty and I am looking ...

**0**

votes

**0**answers

47 views

### About identifying a few diagrams

Please have a look at these beautiful seminar slides,
https://math.berkeley.edu/~bernd/coimbra1.pdf
Can someone kindly identify the algebraic description of the spectrahedron that is drawn on slide ...

**5**

votes

**3**answers

327 views

### Euler-Lagrange equations and Bellman's principle of optimality

One method to optimize the integral
$$\int_{\mathcal T} L(t,x,\dot{x}) \; dt $$
of a functional over a curve is the calculus of variations, which leads to ordinary differential equations: the ...

**1**

vote

**0**answers

55 views

### Hessian matrix positive definiteness (concavity test) [closed]

I have a rather simple scenario based optimization problem:
Maximize
$$
Q_1{_s}(A_1{_s}-Q_1{_s}-bQ_2{_s})+ Q_2{_s}(A_2{_s}-Q_2{_s}-bQ_1{_s})-[(Q_1{_s}-K_1)^+ + (Q_2{_s}-K_2)^+]c
$$
subject to ...

**3**

votes

**1**answer

113 views

### Global minimum of nonlinear least square

We have a continuous and differentiable function $f(\cdot)$ that maps from $R^n$ to $R^n$. We are trying to solve a nonlinear least square problem:
Minimize $J(x)=\Vert f(x)-z\Vert^2$
subject to box ...

**3**

votes

**1**answer

102 views

### Choosing $K$ “centers” from the space of permutations

Let $\Pi$ denote the space of all permutations of $\{1,\dots,n\}$, and let $d(\cdot,\cdot)$ be a metric on $\Pi$. Suppose I am given a large integer $K$ and I have to select $K$ permutations ...

**0**

votes

**0**answers

49 views

### Separable Least squares - is there a notion of conjugate directions?

I have a general question.
Suppose I have the following to optimize
$$\|Y-A(\mathbf{x})B(\mathbf{y})\|^2$$
where $Y$ is a vector, $A(\mathbf{x})$ is a matrix that depends on a vector $\mathbf{x}$ in a ...

**1**

vote

**1**answer

43 views

### On optimizing a function whose projection and projected vector go through a linear transformation

Assume the two sets of vectors $\{\mathbf{a}_1,\ldots,\mathbf{a}_N\}$ and $\{\mathbf{b}_1,\ldots,\mathbf{b}_N\}$ of equal length. My goal is to find the optimum matrix $\mathbf{C}$ to the following ...

**0**

votes

**0**answers

28 views

### Maximisation of a discrete linear function

I am trying to maximise the function
$$
Q=\sum_i{\alpha _{i}x_{i}}
$$
subject to the constraint
$$
W<=\sum_i{\alpha _{i}w_{i}x_{i}}
$$
By changing $\alpha _{i}$ subject to $ 0<=\alpha ...

**0**

votes

**0**answers

63 views

### Curves in $\mathfrak{su}(n)$ with specific property

Consider a curve $\gamma_s= U_s^{\dagger} b U_s$ in $\mathfrak{su}(n)$ where $U_s$ is a smooth curve on $SU(n)$ (starting at $U_0 = \mathbb{I}$) and nonzero $b\in \mathfrak{su}(n)$ and $s \in[0,T]$ ...

**0**

votes

**0**answers

26 views

### Equivalence between multiclass SVMs, power diagrams, and constrained $k$-means

Apologies in advance for the long post:
Suppose we have a collection of points $\mathbf{p}_1,\dots,\mathbf{p}_n$ in $\mathbb{R}^d$, and we consider the following three ways of partitioning these ...

**10**

votes

**2**answers

284 views

### More general than semidefinite program?

I was TAing my convex optimization class and explaining that Linear Programs are a special case of Second Order Cone Programs, which are themselves special cases of Semidefinite Programs. My question ...

**4**

votes

**1**answer

253 views

### Kalman filters and stock price prediction

Could someone be so kind as to direct me to a good source that would explain time series (more specifically) stock price prediction using Kalman filters, Extended kalman filters or particle filters. ...

**1**

vote

**0**answers

89 views

### approximation of rational functions

Suppose $\hat{p}/\hat{q}$ and $p/q$ are two rational functions where $p,q,\hat{p},\hat{q}$ are of degree $n$. Suppose they satisfy that $|p(z)/q(z) - \hat{p}(z)/\hat{q}(z)| < \epsilon$ for any $z$ ...

**0**

votes

**1**answer

100 views

### Is the linear production game a convex game?

In cooperative game theory, the linear production game (LPG) is defined by letting the characteristic function have the form of a linear programming problem.
Does anyone know if the LPG is a convex ...

**0**

votes

**0**answers

18 views

### Can MDPs over functions be solved?

I understand that dynamic programs are difficult to be solved in general. However I have an MDP, for which intuitively I have a solution, I am curious to know if there is a formal approach to get a ...

**0**

votes

**0**answers

47 views

### Applications of systems with multiple time

A dynamical system with multiple time is an action of a group $\mathbb{Z}^d$ or $\mathbb{R}^d$ on a metric space.
I am interested in informative examples and applications of such systems. I know ...

**1**

vote

**0**answers

18 views

### Deterministic global solution to find the Optimal-knot placements for continuous piecewise linear functions to fit nonlinear data

I have been searching lately for a deterministic global technique to linearize a nonlinear function with continuous piecewise linear regions.
I've a univariate nonlinear function y=f(x). where f(x) ...

**2**

votes

**0**answers

51 views

### Is there a name for this variant of the MST and the TSP?

Suppose I am given a weighted graph $G$ that contains a "start vertex" $v_0$, and my goal is to construct a set of paths that all originate at $v_0$ and touch all of the vertices of $G$, with as ...

**0**

votes

**0**answers

20 views

### Bounding the error of the optimal solution from an approximated objective

Let $f$ be a smooth convex function defined in a bounded region $X$ and its Hessian is bounded $m I \le |\frac{\partial^2 f(x)}{\partial x^2}| \le M I$ for some $M>m>0$. Let $x^*=argmin_{x\in X} ...

**2**

votes

**0**answers

100 views

### Are singular critical points isolated for control systems on compact semisimple Lie groups

Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form:
$\frac{d U_t}{dt} = (A + w(t)B)U_t$
where $A,B \in \mathfrak{su}(n)$ generate the ...

**1**

vote

**0**answers

31 views

### solution of an infinite horizon optimization problem

Give the following formulation:
$\min_{\{x_s(t):\forall s,t\}} \sum_{s \in \mathcal{S}} \mathbf{1}\left(\lim_{T\rightarrow \infty} \frac{1}{T} \sum_{t=1}^T \frac{y_s(t)}{x_s(t)}\leq 1\right)$
$s.t. ...

**0**

votes

**1**answer

57 views

### Solving a nonlinear optimisation problem

I have the following nonlinear optimisation problem arising in my model.
$$\min \sum_{k=0}^{N-1} (\tau-t_k)^+\quad \text{ s.t. } {\mathbf{x}^\top\mathbf{w}\le W,\ \mathbf{x}\ge0}, t_k=t_{k-1}+x_k ...

**1**

vote

**2**answers

194 views

### The term for problems “like” Brachistocrone?

Is there a commonly-accepted umbrella term for infinite-dimensional calculus problems where the goal is to compute an optimal geometric path between a pair of points? Three examples of this would be ...

**2**

votes

**2**answers

107 views

### Orthogonal Procrustes problem with Absolute Values

Given a non-negative matrix $T$, how to find a orthogonal matrix $O$ minimising
$$ \left\lVert \, |O| - T \right\lVert_F,$$
where $\lVert \cdot \lVert_F$ denotes the Frobenius norm, and $| \cdot |$ ...

**2**

votes

**0**answers

86 views

### Solve non-linear Optimization Problem [closed]

I have to find $x$ that minimizes: $$ \sum_{k}(x^H\textbf A_kx - b_k)^2$$ where $A_k$ are 4 x 4 positive definite matrices ($A_1, A_2,...A_k$), $x$ is 4 x 1 vector and $b_k$ are scalars ...

**1**

vote

**0**answers

160 views

### A path optimisation problem

Consider a graph of $n$ nodes randomly located in $[0,1]^2$. Each node moves following a path randomly chosen from the set of all possible paths. Regard nodes as attackers. A policeman seeks an ...

**0**

votes

**0**answers

22 views

### Maximizing modular function subject to supermodular constaint

I'm trying to solve a constrained optimization problem with submodular functions and get some nice properties of the solution. Unfortunately, I think I am in a setting where Topkis' theorem does not ...

**3**

votes

**1**answer

121 views

### Singular curves of affine distributions on a Lie group

Are there any results about the rigidity of singular curves of rank 1 affine distributions on a connected compact Lie group?
Specifically the case of a right invariant affine distribution: $D_{U} = ...

**2**

votes

**0**answers

70 views

### Learning rule for recurrent neural network with flexible time steps

Summary: I want to train a recurrent network to output some answers, but the recurrent network is allowed to re-iterate through itself a flexible number of times for each input-output pair.
Why this ...

**2**

votes

**0**answers

49 views

### Attainability of Global Optima In Optimal Control

Given a manifold $M$ and a set of smooth functions of one real variable $\mathcal{A}$ and a 'control system' type first order differential equation:
$\frac{d x(t)}{dt} = F(x,u)$
one can consider the ...

**2**

votes

**0**answers

76 views

### A (non-convex) minimization quadratic programming problem with d constraints

Minimize $0<\omega_{dd}<2$ subject to
$$\sum_{j=1}^{d}(\omega_{dj} - \omega_{ij})^{2} \geq 4, i=0,1,...,d-1,$$
where $-2<\omega_{ij}<2$ is known for $0 \leq i \leq d-1$ and $1 \leq j \leq ...

**1**

vote

**0**answers

100 views

### For of a special case of $Ax>=b$, are there always integer solutions?

The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$.
The output is a matrix $A(n\times n)$ ...

**4**

votes

**2**answers

94 views

### Questions concerning convergence rate of Iterated Projections

Assume for simplicity $C_1,C_2,...,C_n\subset \mathbb{R}^m$ to be closed and convex subsets with $\underset{i=1}{\overset{n}{\bigcap{}}}C_i\neq\emptyset$.
Let $x^0\in\mathbb{R}^n$ and define the ...

**2**

votes

**0**answers

53 views

### Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$:
$V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$
...

**1**

vote

**0**answers

71 views

### Fréchet differentiability of functional defined by a integral [closed]

I want to prove that if the functional $I: \mathcal{C}^1[t_0,t_f] \rightarrow \mathbb{R}$ defined by
$$
I(x) = \int_{t_0}^{t_f} F(x, \dot{x},t)\,dt
$$
is Fréchet differentiable if $F$ is ...

**2**

votes

**0**answers

79 views

### About optimizing a convex function on a hypercube

Given a real valued convex function $g$ on $[-1,1]^n$, let $f$ be the restriction of it on the hypercube $\{-1,1\}^n$. I want to find a vertex on the hypercube $\{-1,1\}^n$ on which either (1) $f$ ...

**5**

votes

**2**answers

135 views

### The use of modules in control theory

So far I have seen the use of vector spaces in control theory and other notions from linear algebra; So I wonder if there's a use of this abstraction of modules over rings in control theory? any ...

**0**

votes

**0**answers

63 views

### Law of motion when initial condition is perturbed

We know how to find the law of motion (Ito process) of the value function:
$$V_t(x)=E\Big{[}\int^{T}_te^{-r (s-t)}f(s,X_s)ds+e^{-r (T-t)}g(T, X_{T})|\mathcal{F}_t\Big{]}$$
such that
...

**3**

votes

**1**answer

42 views

### Parameter uncertainties in LQR

I have a question about LQR.
I apply optimal controller by solving Ricatti equation based on normal plant. Suppose that I have one or few parameter variations in the plant that changes some values in ...

**3**

votes

**2**answers

122 views

### Level sets on $SU(4)$

Given $G \in SU(4)$, what are the level sets of the function $F:SU(n)\rightarrow \mathbb{R}$ defined by $F(V) = |tr(G^{\dagger}V)|^2$?
Can they be written only in terms of abstract linear maps, not ...

**2**

votes

**1**answer

207 views

### Simultaneous integral equation on $SU(n)$

Consider a smooth curve $U_s:[0,T] \rightarrow SU(4)$ which solves:
$\frac{d U_s}{ds} = (a + w(s)b)U_s$
for some given $a,b \in \mathfrak{su}(4)$ (which generate $\mathfrak{su}(n)$) and a smooth ...

**0**

votes

**0**answers

46 views

### Continuous time dynamic programming: Quadratic guess for value function

In a control problem like so:
$$J = min \int_0^{t_f} Qx^2 + Ru^2 dt $$
$$\dot{x} = Ax + Bu$$
$$x(0) = x_0$$
The regular Linear Quadratic Regulator is attained by asssuming that the optimal value ...

**10**

votes

**2**answers

244 views

### A generalization of Chebyshev polynomials

What is the monic polynomial $p(x)$ of degree $n$ which minimizes $\max_{x \in [-1,1]} |p(x)|$? The answer is the Chebyshev polynomial, and its largest value on $[-1,1]$ is $1/2^{n-1}$.
Now suppose ...