**-1**

votes

**1**answer

46 views

### About the critical points of quasi-convex functions

What do we know about the structure of critical points of quasi-convex functions?
I am looking for statements like "the critical points of a quasi-convex function are always either a global minima ...

**0**

votes

**0**answers

42 views

### Conditions for supremum and conditional Expectation to commute

I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and have (for simplicity) two stopping times $\tau_1$ and $\tau_2$ such that $\tau_2 \leq \tau_1$ and $U:\Bbb R\...

**3**

votes

**1**answer

129 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 $\pi_1,\...

**0**

votes

**0**answers

14 views

### Under which conditions discretization of convex\concave function is submodular?

Say, I have $f(x)$ with $x \in [0,1]$, then by discretization I mean $f(x_h)$ with $x_h \in \{0, h, 2h, \dots, 1\}$.
I know about Lovasz extension, but it works in other way: given discrete function ...

**2**

votes

**2**answers

126 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 $| \cdot |$ ...

**0**

votes

**0**answers

38 views

### optimize a Quadratic Matrix Programming with multi-spherical constraints

I have got the following quadratic problem restricted on the Cartesian product of Euclidean spheres.
$\underset{X \in \mathbb{R}^{n\times 3}}{\text{min}}$ $Q(X) = \frac{1}{2} Tr(X^TA X) + Tr(B^T X)$
...

**0**

votes

**1**answer

422 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 ...

**2**

votes

**2**answers

99 views

### Fixed point iteration on symmetric biconvex function

Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not ...

**1**

vote

**0**answers

85 views

### Optimal Control / Hamilton-Jacobi-Bellman Equation

I face the following optimal control problem: let $X=(X_{1},X_{2})^{\top}$ be a controlled (Ito-)process with dynamics
$dX_{1}(t)=\big(X_{1}(t)\,\mu_{1} + a(t)\big)\;dt + X_{1}(t)\,\sigma_{1}\;dW_{1}(...

**2**

votes

**1**answer

73 views

### Optimal control / Portoflio optimization: Maximize expected utility of total consumption

I came across a portoflio optimization problem, where I need to solve for optimal investment and consumption processes, such that the expected utility of total consumption and terminal wealth is ...

**0**

votes

**1**answer

94 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 ...

**7**

votes

**5**answers

1k views

### Robust black box function minimization with extremely expensive cost function

There is an enormous amount of information about the common applied math problem of minimizing a function.. software packages, hundreds of books, research, etc.
But I still have not found a good ...

**2**

votes

**1**answer

34 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 ...

**7**

votes

**5**answers

1k views

### solving Lyapunov-like equation

The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...

**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 ...

**6**

votes

**1**answer

1k views

### Difference between 'generalized gradient' and 'subgradient' ?

Hi, I am wondering what the difference between 'generalized gradient' and 'subgradient' of a (potentially non-differentiable) convex function 'f' is.
The generalized gradient I am interested in is ...

**14**

votes

**4**answers

483 views

### Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant
$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i)
$$
over all $a_0,\dots,a_{n-1}$ such ...

**2**

votes

**1**answer

37 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 x_{...

**8**

votes

**1**answer

270 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 ...

**5**

votes

**3**answers

350 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 Euler-...

**1**

vote

**0**answers

72 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 ...

**1**

vote

**1**answer

177 views

### Max Absolute Difference of Expectations under Change of Measure

Given a sample space $\Omega=\{ 1,\cdots,N \}$, a random variable $x$ defined on $\Omega$ that takes value $x_1,\cdots,x_N$, and a set of strictly positive real numbers $w_1,\cdots,w_N$. Define for ...

**0**

votes

**0**answers

68 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 ...

**1**

vote

**1**answer

193 views

### Covering max flow arcs by arc disjoint paths

Let $(N,A,s,t,u)$ be a network with node set $N$, arc set $A$, source $s\in N$, sink $t\in N$ and capacity vector $u\in\{1,2,\ldots,T\}^A$, and let $x=(x_a)_{a\in A}$ be a maximum $(s,t)$-flow. Is it ...

**0**

votes

**0**answers

51 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 ...

**1**

vote

**0**answers

57 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 $Q_1{...

**3**

votes

**1**answer

136 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 ...

**9**

votes

**4**answers

541 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 ...

**0**

votes

**0**answers

52 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 ...

**0**

votes

**1**answer

101 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 ...

**1**

vote

**1**answer

45 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

29 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 _{i}<=1$...

**0**

votes

**0**answers

65 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 $s \in[0,T]$ ...

**2**

votes

**2**answers

186 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\ \...

**10**

votes

**2**answers

291 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 ...

**0**

votes

**0**answers

27 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 ...

**4**

votes

**1**answer

298 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

90 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

**0**answers

19 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 ...

**11**

votes

**4**answers

4k views

### “You can't push a rope” [closed]

"You can't push a rope" is a wisdom saying that some engineering teachers pass along to their students. Since I'm not an engineer, I can only guess at what they mean, but it sounds to me like code ...

**0**

votes

**0**answers

23 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 ...

**0**

votes

**0**answers

21 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} ...

**0**

votes

**0**answers

48 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 ...

**4**

votes

**0**answers

201 views

### stochastic control / geometric mean

Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...

**1**

vote

**0**answers

21 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

52 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 ...

**3**

votes

**1**answer

373 views

### A min-max formula for depth of the origin in a convex set

Depth is defined as the distance to the boundary of a set, i.e., $\operatorname{depth}(x, C) = \operatorname{dist}(x, \mathbb{R}^n \backslash C)$.
Let $C$ be a convex set that contains the origin. I ...

**1**

vote

**0**answers

32 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 \...