# Questions tagged [oc.optimization-and-control]

Operations research, linear programming, control theory, systems theory, optimal control, game theory

**0**

votes

**0**answers

10 views

### Reference request: Numerical methods for Hamilton-Jacobi-Bellman equations with state constraints

I have two questions on numerical methods for solving Hamilton-Jacobi-Bellman (HJB) equations with state constraints.
Consider an optimal control problem given by
$$
v(x) = \max_{\{u(t)\}_t} \int_o^\...

**0**

votes

**0**answers

45 views

### Techniques for minimizing function [on hold]

How does one obtain $W^*$ from Eq. (7) in
Shichao Zhang et al, Efficient kNN Classification With Different
Numbers of Nearest Neighbors, IEEE Transactions on Neural Networks and Learning Systems ...

**1**

vote

**0**answers

37 views

### Markovian Control in a stochastic control problem

I have a very simple control problem.
\begin{align*}
V(t,X) &= \sup_{(C_t)_t} E_t \left[ |X_t| 1_{C_t = 0} + C_t \right] \\
\text{ s.t. } & d X_t = C_t d B_t, \quad X_t \in [-1,1], C_t \ge 0,...

**10**

votes

**3**answers

572 views

### Famous theorems that are special cases of linear programming (or convex) duality

The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any ...

**0**

votes

**0**answers

21 views

### Maximizing the sum of piecewise linear functions using approximate, differentiable functions

Given an arbitrary (sorted) set of numbers $\{c_1,\dots,c_n\}$, define for each number the piecewise continuous linear function
$$
f_i(x) =\begin{cases}
x & 0\leq x\leq c_i \\
0 & ...

**0**

votes

**0**answers

29 views

### Matlab book request for optimal control

My subject is optimal control of PDE and I want to put optimal control problems in Matlab to be solved but I don't know how to do this. I would be grateful if you could give me some references in that ...

**1**

vote

**0**answers

39 views

### Curvature of projection function onto a smooth curve

Suppose we have a smooth curve $C$ lying in $\mathbb{R}^2$, and let us consider the orthogonal projection function $P_C(x)$ onto the curve, described by
$$P_C(x) = argmin_{y \in C} \Vert x - y \Vert$$...

**0**

votes

**0**answers

30 views

### Reference: Stochastic Optimal Control with cost functional

There are a variety of control problems for controlled diffusions $X_t^u$, with the terminal cost given by
$$
J(u)\triangleq \mathbb{E}\left[g(X_T,u)+\int_0^t h(X_t,u_t)ds\right],
$$
function $g$ and ...

**0**

votes

**1**answer

32 views

### Minimizer for Mean-Variance Portfolio Optimization [on hold]

Let $\lambda \in (0,\infty).$ Does there exists a minimizer for the set
$$
\{ -\text{E}[X] + \lambda \text{Var}[X],\; X \in L^2(\Omega,\mathcal{F},P) \} ?
$$

**1**

vote

**1**answer

71 views

### Literature on the controllability of networks under attack

I would like to request your advice on a problem arising from my research in the life sciences. Consider a modular, sparse weighted network which is partially controllable in the sense that some ...

**4**

votes

**1**answer

72 views

### Optimizing a multivariate symmetric (permutation-invariant) function

Let $\ell$ and $d$ be two integers such that $\ell \le d$.
I would like to find the global maxima of the following symmetric function $f\colon (0,1]^n \to \mathbb{R}$,
$$f(x_1, \ldots, x_n) := \sum_{\...

**1**

vote

**0**answers

40 views

### Optimizing a determinant

Any vector is assumed to be a column vector by default. Suppose $f(\mathbf{x})$ is the $d$-dimensional standard Gaussian density. I am interested in the following optimization problem:
$$
\max_g ~\...

**0**

votes

**0**answers

59 views

### Is dynamic programming suitable for a specific optimization problem?

Let $c,\,\mathcal{P}_0,\,\mathcal{P}_1,\,\mathcal{P}_2,\ldots$ be a sequence of positive real numbers.
Let $N\in\{1,\,2,\,3,\ldots\}$ and let $t\in\{0,\,1,\,2,\ldots\}$, with $N$ and $t$ fixed.
...

**0**

votes

**0**answers

34 views

### optimisation problem for unknown function

I have two variables function $z=f_{a,b}(x,y)$ ($a,b$ - some parameters), however I don't know its formula (I can compute its value for given $x,y$ and $a,b$). I would like to find minimum of this ...

**1**

vote

**1**answer

100 views

### When do supremum and expectation commute?

This is an alternative form of the question in When do maximum and expectation commute?
When we looking for conditions on $G(t,x(t))$ such that
$$
\sup\limits_{t\in [0,N]}E[G(t,X(t))]=E[\sup\limits_{...

**1**

vote

**1**answer

81 views

### About exchanging min and max and correctness of an inequality

Let $v_i \in \mathbb{R}^{n}, \ i=1, \ldots, m, \ \ $ $\mathcal{S}$ a convex polyhedron and $x \in \mathbb{R}^{n}$ be given. Consider the following solution $(s^{*},i^{*})$ to the problem
\begin{...

**1**

vote

**0**answers

21 views

### Pontryagin principle for optimal control problem with a terminal cost involving the control

Let $T >0$ and consider the problem of minimizing
$$
P(v(.)) \triangleq \int_0^T l(x_t,v_t) d t + h(x_T)
$$
over a broad class of control $v(.)$ where
\begin{equation}
\dot x_t = f(x_t,v_t),
\...

**2**

votes

**0**answers

65 views

### What is the most efficient path for a robot without turning radius?

I recently programmed a most efficient path for a robot going from $(x_1, y_1 , \theta_1)$ to $(x_2, y_2)$. The bot does a combination of turning and moving at any given point, based on the difference ...

**0**

votes

**0**answers

21 views

### A question related to parametric linear programming

Consider the following parametric linear problem:
\begin{align}
\min z(t)=c^T x\\
Ax=b(t)\\
0\leq x\leq u.
\end{align}
We know $z(t)$ is a piecewise linear function. Let $x(t_1)$ and $x(t_2)$ be the ...

**0**

votes

**2**answers

144 views

### Is it possible to “solve” iterative (convex/non-convex) optimization problems via learning (one-shot)?

I posted a following question in MSE, but I think it should be posted here in MO. Since I don't know how to transfer the post from MSE to MO, I have pasted the question below. Thank you in advance and ...

**2**

votes

**0**answers

74 views

### A community effort: equilibrium in quitting games [closed]

This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...

**9**

votes

**2**answers

261 views

### Maximum of a quantity for two normal orthogonal vectors in $\mathbb{R}^n$

Let's define for every pair of vectors $u,v\in\mathbb{R}^n$, a quantity as follows:
$$f(u,v) = \sum_{1\leq i,j\leq n}|u_iu_j-v_iv_j|.$$
I want to find:
$$M(n)= \max \{f(u,v): u,v\in \mathbb{R}^n, |u|=...

**4**

votes

**0**answers

139 views

### Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?

Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...

**0**

votes

**1**answer

114 views

### How to solve this optimization problem efficiently? [closed]

Let, $D\in\mathbb{C}^{1\times M}$ is a row vector with $M$ elements
$V\in\mathbb{C}^{3^M\times M}$ is a given matrix
$T$ is a scalar (real and $>1$)
$\textbf{The problem at hand is as follows:}$
...

**2**

votes

**1**answer

63 views

### Echange of Infimum Integral with Pointwise Infimum

Setup
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...

**3**

votes

**1**answer

157 views

### it's convex sequence inequality

A sequence $a_0,a_1,\dots,a_n$ of real numbers is called concave if $a_{0}=0$, and for each $0<i<n$, we have $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.
Find the largest $c(n)$ such that for every ...

**16**

votes

**4**answers

831 views

### What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?

My question is to find the minimum of the following expression:
$$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$
over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...

**0**

votes

**1**answer

75 views

### maximization of a log norm function

Considering the following optimization program:
$$
maximize \ \ \ \log \left( \|x\|_\infty \right)
$$
$$
subject \ to \ \ Ax\leq b, \ x \geq 0
$$
can we rewrite this program as a convex ...

**7**

votes

**2**answers

253 views

### A (linear) optimization problem subject to (linear) matrix inequality constraints

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...

**1**

vote

**1**answer

69 views

### LASSO problem but with a maximization instead of minimization

I have the following optimization problem (like the LASSO problem but with maximization instead of minimization):
$\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...

**2**

votes

**0**answers

170 views

### A parametrization of stable matrices

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is ...

**1**

vote

**1**answer

129 views

### Linear matrix inequality

I have the following linear matrix inequality:
$$F^T P + PF < 0,$$
where $P$ is a positive definite matrix and $F$ is a matrix with appropriate dimension.
Let $Q$ be a positive definite matrix ...

**2**

votes

**0**answers

84 views

### An inequality concerning the solution of a Lyapunov equation

Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...

**0**

votes

**0**answers

53 views

### Using mollifiers (or some other idea) to solve constrained minimax problem

Sorry in advance if this sounds like a more SE question.
Consider a continuously parametrized family of $L$-Lipschitz continuous $f_\theta: X \rightarrow \mathbb R_+$ on a metric space $X=(X,d)$. Let ...

**1**

vote

**0**answers

37 views

### Minimum Preserving Transformations [closed]

If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then
$$
\operatorname{argmin}_{x \in X} f(x)
=
\operatorname{argmin}_{x \in X} g\circ f(x) .
$$
X and Y ...

**9**

votes

**2**answers

389 views

### A trace-constrained maximization problem in the cone of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...

**1**

vote

**0**answers

100 views

### Bounds of Procrustes problem

We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following:
\begin{align}
f(A,B)...

**1**

vote

**1**answer

104 views

### How to solve this system of Matrix equations? (Coupled riccati equations?)

I am trying to solve for K in the following problem:
$ 3I = A_1 + A_2 + A_3$
$ A_1 K A_1 = K_1 $
$ A_2 K A_2 = K_2 $
$ A_3 K A_3 = K_3 $
Where $I$ is the identity, $K, K_1, K_2, K_3, A_1, A_2, ...

**3**

votes

**2**answers

234 views

### A structural optimization problem

I have $n$ objects $O_i$, each of them having $3$ values, $O_i = (A_i, B_i, C_i)$. I am trying to group them into $k$ groups $P_u$ such as $P_u =(A_u, B_u, C_u)$ such that
$$\text{minimize} \quad \...

**0**

votes

**0**answers

52 views

### Stability of an equilibrium for a third order control system with an integral regulator limited by the stop-type element

In what publication is the following theorem prove carried out? The method of the Lyapunov functions is preferable. Thanks.
Consider the system consisting of the controlled object and regulator. The ...

**1**

vote

**2**answers

205 views

### Maximizing a function that is sum of gaussians

Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function
\begin{align}
\mathcal{K}(\mathbf{x},\mathbf{y})=
\alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...

**0**

votes

**1**answer

57 views

### how to impose a terminal condition in a minimisation problem?

Consider the problem of minimising
$$
J(u(.)) \triangleq \int_0^T l(x(t),u(t)) d t + \phi(x(T)), \: T> 0
$$
over a space of controls $\mathcal{U}$
with the constraint
$$
\dot x(t) = f(x(t),u(t)), ...

**0**

votes

**0**answers

91 views

### Minimizing Frobenius norm with sparsity constraints

I am trying to solve the following minimization problem
\begin{equation*}
\begin{aligned}
& \underset{X,Y \in \mathbb{R}^{n\times k}}{\text{minimize}}
& & \| X Y^\top A - B \|_{\text F}^2 ...

**2**

votes

**1**answer

115 views

### Optimal-score partitions

The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...

**0**

votes

**1**answer

62 views

### Boundary controllability of the heat equation and observation

I'm studying the boundary controllability of the heat equation
\begin{array}{c}
y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\
y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\
u(...

**3**

votes

**4**answers

221 views

### References for control-affine theory

In Control theory from the geometric viewpoint by Agrachev and Sachkov, the authors mention the concept of control-affine (affine in control $u_i$) systems:
$$\dot{x} = (f_0 + \sum_{i=1}^{m} u_i f_i)x ...

**0**

votes

**0**answers

34 views

### Should penalty and Lyapunov drift be normalised in drift-plus-penalty optimisation?

I want to use drift-plus-penalty method to optimize control decisions in the system that I am considering in my research. In particular, I want to minimize penalty function (which in my case is ...

**1**

vote

**0**answers

38 views

### Lower semicontinuity of proximal gradient descent sequence

I'm using an alternate optimization scheme to minimize a function $F(a,b,c) = G(a,b,c) + H(c)$ where $G$ is continuous and differentiable, $H$ is not differentiable but lower semi-continuous and ...

**1**

vote

**0**answers

49 views

### An “almost separable” optimization problem on a graph

I am trying to solve an unconstrained optimization problem with the following properties: I am given a graph $G=(V,E)$ with functions $f_{ij}:\mathbb{R}^2\to \mathbb{R}$ for each edge $(i,j)$. I am ...

**3**

votes

**1**answer

87 views

### Shortest Manhattan-norm paths among disjoint rectangles

I am looking for the fastest possible algorithm for solving the following problem: I am given a collection of disjoint axis-aligned rectangles in the plane, and I need to pre-process these rectangles ...