Questions tagged [oc.optimization-and-control]
Operations research, linear programming, control theory, systems theory, optimal control, game theory
0
votes
0answers
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
$$
v(x) = \max_{\{u(t)\}_t} \int_o^\...
0
votes
0answers
45 views
Techniques for minimizing function [on hold]
How does one obtain $W^*$ from Eq. (7) in
Shichao Zhang et al, Efficient kNN Classification With Different
Numbers of Nearest Neighbors, IEEE Transactions on Neural Networks and Learning Systems ...
1
vote
0answers
37 views
Markovian Control in a stochastic control problem
I have a very simple control problem.
\begin{align*}
V(t,X) &= \sup_{(C_t)_t} E_t \left[ |X_t| 1_{C_t = 0} + C_t \right] \\
\text{ s.t. } & d X_t = C_t d B_t, \quad X_t \in [-1,1], C_t \ge 0,...
10
votes
3answers
572 views
Famous theorems that are special cases of linear programming (or convex) duality
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any ...
0
votes
0answers
21 views
Maximizing the sum of piecewise linear functions using approximate, differentiable functions
Given an arbitrary (sorted) set of numbers $\{c_1,\dots,c_n\}$, define for each number the piecewise continuous linear function
$$
f_i(x) =\begin{cases}
x & 0\leq x\leq c_i \\
0 & ...
0
votes
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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 $f\colon (0,1]^n \to \mathbb{R}$,
$$f(x_1, \ldots, x_n) := \sum_{\...
1
vote
0answers
40 views
Optimizing a determinant
Any vector is assumed to be a column vector by default. Suppose $f(\mathbf{x})$ is the $d$-dimensional standard Gaussian density. I am interested in the following optimization problem:
$$
\max_g ~\...
0
votes
0answers
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
0answers
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
1answer
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
$$
\sup\limits_{t\in [0,N]}E[G(t,X(t))]=E[\sup\limits_{...
1
vote
1answer
81 views
About exchanging min and max and correctness of an inequality
Let $v_i \in \mathbb{R}^{n}, \ i=1, \ldots, m, \ \ $ $\mathcal{S}$ a convex polyhedron and $x \in \mathbb{R}^{n}$ be given. Consider the following solution $(s^{*},i^{*})$ to the problem
\begin{...
1
vote
0answers
21 views
Pontryagin principle for optimal control problem with a terminal cost involving the control
Let $T >0$ and consider the problem of minimizing
$$
P(v(.)) \triangleq \int_0^T l(x_t,v_t) d t + h(x_T)
$$
over a broad class of control $v(.)$ where
\begin{equation}
\dot x_t = f(x_t,v_t),
\...
2
votes
0answers
65 views
What is the most efficient path for a robot without turning radius?
I recently programmed a most efficient path for a robot going from $(x_1, y_1 , \theta_1)$ to $(x_2, y_2)$. The bot does a combination of turning and moving at any given point, based on the difference ...
0
votes
0answers
21 views
A question related to parametric linear programming
Consider the following parametric linear problem:
\begin{align}
\min z(t)=c^T x\\
Ax=b(t)\\
0\leq x\leq u.
\end{align}
We know $z(t)$ is a piecewise linear function. Let $x(t_1)$ and $x(t_2)$ be the ...
0
votes
2answers
144 views
Is it possible to “solve” iterative (convex/non-convex) optimization problems via learning (one-shot)?
I posted a following question in MSE, but I think it should be posted here in MO. Since I don't know how to transfer the post from MSE to MO, I have pasted the question below. Thank you in advance and ...
2
votes
0answers
74 views
A community effort: equilibrium in quitting games [closed]
This thread is in the spirit of the polymath project:
a combined effort of the community to solve a difficult open problem.
It is an activity of the European Network for Game Theory
whose goal is to ...
9
votes
2answers
261 views
Maximum of a quantity for two normal orthogonal vectors in $\mathbb{R}^n$
Let's define for every pair of vectors $u,v\in\mathbb{R}^n$, a quantity as follows:
$$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
0answers
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
1answer
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
1answer
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
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
3
votes
1answer
157 views
it's convex sequence inequality
A sequence $a_0,a_1,\dots,a_n$ of real numbers is called concave if $a_{0}=0$, and for each $0<i<n$, we have $a_i\geq\dfrac{a_{i-1}+a_{i+1}}{2}$.
Find the largest $c(n)$ such that for every ...
16
votes
4answers
831 views
What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?
My question is to find the minimum of the following expression:
$$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$
over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...
0
votes
1answer
75 views
maximization of a log norm function
Considering the following optimization program:
$$
maximize \ \ \ \log \left( \|x\|_\infty \right)
$$
$$
subject \ to \ \ Ax\leq b, \ x \geq 0
$$
can we rewrite this program as a convex ...
7
votes
2answers
253 views
A (linear) optimization problem subject to (linear) matrix inequality constraints
Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
1
vote
1answer
69 views
LASSO problem but with a maximization instead of minimization
I have the following optimization problem (like the LASSO problem but with maximization instead of minimization):
$\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...
2
votes
0answers
170 views
A parametrization of stable matrices
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is ...
1
vote
1answer
129 views
Linear matrix inequality
I have the following linear matrix inequality:
$$F^T P + PF < 0,$$
where $P$ is a positive definite matrix and $F$ is a matrix with appropriate dimension.
Let $Q$ be a positive definite matrix ...
2
votes
0answers
84 views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
0
votes
0answers
53 views
Using mollifiers (or some other idea) to solve constrained minimax problem
Sorry in advance if this sounds like a more SE question.
Consider a continuously parametrized family of $L$-Lipschitz continuous $f_\theta: X \rightarrow \mathbb R_+$ on a metric space $X=(X,d)$. Let ...
1
vote
0answers
37 views
Minimum Preserving Transformations [closed]
If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then
$$
\operatorname{argmin}_{x \in X} f(x)
=
\operatorname{argmin}_{x \in X} g\circ f(x) .
$$
X and Y ...
9
votes
2answers
389 views
A trace-constrained maximization problem in the cone of positive definite matrices
Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...
1
vote
0answers
100 views
Bounds of Procrustes problem
We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following:
\begin{align}
f(A,B)...
1
vote
1answer
104 views
How to solve this system of Matrix equations? (Coupled riccati equations?)
I am trying to solve for K in the following problem:
$ 3I = A_1 + A_2 + A_3$
$ A_1 K A_1 = K_1 $
$ A_2 K A_2 = K_2 $
$ A_3 K A_3 = K_3 $
Where $I$ is the identity, $K, K_1, K_2, K_3, A_1, A_2, ...
3
votes
2answers
234 views
A structural optimization problem
I have $n$ objects $O_i$, each of them having $3$ values, $O_i = (A_i, B_i, C_i)$. I am trying to group them into $k$ groups $P_u$ such as $P_u =(A_u, B_u, C_u)$ such that
$$\text{minimize} \quad \...
0
votes
0answers
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
2answers
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
1answer
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
$$
\dot x(t) = f(x(t),u(t)), ...
0
votes
0answers
91 views
Minimizing Frobenius norm with sparsity constraints
I am trying to solve the following minimization problem
\begin{equation*}
\begin{aligned}
& \underset{X,Y \in \mathbb{R}^{n\times k}}{\text{minimize}}
& & \| X Y^\top A - B \|_{\text F}^2 ...
2
votes
1answer
115 views
Optimal-score partitions
The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...
0
votes
1answer
62 views
Boundary controllability of the heat equation and observation
I'm studying the boundary controllability of the heat equation
\begin{array}{c}
y_{t}=\Delta y\text{ in }\Omega \times (0,T), \\
y=\mathbf{1}_{\Gamma }u\text{ on }\partial \Omega \times (0,T), \\
u(...
3
votes
4answers
221 views
References for control-affine theory
In Control theory from the geometric viewpoint by Agrachev and Sachkov, the authors mention the concept of control-affine (affine in control $u_i$) systems:
$$\dot{x} = (f_0 + \sum_{i=1}^{m} u_i f_i)x ...
0
votes
0answers
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
0answers
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
0answers
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
1answer
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 ...
