Questions tagged [oc.optimization-and-control]
Operations research, linear programming, control theory, systems theory, optimal control, game theory
726 questions
0
votes
0answers
29 views
Lower semicontinuity of a multi-valued map $F:X\to 2^Y$ in term of net
Let $X,Y$ be two Hausdorff spaces and $F:X\to 2^Y$ be a multi-valued mapping. We says that $F$ is lower semicontinuous at $x_0\in X$ if for each $y_0\in F(x_0)$ and any neighborhood $U\in \mathcal N(...
0
votes
0answers
23 views
Differential equations of system dynamics and adjoint dynamics, non-smooth dynamics
I have to solve (numerically) the following system of differential equations resulting from dynamic optimization with variation calculus (Canonical equations).
One of the problems for me is that parts ...
0
votes
0answers
12 views
Minimization of Hamiltonian for input affine with consumption optimal cost function
In my textbook on optimal control the following is stated:
The minimization problem
$$min_{\vec{u}\in [\vec{u}^+,\ \vec{u}^-]} H(\vec{u}) = \vec{r}^\top\cdot |\vec{u}| + \vec{q}^\top(\vec{x}, \vec{\...
1
vote
2answers
54 views
Controllability Gramian asymptotics for small times
Set-up. Consider the following linear controlled system
$$
\dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1)
$$
where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ...
2
votes
0answers
32 views
Strong stability of the wave equation with time depending potential
It is well known that the wave equation with frictional damping
$$\eqalign{
& {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr
...
-1
votes
0answers
102 views
+100
KKT for dual of a quadratically constrained linear program
Let $ \mathcal{P}$ be a linear program with quadratic constraints,
$$ \eqalign{
\mathcal{P}: & \min_{x,t} -t \\
& \text{s.t.} \\
{\color{orange}{\mu_k}}: & a^H_k x + a^T_k x^{*} - x^H ...
15
votes
1answer
198 views
Finding a plane numerically
Suppose I have three large finite sets $\{x_i\}$, $\{y_i\}$ and $\{z_i\}$;
they are obtained by measuring coordinates of a collection of vectors in $\mathbb{R}^3$, but I do not know which triples ...
1
vote
1answer
68 views
Constrained optimization of sum of squares polynomials
Consider the problem
$$
\min p(x) \text{ subject to } g_j(x)\le 0
\quad
p,g_j\in\text{SOS},
\qquad
(*)
$$
i.e. $p,g_j$ ($j=1,\ldots,m$) are sum of squares (SOS) polynomials. Can this problem be ...
5
votes
0answers
75 views
Projection of a function $f\in L^1(\Omega)$ onto a finite dimensional subspace
Suppose $\Omega \subset \Bbb R^n$ be a domain such that $|\Omega|<\infty$, $f\in L^1(\Omega)$. Let $Y= \text{span}\{g_1,\dots, g_k\}$.
Is there a characterization of the set of projections of $f$...
0
votes
0answers
53 views
Maximizing the function $x_1+(k-x_1)x_2+(k-x_1)(k-x_2)x_3+\dots$
$$\begin{array}{ll} \text{maximize} & x_1+(k-x_1)x_2+(k-x_1)(k-x_2)x_3+\dots+(k-x_1)(k-x_2)\cdots (k-x_{n-1})x_n\\ \text{subject to} & \displaystyle\sum_{i=1}^n x_i\leq m\end{array}$$
where $...
4
votes
0answers
47 views
Can a nonlinear dynamical system be rewritten in terms of constraints?
My question is based on thoughts after reading to a specific section in the paper "On Contraction Analysis for Nonlinear Systems" by W. Lohmiller and JJ. Slotine, Section 4.2 Constrained Systems. ...
0
votes
0answers
70 views
Question on h-infinity norm of a system
Consider a control system,
$\dot{x}=Ax+Bu\\
y=Cx$.
Define the transfer function $G(s)=C(sI-A)^{-1}B$.
Then it is claimed that the following definitions of induced norm are equivalent.
$\|G\|_{\...
0
votes
0answers
19 views
Can the partial concavity of the following decomposable objective function be used for optimization?
The problem I am trying to solve is the following:
$$\begin{array}{ll}
\min & f(x)+g(y) \\
\mathrm{s.t.} & y\ge x\ge 0,\\
\ & p\le ax+by\le q,
\end{array}$$
where $a,b,p,q$ are ...
1
vote
1answer
50 views
Measuring how suboptimal control is
Suppose I have a linear dynamical system to control. I use PMP to find necessary conditions for the optimal control of the system wrt to some objective function. Now, suppose that the trajectory I ...
5
votes
0answers
140 views
Rigorous proof of the good regulator theorem
As an applied mathematician with an interest in control theory, I have read several research papers that explicitly use the good regulator theorem of Conant and Ashby 1 which states that:
Every ...
3
votes
1answer
185 views
Best approximation of a compactly supported density by a single Gaussian
Note: This is a follow-up question inspired by a previous (more difficult) question I asked on MathOverflow.
Let $f:\mathbb{R}\to\mathbb{R}$ be a (sufficiently regular, e.g. smooth) probability ...
0
votes
1answer
66 views
Conditioning on an irrelevant variable in a martingale control problem
Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued $\...
1
vote
1answer
117 views
Trajectory leaving a set
Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...
3
votes
2answers
82 views
Expected minimum of a linear function on the unit cube
Let $c\in\mathbb{R}^n$ and let $X_1,X_2$ be two independent uniform samples on the unit cube in $\mathbb{R}^n$. Is there anything at all (in an analytic sense) we can say about the expectation $E\min\...
0
votes
0answers
35 views
Convex optimization problem with really simple submodular structure
I am trying to characterize the solutions to the below convex optimization problem as concisely as possible, where we are given as input a probability vector $\mathbf{p}\in\mathbb{R}^n$ and a positive ...
1
vote
0answers
22 views
Optimal control problem with spike source and “split” state
For $p \in \mathbb{R}$, consider the following problem:
\begin{equation} \label{1}
\begin{cases}
\operatorname{div}(a \nabla u ) = p\delta_{x_0} \quad \text{in } \Omega \\
u=0 \quad \text{on } \...
3
votes
0answers
162 views
Equal principal minors of matrix plus rank-1 and inverse
Given an invertible real matrix $A$ and real column vectors $b$ and $c$.
For which $A$, $b$ and $c$ are all corresponding principal minors of $B = A-bc^T$ and
$A^{-1}$ equal?
According to a ...
6
votes
0answers
249 views
Geometric bang-bang theorem for nonlinear optimal control
The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems ...
0
votes
0answers
31 views
Single step analysis of Augmented Lagrangian Method
I am wondering if there is any single step analysis for the Augmented Lagrangian method. Specifically, the problem is
$$\min f(x) \text { s.t. } A x=b$$
where $f$ is convex, smooth. Such an objective ...
5
votes
0answers
118 views
How to choose phase to give a desired Fourier transform
Cross posted from MSE.
I have a mathematical problem arising from a physics application, which I feel must have been solved before, but I don't know the terminology associated with it. I am looking ...
0
votes
0answers
14 views
Parametric research for the global minimum in a family of polynomial multivariable functions on closed domains
Consider the family of functions:
$$
V(\{x_j\},\{y_j\})=\sum_{j=1}^L\left[\frac{1}{2} x_j^2+ \frac{\beta^2}{2}y_j^2 + \alpha\beta\, x_jy_j \right]
$$
Each member of the family is therefore ...
1
vote
0answers
19 views
Simple monotonicity property for coordinate descent and linear objective functions
Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for ...
-1
votes
1answer
52 views
How to encode minimality constraint into SAT? [closed]
How to encode maximality/minimality constraints in SAT or its variants such as MaxSAT or MinSAT? For example, let us say (x1 OR x2) AND (x2 OR x3 OR x4) AND (x4 OR x5) is a formula. I want its ...
0
votes
0answers
35 views
Stochastic Control: Markovian restriction
Consider a stochastic control problem,
$$v^C(0,x) = \mathbb{E} \Big[\int_0^\tau f(X_t,C_t) d t + (T-\tau)|X_\tau|\Big] $$
where $X_t$ is a weak solution to the SDE
$$dX_t = C_t dB_t, \quad X_0 = x \...
0
votes
0answers
33 views
Unique solution for ODE optimal control
In the basic theory of optimal control we must have a unique absolutely continuous function as a solution to a differential system. I will choose the LQR (Linear Quadratic Regulator problem):
$$\...
0
votes
0answers
14 views
Linear quadratic regulator equivalence of formulations
I don't see why the following three forms of the LQR optimal control problem are equivalent:
For $\begin{cases} x'=Ax+Bu \\ x(t_0)=x_0\end{cases}$ find
$$\min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u)=\...
2
votes
0answers
41 views
Solving Mixed-Integer Non-Linear Optimization Problem
I would like to solve the following optimization problem:
\begin{array}{ll}
\underset{x_{i}\geq0,\, \pi_{i}\in\{0,1\}}{\text{minimize}} & \displaystyle\sum_{i=1}^n x_i\\
\text{subject to} & ...
0
votes
0answers
33 views
A right-inverse property of a regression problem
Disclaimer: This might be a silly question. However, after some days of thought, I could not find a clear/rigorous answer. So I decided to post it here.
Let $Y\in\mathbb{R}^{n\times p}$ and $X\in\...
2
votes
2answers
63 views
Program to solve Optimization Problem
I have an optimization problem, this problem has linear constraints and nonlinear constraints. I solved the linear part by MATLAB but the nonlinear constraints I could not solve it. I downloaded ...
0
votes
0answers
62 views
How to solve optimization with matrix pseudo inverse
Thank everybody in advance. I'd like to solve an optimization problem for a matrix function $f(C)$. However, the matrix pseudo-inverse constraint gives me big troubles.
Vectors $V_{n\times 1}$, $F_{m\...
1
vote
0answers
28 views
Asymptotics for a random set cover problem
Suppose you are given a positive integer $k$ and a probability distribution $f$ on the positive reals. I am interested in the limiting behavior of the following process as $n\to\infty$:
Create an ...
0
votes
0answers
29 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
$$
v(x) = \max_{\{u(t)\}_t} \int_o^\...
11
votes
3answers
623 views
Famous theorems that are special cases of linear programming (or convex) duality
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any ...
0
votes
0answers
33 views
Maximizing the sum of piecewise linear functions using approximate, differentiable functions
Given an arbitrary (sorted) set of numbers $\{c_1,\dots,c_n\}$, define for each number the piecewise continuous linear function
$$
f_i(x) =\begin{cases}
x & 0\leq x\leq c_i \\
0 & ...
0
votes
0answers
44 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 ...
0
votes
1answer
56 views
Solving Problem: LMIs and block matrices
I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find ...
1
vote
0answers
41 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
40 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
37 views
Minimizer for Mean-Variance Portfolio Optimization [closed]
Let $\lambda \in (0,\infty).$ Does there exists a minimizer for the set
$$
\{ -\text{E}[X] + \lambda \text{Var}[X],\; X \in L^2(\Omega,\mathcal{F},P) \} ?
$$
2
votes
1answer
73 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
85 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 $f\colon (0,1]^n \to \mathbb{R}$,
$$f(x_1, \ldots, x_n) := \sum_{\...
1
vote
0answers
50 views
Optimizing a determinant
Any vector is assumed to be a column vector by default. Suppose $f(\mathbf{x})$ is the $d$-dimensional standard Gaussian density. I am interested in the following optimization problem:
$$
\max_g ~\...
1
vote
1answer
170 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
$$
\sup\limits_{t\in [0,N]}E[G(t,X(t))]=E[\sup\limits_{...
1
vote
1answer
99 views
About exchanging min and max and correctness of an inequality
Let $v_i \in \mathbb{R}^{n}, \ i=1, \ldots, m, \ \ $ $\mathcal{S}$ a convex polyhedron and $x \in \mathbb{R}^{n}$ be given. Consider the following solution $(s^{*},i^{*})$ to the problem
\begin{...
1
vote
0answers
36 views
Pontryagin principle for optimal control problem with a terminal cost involving the control
Let $T >0$ and consider the problem of minimizing
$$
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),
\...
