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Questions tagged [oc.optimization-and-control]

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

0
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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 $f$...
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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
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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
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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
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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 & ...
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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
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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
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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), \...