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

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

726 questions

**0**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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**

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**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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**

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**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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),
\...