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

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

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**1**answer

96 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**

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**2**answers

76 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**

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**0**answers

33 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**

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**0**answers

18 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**

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**0**answers

160 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

234 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**

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**0**answers

28 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**

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**0**answers

112 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**

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**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**

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**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**

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**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**

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**0**answers

33 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**

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**0**answers

23 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**

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**0**answers

10 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**

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**0**answers

38 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} & ...