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        Questions tagged [oc.optimization-and-control]

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

        1
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        1answer
        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 $x(0) \in A$, ...
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        2answers
        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\...
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        0answers
        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 ...
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        0answers
        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 } \...
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        votes
        0answers
        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 ...
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        0answers
        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 ...
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        0answers
        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 ...
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        0answers
        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 ...
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        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 ...
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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 ...
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        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 ...
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        0answers
        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 \...
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        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): $$\...
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        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)=\...
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        0answers
        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} & ...

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