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

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

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### Why do bees create hexagonal cells ? (Mathematical reasons)

Question 0 Are there any mathematical phenomena which are related to the form of honeycomb cells ? Question 1 May be hexagonal lattice satisfy certain optimality condition(s) which are related to ...
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### How can I deploy a CG-Steihaug algorithm for trust region sub-problem solving?

I'm studying various optimization methods and on this occasion, I'm trying to tackle the Trust Region Problem by solving the sub-region problem with the Steihaug-CG algorithm in Python. I'm using the ...
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### Does the plane clustered to minimize sum distances^2 to clusters centers ( inertia / “K-means”) produce hexagonal clusters / hexagonal lattice?

"K-means" is the most simple and famous clustering algorithm, which has numerous applications. For a given as an input number of clusters it segments set of points in R^n to that given number of ...
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### Intuition behind Gubinelli derivative

I apologise for the confusion of the following sentences. I'm lazy to give more information about Rough path theory as Is a fairly broad subject. On page 14 of "A Course on Rough Paths With an ...
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### How to solve this stochastic optimization problem?

How one can solve the following stochastic optimization problem? \begin{align} \max\quad& \mathbb{E}[\mathbf{1}^{\mathrm{T}}X]\\ \text{s.t.} \quad& \mathbb{E}[\mathbf{A}X]\leq\mathbf{1}_{m\...
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### SOS polynomials with integer coefficients

A well known theorem of Polya and Szego says that every non-negative univariate polynomial $p(x)$ can be expressed as the sum of exactly two squares: $p(x) = (f(x))^2 + (g(x))^2$ for some $f, g$. ...
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### Eigenvalues of a matrix sum

I have a control system problem, which ends up in that the eigenvalues of the system matrix should have a negative real part, then the system is stable. The system matrix is real but not symmetric. ...
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### Conditions for optimal stationary strategies in MDPs

I have a specific markov decision process (MDP) which is generated from a problem in another domain. What I would like to show is that under the limit of means criterion (no discounting) the optimal ...
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### A minimax optimization with expectation operator

Let $\mathbf{X}$ be an $m\times n$ random matrix with Gaussian i.i.d entries with zero mean and unit variance. How we can think about the following optimization \begin{align} \max_m\mathbb{E}_\mathbf{...
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### $\arg\max$ in the dual norm of the nuclear norm

Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by $$\left \| X\right\| := \max\limits_{i \in \{1, \dots, \min\{m,n\}\} }\sigma_i (X)$$ whereas the nuclear norm is ...
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### Maximum of sum of exponential function

Let $x_1,\dots,x_n$ be a set of given vectors in $\mathbb{R}_{+}^d$. Let $c_1,\dots,c_n$ be given positive constants. I am interested in finding the vectors $w_1,\dots,w_n$ in $\mathbb{R}_{+}^d$ that ...
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### Consistency results of optimal control solutions using direct transcription

I have read a number of books on discrete time solutions to continuous time optimal control problems. What is not clear to me is what consistency results exist with respect to showing the discretized ...
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### Compute weights (analytically or numerically) so that they minimize an integral

I have a measurable function $f:E\to[0,\infty)$, measurable weights $w_1,\ldots,w_k:E\to[0,1]$ with $\sum_{i=1}^kw_i=1$, positive probability densities $q_1,\ldots,q_k:E\to[0,\infty)$ and measurable ...
I am encountering a constrained LS problem with the following structure: $$\text{min}_Q\ \sum_{i=1}^M ||Q_i X_i-Y_i||_F^2$$ $$\text{s.t. }\ Q_M Q_{M-1}\cdots Q_1=I,$$ where $Q_i,X_i,Y_i\in \mathbb{... 1answer 98 views ### Hardness of concave minimization problem I have an optimization problem$\underset{x}{\min} ~ c(x) - k \cdot x$where$c(x)$is a non-decreasing concave function with$c(0) = 0$,$x \in C \subseteq \mathbb{R}^d_{\geq 0}. By non-decreasing, ... 1answer 68 views ### Solve a 2-dimensional optimal control problem via Riccati nonlinear equation Consider the 2-dimensional optimal control problem of the LQR kind $$\min_u \int_0^\infty (x^T Q x + u^TRu) \, dt \quad\text{such that}\quad \begin{cases}\dot x(t) = Ax(t)+Bu(t) \\ x(0) = \... 0answers 106 views ### Compute which of a finite number of integrals is minimal Let (E,\mathcal E,\lambda) be a \sigma-finite measure space; f:E\to[0,\infty)^3 be a bounded Bochner integrable function on (E,\mathcal E,\lambda) and p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3 ... 1answer 177 views ### Generalization of minimal selection theorem Consider a metric space X and a set-valued map F : X \to \mathbb{R}^{n}. We define the minimal selection \begin{equation*} m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\}, \end{... 1answer 48 views ### Connected graphs G with \delta(G) > 1 and long minimum size roundtrips Let G = (V,E) be a finite, connected, simple, undirected graph. By a roundtrip of G we mean a map r:\{0,\ldots,n\} \to V for some n\in\mathbb{N} with the following properties: r is ... 1answer 128 views ### Optimal function existence? what is it? It's a problem abstracted from a real engineering project. I want to find the best curve y=y(x), x \in [0,1]: y doesn't have to be a continuous function. The constraint is$$ L=\int_{0}^{1} \... 0answers 53 views ### Low rank approximation Can we solve low rank approximation problem by using concept of Gröbner basis? I was trying to find it by Macaulay2 but didn't find the answer. I was trying to do by toric ideals as for them Gröbner ... 1answer 89 views ### A close-form solution for a simple quadratic optimization problem Is there any closed form solution for the following optimization problem: \begin{align} &\min_{\mathbf{X},\alpha} \mathrm{Tr}[(\mathbf{A}-\mathbf{B}\mathbf{X})(\mathbf{A}-\mathbf{B}\mathbf{X})^{\... 0answers 54 views ### Linear algebra - For symmetric matrix X\in S^n$, prove the$a^T X a$=$\det X \det(X_{n-1})$, where$a_i$=$(-1)^i M_{in} $[closed] Suppose we have a symmetric matrix X$\in S^n$, and$X_k$denotes the submatrix consists of first$k$rows and columns of X. If$\det X < 0$, but$\det X_1, ..., \det X_{n-1} > 0$. Let$a_i=(-1)^...
Suppose that $b:\mathbb{R}\to\mathbb{R}$ is locally Lipschitz and of polynomial growth. Suppose further that there are constants $C_1,C_2>0$ such that $(x-y)(b(x)-b(y))\leq C_1-C_2(x-y)^2$ for all \$...