# Tagged Questions

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

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### Eigenvalues of a given parametrized matrix.

Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as \begin{align} \mathbf{C}=\left(\mathbf{I}*\frac{1}{\...
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### Union of linear inequalities cover whole space?

We have $n$ variables $a_0,a_1,\ldots,a_n$ such that $a_i\geq a_{i+1}$. There are $k$ sets of linear inequality constraints on the $a_i$. I need to check that any choice of $a_i$ satisfies at least ...
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### How to solve such an optimization problem efficiently？

Given a symmetric positive semi-definite matrix $\mathbf Y$ and a convex set $\mathcal M$ which is a subset of all symmetric positive semi-definite matrices (consider a simple case of $\mathcal M$： a ...
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### An optimization in Markov Chain

We are given two correlated random variables $V$ and $X$ supported over a finite alphabets $\mathcal{V}$ and $\mathcal{X}$. Suppose the marginal $P_V$ and conditional distribution $P_{X|V}$ are ...
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### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
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### Maximizing an integral over a convex region

Let $C$ denote a compact, convex region in the plane containing the origin with unit area, and let $f$ be a probability distribution on $C$. Let $f^\ast$ denote the distribution that maximizes the ...
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### Forcing a set of complex points to be closed under conjugation

I am a PhD student and I have to address the optimization of a real scalar function of complex variables $f(z_1,z_2,\ldots,z_n)$. The function is real valued in my case because each complex $z_i$ ...
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### Generalized Moore Graphs

A generalized Moore graph - as defined by Cerf, Cowan, Mullin and Stanton - is one for which the girth G and diameter D satisfy G ≥ 2D - 1. These graphs retain many of the optimal properties of Moore ...
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### Simplified knapsack problem

There is a problem that I can not solve. Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
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### Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices

I have an engineering back ground. Due to work, I came across this problem \begin{align} &\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\ s.t.~&\left(\mathbf{A}_0+\sum_{i=1}^{K}y_i\mathbf{A}_i\...
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### About the suboptimality of linear estimators

Let $X$ be a random variable and $N$ a Gaussian noise independent from $X$. We observe $Y=X+N$ and want to estimate $X$ based on $Y$ to minimize the mean square error $mmse(X|Y):=E(\hat X(Y)-X)^2$. ...
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### Maximum Principle with Banach Control Space

This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in which,...
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### Delta-convex functions and inner products

A delta-convex (d.c.) function is one which can be written as the difference of two convex functions. The space of d.c. functions includes all C2 functions, and is interesting because it allows many ...
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I was wondering, whether you could point me to some tools with which I could tackle the following "infinite-dimensional linear programming" problem: Notation: $a=1,2,\ldots, A$, $x\in\Omega:=\left\{... 1answer 398 views ### solve non-convex quadratic constrained quadratic programming$\min_{\beta}\beta^{T} A \betas.t. \ \beta^{T} C \beta=1\ and\ \beta\geqslant 0$Here$A,C\in \mathbb{R}^{M\times M}$,$\beta \in \mathbb{R}^{M}$I saw in one paper saying that it could be ... 1answer 620 views ### How to maximize the determinant of a matrix of the form VDV^H Hi, I have a matrix of the form$A=VDV^H$, where$V$is a$M \times 2M$complex matrix,$D$is a$2M \times 2M$diagonal real matrix, thus the dimension of$A$is$M \times M$. My problem is how ... 2answers 225 views ### On a version of gradient descent I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has Lipschitz constant$M$and considers the scheme $$x(t+1) = x(... 1answer 167 views ### To what equal constant in the Gibbs lemma The Gibbs lemma is broadly used in games theory and in mathematical economics (optimal distributions of resourses, Cournot competition e.t.c.). Here it is: Lemma (Gibbs). f_1,f_2,\ldots,f_n be ... 0answers 572 views ### A detail in the proof of the Motzkin-Straus theorem The Motzkin-Straus theorem says that the global optimum of the quadratic program$$\max f(x)=\frac{1}{2} x^{t}Ax,\qquad \mbox{ subject to }\sum x_{i}=1 \mbox{ and } x_{i}\geq 0,$$where A is the ... 2answers 516 views ### Optimal inspection path on a sphere Suppose you would like to "inspect" every point of a unit-radius sphere S \subset \mathbb{R}^3 by walking along a path \gamma on S, but you can only see a distance d from where you stand. ... 0answers 311 views ### A tricky optimization problem over matrices Hi I have the following problem whose solution has lured me for some months now.... All matrices are complex N\times N. Let A be a positive definite matrix with all eigenvalues strictly smaller ... 0answers 99 views ### null controllability of linear wave equation Consider the linear wave equation :$$z_{tt}=\Delta z + k(x) z + h(t) , \; in \; \Omega\times (0,T)$$Are there sufficient conditions on the functions$k(x)$and$h(t)$for which$(z,z_t)$vanish ... 0answers 425 views ### Incremental minimum spanning tree Given a connected graph$G=(V,E)$with a weight function$w:E\to\mathbb{R}$and a subset$E_0\subseteq E$such that the subgraph$(V,E_0)$is connected, I am looking for a sequence$E_0\subseteq E_1\...
Hi I have the following problem (and a conjecture which holds in Matlab). Given an $N\times N$ matrix $H$, represent it by its $2N\times 2N$ real-valued representation (which I will also denote by $H$...