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Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.

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Suppose $m, n, p, n_c\in\mathbb{N}$, $Y\in\mathbb{R}^{m\times p}$. Let $\mathcal{P}_{m, n, p} := \mathbb{R}^{m\times n}\times\mathbb{R}^{n\times p}\times\mathbb{R}^{m\times p}$ and $g:\mathcal{P}_{m, ...
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I am interested in the following problem, which came up while working on this paper about estimating Betti numbers of Kähler manifolds, where we were not able to solve it and had to resort to ...
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My question is related to a very concrete minimization problem that cannot be treated by https://proceedings.mlr.press/v70/jin17a/jin17a.pdf Denote $\Omega:=\mathbb R^d$ with $d\in \{2,3\}$. We adopt ...
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I am looking for comparision results for nonlinear integral Volterra equations with parameters. This was partially motivated by this paper. There, the author establishes, under mild hypothesis, the ...
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Let $f:\Omega \to\mathbb R$ be semi-concave and Lipschitz, where $\Omega\subset\mathbb R^d$ is convex and closed. Consider the minimization problem (assuming the existence of a minimizer) $$\inf_{x\in ...
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I want to establish some useful criteria for uniqueness of solutions to the following: $$Mx=b,\\ \text{subject to}\ ||x(2k-1:2k)||=1, k=1,2,\cdots,5,$$ where $M\in\mathbb{R}^{10\times10},\ x\in\mathbb{...
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Let $S_m=\{-1,1\}^m$ be the hypercube of signs. Define the set of "admissible weights" $W_m$ as the subset of $\{w\in\mathbb{R}_+^m : \|w\|^2=m\}$ with a "support property" of the ...
Veit Elser's user avatar
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We want to get from 0 to 1 on the real axis with a moving point $P(x(t))$, that moves only in the right direction, as soft as possible in a minimum time. We introduce the class $\mathcal{S}$ ...
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I am sorry if the question is quite strange, but I will try. Suppose $X_1$,...,$X_N$ are independent non-negative random variables having values in $[0,N+1]$. The expectation of all of them is $\...
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I have two friendly functions $g,h:\mathcal{X}\subseteq\mathbb{R}^N\to\mathbb{R}$ whose exact properties I'm somewhat flexible on. Maybe for starters they are lower semi-continuous and convex. Their ...
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The Thomson problem on the $S^2$ sphere asks what configuration(s) of $N$ points minimize a particular function which is symmetric under all permutations of its arguments, $$F(x_1,\ldots,x_N) = \sum_{...
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I'm trying to optimize a 2D point $p = (x_p, y_p)$ given a set of 2D points $q_i \in \mathbb{R}^2$ and associated direction vectors $G_i \in \mathbb{R}^2$. Each point $q_i$ is assumed to lie on an ...
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Assume that matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}=\mathbf{A}\mathbf{B}$ are given. I aim to find matrices $\mathbf{E}_1$, $\mathbf{E}_2$, and $\mathbf{E}_3$ such that \begin{align} \...
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I am developing quality coefficients for a specific type of approximation of covariance matrices. I want to to specify meaningful lower bounds (or rather thresholds), for which I would like to use the ...
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I have functions $f : \Bbb{R}^n \to \Bbb{R}_{\geq 0}$ and $g : \Bbb{R}^n \to \Bbb{R}_{\leq 0}$ and would like to find a $\mathbf{x}_* \in \Bbb{R}^n$ where $g(\mathbf{x}_*)=0$ and $f \left( {\bf x}_* \...
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Let $f$ be an $\rho$-weakly convex function, $f:\mathbb{R}^d\to \mathbb{R}$. The Moreau envelope of $f$, $f_\lambda$ for a parameter $0<\lambda<1/\rho$ is defined as $$ f_{\lambda}(x) := \min_{y}...
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Consider two strictly increasing positive sequences $a = \{a_i\}_{i=1}^m$ and $b = \{b_j\}_{j=1}^n$, where the maximum terms satisfy $a_m = M > 0$ and $b_n = N > 0$. Define a composite sequence $...
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Let $U \subset \mathbb{R}^n$ be an open set. Let $\Phi : \bar{U} \to \bar{\Phi(U)}$ be a $C^1$ diffeomorphism defined up to the boundary (i.e., $\Phi$ is a diffeomorphism from an open set containing $\...
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Let $f : \mathbb{R}^n \to \mathbb{R}$ be a locally Lipschitz function, and let $\bar{x} \in \mathbb{R}^n$. Assume the directional derivative $$ f'(\bar{x}; v) := \lim_{t \downarrow 0} \frac{f(\bar{x} +...
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More precisely, assume that $f:X\times Y\to \mathbb{R} \cup \{+\infty \}$ is a function where $Y\subset X$ is compact, $\partial_{x}^{C}f\left( x_{0},y\right) $ is the Clarke subdifferential of at $...
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Let $A$ be a non-singular matrix in $\mathbb{R}^{n \times n}$. Let $S^{n-1}$ denote the surface of the unit $n$-sphere in $\mathbb{R}^{n}$. Suppose we know that there exists an $x \in S^{n-1}$ such ...
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I often notice tankers of the type illustrated in the figure below. The cross section is neither circular nor elliptical. Is it a "notable" geometric shape? Which function or property does ...
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Thanks for experts on the terminology-related questions. Given a Banach space $X$ and an open subset $U\subset X$, a functional $\mathscr{F}:U\mapsto \mathbb{R}$. What is the standard terminology of ...
Silentmovie's user avatar
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Let $\mathcal{H}^n$ be the space of dimension-$n$ Hermitian matrix space. Given $A\in\mathcal{H}^n$, consider the following optimization problem: \begin{align*} &\max f(X,Y)=\text{tr}(XAYA)-\...
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I'm looking for a reference to the solution of trace-ratio linear discriminant analysis with an orthogonality constraint. Consider the following optimization problem where $A$ and $B$ are symmetric ...
Moo's user avatar
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If I have a rectangle ABCD such that A and B touch two points of the outer circle and CD's touches one point of the inner circle, how could the maximum dimensions of the rectangle be calculated? ...
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I am working on MERW (Maximal entropy random walk) for a project. I want to show that given a graph G, there is $\textbf{only one}$ aperiodic markov chain on G that maximises the entropy creation rate ...
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Suppose I have a quadratic program (with positive semidefinite cost matrix) with affine (polytopic) constraints. It is known that the solution to this is piecewise affine, with the ``pieces'' defined ...
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Let $E\subset \mathbb R^2$ be compact, convex and connected. For $p_1,\ldots, p_n>0$ with $$\sum_{i=1}^n p_i=1,$$ and a probability measure $\nu$ supported on $E$ of density $f$, we consider $$\...
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Please consider the following optimization problem: Given a fixed positive natural $n < N$, and a set of functions $f_i$ over a finite domain of nonnegative outputs, s.t. $1 \le i \le N$, then we ...
Jason Hu's user avatar
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Given a smooth function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ and a smooth manifold $\mathcal{M}$. Now, consider the set $$ S(v)=\{x:0\in x\circ(\nabla_{\mathcal{M}} f(x)+N_{\mathcal{M}}(x)+v)\}. $$ ...
ren chong's user avatar
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I have the function $$ g(x,y,t)=\frac{(8x^2y^2+f_+(x,y,t)-\cos(2t))(8x^2y^2(1+(x+y)^2)+(x+y)^2(f_-(x,y,t)-\cos(t))+4xy(x+y)\sin(2t))}{64x^4y^4(1+(x+y)^2)} $$ with $$ f_{\pm}(x,y,t) = 1+2x^2+2y^2\pm4xy\...
Guoqing's user avatar
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I am working on a matrix optimization problem, and the constraints are difficult to handle. The constraints are in the following form, \begin{align} \text{Given: } &b \in \mathbb{R}^n \text{ , and ...
zycai's user avatar
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My apologies if this question is not proper for this site, but I could not figure out the following. Can anyone provide insight? It is almost certain that stochastic differential equations (SDEs) can ...
Creator's user avatar
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Motivation for this problem This problem arises from the fact that the derivative of the gravitational force (tidal force) in the $z$-direction between two objects $A$ and $B$, which have equal ...
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1. On the $L^\infty$ calculus of variations: The field known as the $L^\infty$ calculus of variations is a relatively new field that concerns itself with minimising functionals involving the supremum ...
Nate River's user avatar
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This is a close relative of the following problem. Let $\Omega$ be an open, bounded subdomain of $\mathbb R^n$ with smooth boundary, and $f_i \in W^{1, \infty} (\Omega)$ a sequence of functions ...
Nate River's user avatar
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For any integers $i$ and $j$ such as $1\le i<j\le6$, let $x_{ij}$ be a nonnegative real number. Is it true that, given the condition $$\sum_{1\le i<j\le6}x_{ij}^2=1,$$ the sum $$\sum_{1\le i<...
Iosif Pinelis's user avatar
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Is the smallest root $x$ of $$ 10x^{3}-30x^{2}+\left(30-2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}\right)x\\ +2\sum_{1\le i<j\le6}\cos^{2}\alpha_{ij}-\sum_{1\le i<j<k\le6}\cos\alpha_{ij}\cos\...
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3 answers
2k views

Is it true that the smallest root $t$ of the polynomial $$ 20 t^3 - 30 t^2 + (12 - 4 \cos^2 \alpha - 4 \cos^2 \beta - 4 \cos^2 \gamma) t + \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 2 \cos \alpha \...
Venus's user avatar
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Is it true that \begin{gather} \min\left(\lambda_{\min}(M_{12}), \lambda_{\min}(M_{13}), \lambda_{\min}(M_{14}), \lambda_{\min}(M_{15}), \lambda_{\min}(M_{23}), \\ \lambda_{\min}(M_{24}), \lambda_{\...
Jasmine's user avatar
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Suppose we have difficult peak fitting problems where the the users wish to fit asymmetric peaks to their experimental data by the least squares method. One such function is illustrated below: Here $...
ACR's user avatar
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Assume that $ f: \mathbb{R} \to \mathbb{R}_+ $ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $ 0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
Alireza Bakhtiari's user avatar
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Let's say we are considering the following model: $$ (\beta^{\star},f^{\star}) := \arg\min_{\beta,f \in \mathcal{F}} \mathbb{E}[\left(Z_i - f(X_i, E_i) - \beta^\top \boldsymbol{\tau}_{i,E_i}\right)^2|...
Frédéric Chopin's user avatar
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Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
Fawen90's user avatar
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$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\...
Fawen90's user avatar
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2 votes
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Let $(\mathcal{M}, d)$ be a metric space. Let $k \in \mathbb{N}$. Let $[\mathcal{M}]^k$ be the set of $k$-subsets of $\mathcal{M}$. Consider the following problem: $$ \operatorname*{argmin}_{\mathcal{...
user76284's user avatar
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Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
Fawen90's user avatar
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I have a problem with logical constraints (either-or constraints). I know that it can be solved by either big-M or complementary formulations. However, i do not want to convert it into mixed-integer ...
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I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
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