Questions tagged [nonlinear-optimization]

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

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Can a real quartic polynomial in two variables have more than 4 isolated local minima?

This question: "Can a real quartic polynomial in two variables have at most 4 isolated local minima?" came up in this post on Math SE but with no answer so far. Finding examples of 4 ...
Jap88's user avatar
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5 answers
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Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?

Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows: $$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$ For example, if $m=3$, the matrix is $$\begin{pmatrix}6 & 20 & 6& 0 ...
user42804's user avatar
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26 votes
1 answer
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Finding the closest matrix to $\text{SO}_n$ with a given determinant

$\newcommand{\GLp}{\operatorname{GL}_n^+}$ $\newcommand{\SLs}{\operatorname{SL}^s}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\Sig}{\Sigma}$ $\newcommand{\id}{\text{Id}}$ $\newcommand{\...
Asaf Shachar's user avatar
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14 votes
3 answers
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How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?

Related question asked by me on Math SE a few days ago: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$? A few days ago, somebody asked How to prove $ \mathrm{e}^x\left|\...
Maximilian Janisch's user avatar
14 votes
2 answers
948 views

Conjecture on maximum of symmetric combinatoric function

A curious symmetric function crossed my way in some quantum mechanics calculations, and I'm interested its maximum value (for which I do have a conjecture). (The question was first asked at math.SE, ...
Mario Krenn's user avatar
13 votes
1 answer
2k views

An intuition for three different types of subgradients (proximal, regular, limiting)

I'm having a bit of difficulty getting my head around the different types of subgradients we're currently covering in a nonsmooth optimisation class I'm taking. These subgradients are (assume $x \in$ ...
Ben Stott's user avatar
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11 votes
2 answers
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Quadratic Farkas' Lemma?

The Farkas Lemma says that if a system of linear inequalities implies yet another linear inequality, then this last inequality can be obtained by taking a positive linear combination of the ...
Seva's user avatar
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11 votes
2 answers
339 views

A (reverse)-Minkowski type inequality for symmetric sums

Let $(u_1, u_2, u_3, u_4)$ and $(v_1, v_2, v_3, v_4)$ be vectors in $\mathbb R_+^4$. Is the following inequality true? \begin{align*} \left(\sum_{{[4] \choose 3}} \sqrt{u_i u_j u_k}\right)^{2/3} + \...
VSJ's user avatar
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Is there a "formula" for the point in $\text{SO}(n)$ which is closest to a given matrix?

$\newcommand{\Sig}{\Sigma}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\distSO}[1]{\dist(#1,\SO)}$ $\newcommand{\distO}[1]{\text{dist}(#1,\On)}$ $\newcommand{\tildistSO}[1]{\operatorname{...
Asaf Shachar's user avatar
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11 votes
0 answers
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$L_2$ minimizing makespan vs. $L_\infty$ minimizing makespan

There are $n$ positive real numbers. We partition these numbers into $m$ parts, the size of each part is the sum the numbers in this part. Maximum size of the parts is called a makespan of a partition....
kakia's user avatar
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10 votes
2 answers
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Maximize $f(0)+\cdots+f(n-1)$ subject to $f(x)f(y) + f(x+y) \leq 1$

Suppose $f:\mathbf{N} \to [0,1]$ satisfies $$f(x)f(y) + f(x+y)\leq 1\qquad(1)$$ for all $x,y$. Let $$d_n = \frac{1}{n} \sum_{x=0}^{n-1} f(x).$$ It is easy to prove that $$\limsup d_n \leq 1/\varphi,$$ ...
Sean Eberhard's user avatar
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5 answers
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Software for rigorous optimization of real polynomials

I am looking for software that can find a global minimum of a polynomial function over a polyhedral domain (given by, say, some linear inequalities) in $\mathbb R^n$. The number of variables, $n$, is ...
Boris Bukh's user avatar
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8 votes
5 answers
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Prove that this expression is greater than 1/2

Let $0<x < y < 1$ be given. Prove $$4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Big[ \sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}[\pi(y-x)] \Big] \geq \frac{1}{2}$$ I have been working on this ...
Seaweed's user avatar
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8 votes
5 answers
471 views

Nearest matrix orthogonally similar to a given matrix

Given $A,B\in\Bbb R^{n\times n}$ is there technique find $$\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_F\mbox{ or }\min_{ T\in O(n,\Bbb R)}\|A-TBT^{-1}\|_2$$ within additive approximation error in $\...
Turbo's user avatar
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8 votes
1 answer
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How many saddle points can a quartic polynomial in two real variables have? All 9?

By Bézout's theorem a quartic polynomial $p(x,y)$ can have at most 9 isolated critical points. Can all of them be saddle points? In case of a cubic polynomial there is a mechanical way to answer this ...
Pavel Kocourek's user avatar
8 votes
1 answer
2k views

Finding a point maximizing the minimal distance to a set of points

Given a set of of $N$ points $\{\mathbf x_i \in \mathcal{S}^d\}_{i = 1, \ldots, N}$, where $\mathcal{S}$ is a set of possible values, how can I find the point $\mathbf x^*$ that maximizes the minimum ...
Wookai's user avatar
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8 votes
2 answers
324 views

Projecting onto space of matrices with spectral radius less than one

Consider the space $$ S = \left\{ A \in \mathbb{R}^{n \times n} : \mathrm{SpectralRadius}(|A|) \leq 1 \right\}$$ where $|A|$ is the entry-wise absolute value. Given a matrix $M \in \mathbb{R}^{n\times ...
CComp's user avatar
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8 votes
1 answer
491 views

Higher dimensional scutoids?

The recent discovery of scutoids in biological structures is fascinating. Two scutoids are depicted below (from Scientists Have Discovered an Entirely New Shape, And It Was Hiding in Your Cells), each ...
Tom Copeland's user avatar
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8 votes
0 answers
277 views

The busy Star Guardian

On an infinite plane, the Prime Star has disintegrated into four constituent stars, the North Star, the South Star, the East Star and the West Star, each traveling at a constant speed of $1$ in their ...
Eric's user avatar
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8 votes
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192 views

Concavity of product and ratio of sums

Apologies if this question is not appropriate for MathOverflow. I have asked at Math.StackExchange without success. Consider the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ defined as $$ f(x)=\...
user_lambda's user avatar
8 votes
0 answers
196 views

Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$

Does anybody know an algorithm to solve the following matrix equation? $$X^{-1}=\sum_{i=1}^n D_i X A_i$$ where $D_i$s are diagonal and $A_i$s are symmetric matrices. It would be great to have an ...
Reshad Hosseini's user avatar
7 votes
1 answer
989 views

Can a cubic polynomial in two real variables have three saddle points?

Is there a cubic polynomial $c(x,y)$ with exactly 3 saddle point critical points? In other words, can a cubic polynomial in two variables have three critical points, all of which are saddle points? ...
Pavel Kocourek's user avatar
7 votes
1 answer
362 views

An elementary inequality for three complex numbers

The following problem arose in asymptotic analysis of difference equations. Numerical maximization suggests that for all nonzero complex numbers $a,b,c$ we have $$h\big(r(a,b,c),r(b,c,a),r(c,a,b)\big)...
Iosif Pinelis's user avatar
7 votes
2 answers
1k views

What is Young measure?

I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem: Theorem. Assume ...
Devashish Sonowal's user avatar
7 votes
2 answers
1k views

Maximum average Euclidean distance between $n$ points in $[-1,1]^n$

For my research I have designed a metric that is based on the average Euclidean distance between $n$ points in the $n$-dimensional hypercube $[-1,1]^n$. However, I have a hard time finding the maximal ...
Simon's user avatar
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7 votes
1 answer
303 views

Surprisingly simple minimum of a rational function on $\mathbb R_+^n$

Motivation: The following problem has occurred in a study of energy dissipation in a chain of coupled, damped oscillators. The problem: Let me define specific rational functions $f$, $g$, and $...
Dierk Bormann's user avatar
7 votes
1 answer
397 views

Generalization of the equilateral triangle?

I consider points in the two-dimensional plane. An equilateral triangle is a set of three points in the plane which are equidistant. Suppose now I have $n$ points $x_1,...,x_n$. What is the ...
user16215's user avatar
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7 votes
2 answers
509 views

Proving an infinite norm minimization problem has finite support (non-convex p-norms)

Consider an optimization problem over infinite variables: $$ \begin{align} \min_{x}~& {\left\lVert{x}\right\rVert }_p \\ \text{s.t}~& \left\langle x, a_n\right\rangle \ge 1~,~\forall n=1,\...
Itay's user avatar
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7 votes
1 answer
350 views

Is the solution of this optimization problem always positive semidefinite?

We are given a set of unit vectors $U \subset \mathbb{C}^n$ which spans the space $\mathbb{C}^n$. Given another unit vector $x$, consider then the following optimization problem: $$ \sup_H \left\{ x^*...
F.G.'s user avatar
  • 73
7 votes
1 answer
470 views

Generalized Rayleigh-quotient gradient flow on Grassmannian

The following theorem appears without proof in : Helmke, Uwe, and John B. Moore. Optimization and dynamical systems. Springer Science & Business Media, 2012. Let $A$ be a symmetric $n\times n$ ...
mystupid_acct's user avatar
7 votes
2 answers
565 views

Determining if a quadratic form is non-negative if variables are non-negative

Let $f(x_1,\dots,x_n) = \sum_{1 \le i \le j \le n} c_{i,j}x_ix_j$ be a homogeneous quadratic form. Is there a quick-ish way to determine whether $f(x_1,\dots,x_n) \ge 0$ for all $x_1,\dots,x_n \ge 0$? ...
mathworker21's user avatar
7 votes
2 answers
3k views

Quadratically constrained linear program (QCLP) over $x$ with the linear constraint $x = Az$

I have a problem that looks very much like a norm-constrained linear program, but with an extra constraint that is unusual for me. The problem is the following. Given a matrix $A$ and a vector $w$, $$...
redfly10's user avatar
7 votes
3 answers
678 views

How to solve such an optimization problem

I encounter the following optimization problem, but I can't solve it. Given $N$ variables satisfying $0 \leq x_1 \leq x_2 \leq x_3 \leq ... \leq x_N \leq 1$ and an integer $K$ no large than $N$, find ...
peng's user avatar
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7 votes
0 answers
454 views

Analytic expression for the Tsirelson bound of the I3322 inequality?

Finding Tsirelson bounds for Bell inequalities is a well-loved problem in quantum information theory. A famous case where it is still open is for the I3322 inequality. In this paper Pál and Vértesi ...
Mateus Araújo's user avatar
7 votes
0 answers
215 views

Characterizing matrices with rank constraint

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...
Turbo's user avatar
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6 votes
2 answers
2k views

Find minimum-area ellipse which encloses two ellipses

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...
Dave's user avatar
  • 61
6 votes
2 answers
701 views

Can we decompose a polynomial into difference of convex polynomials?

Given a multivariate polynomial $p(x_1, ..., x_n)$ on $\mathbb{R}^n$, can we always decompose it into the difference of two convex polynomials? i.e., is there a pair of convex polynomials $f$ and $g$, ...
slwang's user avatar
  • 81
6 votes
1 answer
660 views

How to solve optimization problems on manifolds?

I encountered some optimization problem on the special Euclidean group SE(3) at work and wonder how to solve it. The current approach of my colleagues was to use a local parametrization of the ...
Markus Sprecher's user avatar
6 votes
2 answers
8k views

How to fit the parameters of differential equations with known data?

I have the following data from chemical kinetics research to fit the parameters of ordinary differential equations: $$ \left[ \begin{array}{ccccccc} \text{No.}& t & y_1(t)&y_2(t) & ...
LCFactorization's user avatar
6 votes
1 answer
1k views

Solve equation with matrix variable

I want to solve a matrix $\Omega$ from a equation $\sum_k (\Omega + \Theta_k)^{-1} = Q$. The $Q$ and $\Theta, \forall k=1,\ldots,K$ are known, and are positive definite matrices. $\Omega$ also has to ...
Wei Liu's user avatar
  • 173
6 votes
1 answer
532 views

Bound on the sum of arguments

Problem: Show that for all real $s,t,u$ and all complex $z$ with $|z|<1$ one has $$(*)\qquad \arg\frac{1-zf(s-u)}{1-zf(s+u)} +\arg\frac{1-zf(t+u)}{1-zf(t-u)}<\pi, $$ where $f$ is the ...
Iosif Pinelis's user avatar
6 votes
1 answer
155 views

Adding constraints as penalty with $\| \cdot \|_0$ norm

In the paper Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries (page 2), the authors rewrite the minimization problem \begin{align} \min_{\alpha \in \mathbb R^k} \| \...
TheWaveLad's user avatar
6 votes
1 answer
574 views

Combinatorial equation system with exponentially many equations in quadratic many variables

A certain question on graph theory (about the existance of graphs with a certain coloring inherited by perfect matchings) can be translated into the satisfiability problem of a certain set of ...
Mario Krenn's user avatar
6 votes
1 answer
5k views

max-min optimization problem

I'm curious if there is any nice way to approach solving the following kind of optimization problem. Given a $n \times m$ matrix $A = (a_{ij})$, I want to solve \begin{align*} & \max_{c}\min_{1 \...
Mark Hubenthal's user avatar
6 votes
0 answers
251 views

Concavity of a function implicitly defined by a polynomial

Consider the following system of $n$ equations: \begin{equation}f_j^2 = x_j^2\sum_{i=1}^n A_{ij} f_i \tag{$\star$} \end{equation} where $A_{ij}\geq 0$ are known constants and where $x_j>0$ for ...
user_lambda's user avatar
6 votes
0 answers
77 views

Bounding volume of cell in complement of zero set

I am given an integer polynomial $f \in \mathbb{Z}[X_1, \ldots, X_n]$ of bounded degree and bounded coefficient size. The polynomial's zero set partitions $\mathbb{R}^n$ into cells. What I am looking ...
eins6180's user avatar
  • 1,302
5 votes
5 answers
615 views

Elementary inhomogeneous inequality for three non-negative reals

I need the following estimate for something I am working on, but I don't immediately see how to establish it. For $x, y, z \in \mathbb{R}_{\ge 0}$, show that $$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...
BPN's user avatar
  • 543
5 votes
2 answers
490 views

Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$

Look at the expression $$ f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1. $$ The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
Kurisuto Asutora's user avatar
5 votes
4 answers
2k views

Levenberg's original article "A method for the solution of certain problems in least squares"

Does there exist any digital copy of the original article (or a transcript) K. Levenberg, A method for the solution of certain problems in least-squares, Quart. Appl. Math. 2 (1944): 164-168? It is (...
Pierre Schroeder's user avatar
5 votes
4 answers
2k views

Differentiability of eigenvalue and eigenvector on the non-simple case

Let $h:\mathbb{R}^n\to\mathbb{R}^m, n>1$ be a twice continuously differentiable function and $J_h:\mathbb{R}^n\to\mathbb{R}^{m\times n}$ be its jacobian matrix. Let us consider the functions $A(x):=...
Shake Baby's user avatar
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