Questions tagged [nonlinear-optimization]

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

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Under which condition, such that all second-order critical points satisfy $\sum_j\cos(\theta_i-\theta_j)>0$ for all $i\in[n]$?

Consider the following non-convex function $$E(\theta):=-\sum_{i,j}A_{ij}\cos(\theta_i-\theta_j)$$ where $A$ is a symmetric, diagonal-free matrix whose non-diagonal element are $\pm 1$. In other words,...
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Optimality condition for strongly convex function under sparsity constraint

Let $f: \mathbb{R}^p \to \mathbb{R}$ be a $2s$-sparse strongly smooth, $2s$-sparse strongly convex and twice differentiable function. In other words, there exists positive constants $\alpha, L >0$ ...
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How to control the angles of Kuramoto model by controlling its order parameter?

Consider Homogenous Kuramoto model in this paper. In theorem 3.1, the author derive condition on $A$ such that all second-order critical points of $E(\theta)$ are in two opposite quadrants, by saying ...
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Numerical partial differentiation of a convolution product with FFT

How can one numerically calculate the partial derivatives of a convolution function, particularly when the closed-form or analytical expressions of the derivatives are not readily available? I am ...
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Numerical estimation of partial derivatives of convolved functions when closed forms do not exist

Summary: Some peak functions are convolutions which may not have a closed form solution. A classical example can that of a Voigt which is a convolution of a Lorentzian and a Gaussian, followed by ...
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How to derive a lower bound of a MinMax inequality?

Let $x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]$ where $\alpha$ is a fixed angle $\in(0,\pi/2)$. The goal For a fixed $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}\in\{-1,+1\}$, verify whether it ...
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Efficient algorithms to find the global minimum of a non-convex quadratically-constrained quadratic program

I am working on a problem involving a non-convex quadratically-constrained quadratic program and am seeking efficient algorithms to find its global minimum. The problem is structured as follows: Fix ...
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How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let $$ \begin{matrix} F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\ (x,z,p) \mapsto F(...
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Seeking help with a matrix optimization problem involving matrix exponentiation

I'm working on an optimization problem where I need to find matrices $P$, $Q$, and $C$ that minimize the norm of the difference between a given matrix $A$ and another matrix defined as $e^{P(Q + Q^T - ...
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Do we have tetration uniqueness by $ A = \inf \sum_n a_n^2 $?

Let $f$ be a real analytic (on at least $|x|<2$) and real solution of the functional equation $f(0) = 1,f(x+1) = \exp(f(x))$. For the existence of such $f$, see here. Then $$ f(x) = \sum_n a_n x^n ;...
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Optimization: Determine the categorical pmf that maximizes the objective function

Let $T$ denote a $J$-component categorical random variable with pmf $$ \mathsf P(T=t_j)=w_j,\quad j=1,2,\dots,J, $$ where $t_j\in[0,t_\max]$, $t_\max>0$. I came across a problem that seeks to ...
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Which penalization works for this optimisation problem?

Let $m,n$ be given integers. Set $K:=[0,1]^{mn}$ and define the function $L:K\times\mathbb R_+^m\times\mathbb R_+^n\to\mathbb R$ as follows : for $z=(z_{i,j})\in K$, $a=(a_i)\in\mathbb R_+^m$, $b=(...
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Primal optimal attained implies dual optimal attained

Given some optimization problem $$\min_{x \in S \subset \mathbb{R}^n} f_0(x) \quad \text{s.t.} \quad f_i(x) \leq 0, \quad 1\leq i\leq m$$ we can find the dual problem $$\max_{\lambda\in\mathbb{R}^m} g(...
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How to integrate an indicator function/constraint into the cost function of a linear program?

I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$. In $F_2$, I want it to be included only when its expression ...
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Maximization of $\ell^2$-norm

Consider for $r,c>0$ the set $$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid \|x\|_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$ Then I can show that $\inf_{x \in X_{r,c}} \|x\|_2 = 0.$ But is it ...
SequenceGuy's user avatar
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Reference for article that introduces and motivates different notions of subdifferentials

I saw a tutorial/expository journal article a while ago that focused on introducing intuitively different notions of subdifferentials appropriate for general nonlinear optimization. I forgot the ...
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Solving optimization problem restricted to non-convex subset

I have two continuous random variables that have pdfs with respect to the Lebesgue measure $p_{-1}(x), p_1(x)$. Let $m(x) := \frac{p_{-1}(x)+p_1(x)}{2}$ be a mixture of these two distributions. Let $B(...
Mark Schultz-Wu's user avatar
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Bound the distance between two vectors on the probability simplex

Let $a,b$ be two vectors with strictly positive elements and $\delta = 1 - \frac{\langle a,b \rangle}{\|a\|\|b\|}$. Bound the following optimization problem as a function of $\delta$ $$\sup_{x>0} \...
good bandit's user avatar
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Positive semidefinite maximum principle

Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Let $\mu$ be a Borel probability measure on $M_n(K)$ supported on a compact set $C$ of positive semidefinite matrices with $\mathbf{0}\not\in C$. ...
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Convergence bound for zero-order optimization method

I would like to understand the error bound for a particular zero-order optimization method: (stochastic) difference method. To solve an nonsmooth optimization problem $min_x G(x)$ where $G$ is only a ...
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Error bound for stochastic gradient descent method

To solve an optimization problem $\min_x G(x)$ using standard stochastic gradient descent method, we let $x_0$ be the initial point and $x_k$ be the $k$-th point such that \begin{equation} x_k = x_{k-...
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optimization over moving domains

Let $A, B$ be Banach spaces, and for any $a\in A$, $B_a\in B$ is a measurable subset. Consider the following optimization problem: $$L(a)=\inf_{b\in B_a}\ell(b),$$ where $\ell(b)$ is a infinite-times ...
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nonlinear equation problem

Can you please help me solve the following nonlinear equation to determine the value of the vector $z$ : $$ \boldsymbol{a}=\boldsymbol{z}^{2} \odot \boldsymbol{K}*\boldsymbol{z}^{-1}$$ Where: $\...
Iman Nodozi's user avatar
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LICQ vs MFCQ who is stronger [closed]

I want to ask you which constraint is stronger: MFCQ or LICQ.
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Optimization over permutation

The Problem This is the problem I am working on: Given a set $X = \{x_1, x_2, \cdots , x_n\}$ in a metric space, find an optimal ordering $\pi : X \rightarrow X$ that maximizes the following objective ...
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Argmax of a function of $n$ variables under linear constraint

(I start by saying that the tags are probably not accurate but I didn't know what to put, so if someone knows what I could tag this question with, let me know in the comments and I'll provide to edit ...
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Maximal entropy distribution on three variables knowing its marginals on any two

Observation 0: Given a finite set $X$, the probability distribution on $X$ with highest entropy is the uniform one. This is well known. Observation 1: Given two finite sets $X,Y$ and two probability ...
Gro-Tsen's user avatar
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Can we say this nonlinear integer programming problem is NP-hard?

I was wondering if the following nonlinear integer programming problem is NP-hard or not. $$\max_{x_i \in \{0,1\}} \frac{\sum_{i=1}^{n}a_i x_i}{\sqrt{\sum_{i=1}^{n}b_i x_i}}$$ such that $\sum_{i=1}^{n}...
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Detecting linear operator from actions of powers on subspace

Say I have a sequence of linear operators $A_1,...,A_n$ on a (real) vector space $V_1$. I suspect that there's a second vector space $V_2$, and an operator $A$ on $V_1\oplus V_2$, such that $A_i=(\...
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Numerical solution to partially-free boundary optimization problem

Background First of all, I'm a PhD physicist working in numerical analysis, so I apologize for possible easy-to-spot mistakes (they're most likely not that easy for me). The problem I'm trying to ...
Mauro Giliberti's user avatar
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Weighted Least squares with Multiple Unknowns and Iterations

I am currently working on a problem involving the minimization of the $\chi^2$ deviation between a model matrix ($C_\text{model}$) and a measured matrix ($C_\text{measured}$). by finding the best-fit ...
Elaf Salah's user avatar
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Min problem on integers

Let $n$ be any integer greater than $2^{10^6}$. Given any $s\le (\log_2 n)/1000$ integers $1=q_1\le q_2\le \cdots q_{s-1}\le q_s=n$. Prove that $$\min_\ell\left(\sum_{i=1}^\ell q_i\right)\left(\sum_{i=...
Nader Bshouty's user avatar
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Relationship of optimal solutions between the total function and the sub function

This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
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A dilemma in a lemma regarding Hexagonal Fuzzy approximation of parabolic fuzzy number

I was reading Nayagam & Murugan - Hexagonal fuzzy approximation of fuzzy numbers and its applications in MCDM paper regarding hexagonal fuzzy approximation. In that, the statement of Lemma 3.1 and ...
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Does the value function of a quadratic program stay convex when adding constraints?

I am interested in the value function of a quadratic program of the form $$ v(y)=\min_x \frac{1}{2} x^\top Q(y) x, $$ subject to a linear equality constraint $$ E(y)x=d(y), $$ and a linear inequality ...
user_lambda's user avatar
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Computational comparison in solving two optimization problems

Can I get some inputs on whether the following two optimization problems are computationally the same, or one of the problems is easier to solve computationally than the other, such as, finding their ...
muddy's user avatar
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How to solve mckp (multiple-choice knapsack problem) problem with non-linear constraint

How to solve the below optimization problem? $P$ is a probability matrix, $0\le P_{ij}\le 1$. Are there any developed tools to solve this? Thanks a lot. \begin{equation*} \begin{aligned} &\...
Yi-Yu Peng's user avatar
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An optimization problem defined by optimal transportation

Let $\mu,\nu$ be two probability measures on $E:=\mathbb R^d$ and denote by $\gamma:=\mu\otimes \nu$ the product measure on $E\times E$. Define $\mathcal K$ to be the class of continuous functions $k:...
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An integer optimization problem on the simplex

For $K \geq n$ and some $\sigma_i > 0$, I am looking for a solution to the following optimization problem: \begin{equation} \underset{\begin{smallmatrix} t_1, \cdots, t_n \in \mathbb{N}^{*} \\ \...
Titouan Vayer's user avatar
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Minimizing a ratio related to scalar subset sums

Let $a = (a_1, a_2, \cdots, a_n)$ be non-zero real numbers. Let $[n] = \{1,2,\cdots,n\}$ be a set of indices. Define the maximum absolute subset sum of the array $a$ as: $$\mathrm{MASS}(a) = \max_{T \...
Nango's user avatar
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$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer

Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ : $$f(n) = \frac{n^2 + n + 4}{2}$$ so $$ \begin{split} f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\ f(...
mick's user avatar
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Maximizing the ratio of multilinear polynomials

Consider two multilinear Polynomials $A(x_1,x_2,x_3,\dotsc,x_n)$ and $B(x_1,x_2,x_3,\dotsc,x_n)$ of $n > 2$ variables $x_i \in \mathbb{R}$ and their ratio \begin{equation*} F(x_1,x_2,x_3,\dotsc,...
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Constrained trace optimization with relavance to optimal asset selection

Let $D$ and $Q$ be two real $m\times m$ diagonal matrices given $$ D=\left(\begin{array}{cccc} d_1 & 0 & \cdots & 0\\ 0 & d_2 & \cdots & 0\\ \vdots & \vdots & \ddots &...
hopeless's user avatar
3 votes
1 answer
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Interesting question about the Thomson problem for arbitrary number of electrons

This question is crossposted from here I believe this is a pretty hard question and so I decided to repost the question in the Math Overflow forum. If there is something wrong with doing this, I am ...
Rodrigo's user avatar
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4 votes
1 answer
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Equivalence of bounded gradient flow solutions and uniformly bounded gradient descent trajectories for definable functions

I am reading paper [1] by C. Josz regarding the global convergence of the gradient method. The main result is the following: $\textbf{Theorem}$: For a definable differentiable function $f : \mathbb{R}^...
Andreea M's user avatar
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2 answers
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Any idea of solving an optimization problem with cubic constraints?

I have the following optimization problem with cubic constraints, which is hard to solve. Are there any ideas, or related references, of solving such a problem? $$ \begin{array}{ll} \underset {y, z} {\...
Erik's user avatar
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3 votes
0 answers
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How many local maxima can $(x_1,\dots,x_r)\mapsto\|x_1A_1+\dots+x_rA_r\|_\infty/\|(x_1,\dots,x_r)\|_2$ have for Hermitian $A_1,\dots,A_r$?

Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Suppose that $A_1,\dots,A_r\in M_n(K)$ are all Hermitian. Define a function $f_{A_1,\dots,A_r}:\mathbb{RP}^{n-1}\rightarrow[0,\infty)$ by setting $$f_{...
Joseph Van Name's user avatar
1 vote
0 answers
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On an optimization question

Suppose we have a square matrix $M=(1-z)A+zB$ where $A,B$ have integer entries from $\{0,1\}$ with $\det(A)+\det(B)=1$ and $\det(A),\det(B),per(A),per(B)\in\{0,1\}$ and we want to find $z\in[0,1]$ ...
Turbo's user avatar
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2 votes
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How many strict local minima can a quintic polynomial in two real variables have?

A quadratic or cubic polynomial (in two variables) can have at most one strict local minimum. A quartic polynomial can have up to five strict local minima [1]. So, how many strict local minima can a ...
Pavel Kocourek's user avatar
8 votes
1 answer
640 views

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

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