Questions tagged [nonlinear-optimization]

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

Filter by
Sorted by
Tagged with
1 vote
0 answers
24 views

How can we calculate the Euler-lagrange equations?

In this paper https://arxiv.org/pdf/1907.09605.pdf \ let $\Omega \subset \mathbb{R}^n$ with $n \geq 1$ be a bounded Lipschitz domain with boundary $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
2 votes
1 answer
291 views

Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?

For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it. Problem definition: Let $f(\xi) \in \...
1 vote
1 answer
274 views

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 ...
1 vote
1 answer
112 views

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 ...
1 vote
1 answer
119 views

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} \...
0 votes
1 answer
66 views

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 $\alpha, L >0$ ...
2 votes
2 answers
287 views

Reference request on computational schemes for $\inf_{x\in\Omega^n}\sup_{y\in\mathbb R^n}F(x,y)$

Let $\Omega\subset \mathbb R^d$ be compact, $\rho$ be a density function on $\Omega$ and $p_1,\ldots, p_n\in (0,1)$ be weights satisfying $\int_{\Omega}\rho(z)dz=1=\sum_{k=1}^n p_k$. We consider the ...
4 votes
1 answer
239 views

Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?

This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian $$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$ are ...
1 vote
0 answers
57 views

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,...
0 votes
0 answers
22 views

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 ...
1 vote
0 answers
44 views

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 ...
0 votes
0 answers
30 views

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 ...
2 votes
1 answer
347 views

A "nice" (but non-definite) quadratic programme

For integers $n\geq k>0$, let $f$ be the following quadratic form: $$f(x_1,\ldots,x_n)=\sum_{i=1}^n\sum_{j=0}^{k-1}x_ix_{i+j\bmod n}.$$ Is it true that the minimum of $f$ over the unit simplex is ...
1 vote
1 answer
85 views

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(...
2 votes
1 answer
290 views

How to solve an optimization problem whose optimization variable is a function?

I would like to find an optimal probability density function (PDF) $f$. Given $b$, $$ \begin{array}{ll} \underset {f} {\text{minimize}} & C \\ \text{subject to} & 1 + \frac{b}{x} \displaystyle\...
0 votes
0 answers
35 views

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 - ...
0 votes
0 answers
145 views

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 ;...
3 votes
3 answers
453 views

Looking for a very particular kind of non-convex functions

I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously, It should be at least twice differentiable. It should have a ...
0 votes
1 answer
71 views

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 ...
4 votes
1 answer
153 views

Gap to fill in the Aubin–Ekeland proof of the mountain-pass theorem

Working through the proof of the mountain-pass theorem given in Applied Nonlinear Analysis by Aubin & Ekeland, at what seems to be a critical point of the proof (the top of page 274) they refer to ...
0 votes
1 answer
474 views

How do I get an analytical solution to this nonlinear equation?

I posted this question over on Math Stack Exchange (link), but have not received a response. I'm wondering if it's too complicated for that audience, so I'm posting it here in the hopes that someone ...
0 votes
1 answer
102 views

An optimization problem with variables on the exponential of a complex number

$$\min_t \quad\operatorname{Re} \sum\limits_{i = 1}^N {\left( {{e^{ - j2\pi {f_i}t}}{r_i}} \right)}, $$where $\operatorname{Re}$ refers to get the real part of a complex number, $\{f_i\}$ is an ...
0 votes
0 answers
90 views

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(...
3 votes
1 answer
156 views

Numerical scheme for convex optimization

Given $(e_n)_{-N\le n\le N}\in\mathbb R^{2N+1}$ and $-1<x<1$, solve \begin{eqnarray} &&\max_{(q_n)_{-N\le n\le N}\in\mathbb R^{2N+1}_+}~ \sum_{n=-N}^N (e_n-\log(q_n))q_n \\ \mbox{s.t.} &...
0 votes
0 answers
25 views

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=(...
2 votes
1 answer
149 views

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}...
0 votes
2 answers
277 views

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} {\...
0 votes
1 answer
114 views

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 ...
4 votes
2 answers
626 views

Difference between Chebyshev first and second degree iterative methods

Consider linear equation $Au = f$. We want to solve it with iterative method (assuming $A$ is good). First order iterative method is: $$ u^{k+1} = u^k - \alpha_{k+1}(Au^k - f), $$ The second degree ...
0 votes
0 answers
70 views

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-...
4 votes
1 answer
210 views

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}^...
1 vote
1 answer
160 views

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 ...
2 votes
2 answers
73 views

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 ...
4 votes
1 answer
789 views

Lagrange-Newton minimization and initial value for the Lagrange multipliers

Short version. How to choose the initial values for the Lagrange multipliers in the Lagrange-Newton equality-constraint minimization method? Introduction. The problem to solve is \begin{equation} \...
0 votes
0 answers
33 views

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(...
1 vote
0 answers
78 views

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 $\mathbf{0}\not\in C$. ...
0 votes
0 answers
26 views

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 ...
1 vote
1 answer
81 views

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 ...
0 votes
1 answer
139 views

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: $\...
-1 votes
1 answer
693 views

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 ...
1 vote
0 answers
53 views

LICQ vs MFCQ who is stronger [closed]

I want to ask you which constraint is stronger: MFCQ or LICQ.
2 votes
1 answer
141 views

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 ...
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 ...
2 votes
1 answer
136 views

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 ...
1 vote
1 answer
125 views

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 ...
3 votes
1 answer
73 views

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=(\...
0 votes
0 answers
20 views

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 ...
1 vote
0 answers
30 views

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 ...
3 votes
1 answer
222 views

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=...
4 votes
3 answers
348 views

Eigenvectors that are tensor products?

Consider a fixed $N\times N$ positive definite symmetric matrix $A$. Assume $N=d^r$ for some $d,r\geq 1$. I wonder if one can find a closed formula for the maximizer/maximum of the function $$f(x):=\...

1
2 3 4 5
13