# Questions tagged [nonlinear-optimization]

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

625
questions

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### Inequality Involving Concave Monotonic Function

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$. ...

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### When can a point be reconstructed from relative angle measurements?

Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...

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### A question on the optimisation problem and FWL theorem

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|...

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### Minimizing a Function with Nonlinear Constraints [migrated]

I am trying to minimize the function $f(x, y) = \frac{1}{x} + \frac{1}{y}$ subject to the constraints:
$$
x + \frac{x}{x + y} y - c \leq 0,
$$
$$
x \geq 0,
$$
$$
y \geq 0.
$$
I have attempted to use ...

0
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0
answers

55
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### A maximisation problem : finite or not?

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 ...

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### Extension of this maximisation problem : finite or not?

$\mathcal M$ 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 $\...

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36
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### $k$-subset with minimal Hausdorff distance to the whole set

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{...

2
votes

1
answer

189
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### Does this maximisation problem admit a finite upper bound?

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 ...

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33
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### How to formulate piecewise quadratic function optimization without introducing binary variables?

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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### Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX

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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### How to prove the convergence of Gechberg-Saxton algorithm?

I just have a problem that Gerchberg-Saxton algortihm is no worse than the previous iteration
but not sure whether it is convergent.

2
votes

1
answer

114
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### Equivalence of minimizing trace and determinant over matrix quadratic form in multivariate regression

Consider the multivariate regression model
$$Y = XB + E$$
where $Y$ is $n \times p$ and corresponds to the dependent variables, $X$ is $n \times k$ and corresponds to the independent variables, $B$ is ...

2
votes

0
answers

69
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### Gradient descent over the set of complex symmetric matrices

In the course of my research (somewhat related to compressive sensing), I am trying to determine a complex, symmetric matrix $L$ (i.e. $L = L^T$) through the following optimization formulation:
$$ \...

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1
answer

71
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### Clarification about this optimisation problem

Good morning everybody. First of all, I apologise to ask here the same question I asked on MSE three days ago, but I am in fact re-asking since I obtained no relevant advice. Perhaps I will hear some ...

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46
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### Generalizations of Berge's maximum theorem

I have a parameterized optimization problem
\begin{eqnarray}
\max_{x\in D(\theta)} f(x,\theta).
\end{eqnarray}
Assumptions of the standard Berge's maximum theorem are satisfied, so the value function $...

1
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0
answers

50
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### 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 $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...

1
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0
answers

60
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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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votes

1
answer

92
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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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votes

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answers

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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 ...

1
vote

1
answer

141
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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 ...

1
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1
answer

294
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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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0
answers

46
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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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0
answers

33
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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 ...

1
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1
answer

94
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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(...

0
votes

0
answers

46
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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 - ...

0
votes

0
answers

152
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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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votes

1
answer

72
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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 ...

0
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0
answers

27
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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=(...

0
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0
answers

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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(...

0
votes

1
answer

140
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 ...

1
vote

1
answer

169
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

82
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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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votes

0
answers

35
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

1
answer

177
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} \...

1
vote

0
answers

87
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

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answers

27
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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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0
answers

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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-...

1
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1
answer

83
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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 ...

0
votes

1
answer

139
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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:
$\...

1
vote

0
answers

59
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### LICQ vs MFCQ who is stronger [closed]

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

2
votes

1
answer

155
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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 ...

2
votes

1
answer

169
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

147
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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 ...

2
votes

1
answer

221
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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}...

3
votes

1
answer

80
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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=(\...

0
votes

0
answers

21
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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 ...

1
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0
answers

32
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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 ...

3
votes

1
answer

229
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=...

0
votes

0
answers

55
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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}^...

1
vote

1
answer

192
views

### 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 ...