For questions about optimization and optimization problems with non-linear restraints.

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10 views

Non linear Programming Problem

I am struggling with the following question: Solve the following programing problem: max $f(x_1,x_2)$= $ \sqrt{(x_1 + 1) (x_2+1)} $ subject to $x_2-(x_1-1)^2 \leq 0 $; $x_1+x_2 \leq 7 $; $x_1, x_2 ...
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0answers
16 views

Question regarding the solution of a nonlinear algebra equation

I want to prove for a nonlinear equation $x=f(x)$, the solution for this equation $x_0$ will maximize the first order of $f(x)$. i.e. $x_0=argmax(f'(x))$. Here are what we have $f(x)$ is a ...
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0answers
18 views

Separate a list of spheres into several lists, each contained in a sphere with a radius no larger than specified.

I have a list of arbitrary spheres, what I want to end up with is that list separated into a number of groups, where spheres in each group all fit into thier specific larger sphere. The limitation is, ...
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0answers
25 views

Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using ...
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1answer
27 views

Question regarding KKT conditions in optimization

Following is Proposition 3.3.7 in Bersekas' Nonlinear Programming. Let $x^*$ be the local minimum of the problem: $$\text{Minimize }\; f(x) $$ $$ \text{subject to: }\ h_j(x) = 0, ...
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58 views

Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
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0answers
32 views

Fix points and stability [closed]

Find fixed (equilibrium) points of the map $f(x) = x^3 − \dfrac{x}{9}$ and classify their stability: asymptotically stable (attracting), unstable (repelling), neither. Illustrate your answer by the ...
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1answer
228 views

Optimization problem with ratio objective

I need to solve the following optimization problem $$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad \|x\|_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$ ...
7
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1answer
290 views

Minimization of $\sum \frac{1}{n_k}\ln n_k >1 $ subject to $\sum \frac{1}{n_k}\simeq 1$

Looking at an algorithm for minimizing $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} = 1$ in which $n_k$ are positive and in general non-sequential integers, I ...
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0answers
15 views

Solving many independent non-linear systems simultaneously

I'm working on solving lots of systems of nonlinear equations. Luckily, the non-linear equation is the same, but the parameters are different: $$ f(\vec{x}_0; c_0) = 0\\ f(\vec{x}_1; c_1) = 0\\ ...
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0answers
10 views

Quasiconcave condition for a power function

Let $f(x, y)= (ax^2+by^2)^n$ where $a, b, n$ are positive, $x, y\in \mathbb{R}$. What is the condition of $n$ so that $f(x, y)$ is a quasiconcave, and concave function? My idea is only calculate ...
2
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1answer
45 views

min : sum of L2 norm and squared-L2 norm.

Is there a closed-form solution of the following convex problem: $$\min_x \| x - u \| + C \| x - v \|^2$$ where $\| \cdot \|$ is the L2 norm.
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13 views

find of value Zx and Zy

I try to solve this Z^2+ZY+XY=0, and find the value of ZX and ZY, but couldn't find any hint in the web neither in a couple of calculus books for this particular equation. I have no clue about how to ...
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0answers
32 views

mathematical optimization using the apache common-maths3 library

I have been asked to implement a logistic mixture model in java. I have expressed the likelyhood of the modele and i want to use the Classifying Expactetion-Maximization(CEM) algorithm to estimate ...
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2answers
315 views

Convex Optimization of quadratic function with inequality constraints

How would I solve the following problem? $$\min_{x\in\mathbb{R}^n} x^T A x$$ subject to the constraints $$x_i\geq 1,\,i=1,\dots,n,$$ where A is positive semidefinite and symmetric. Is it possible to ...
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1answer
35 views

Why don't we go beyond the Hessian in multivariate optimization?

In univariate optimization, we perform the first derivative test to identify stationary points and the second derivative test to classify the stationary points as minima, maxima and inconclusive. When ...
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1answer
41 views

Constrained Optimization : Minimize sum of dot products

I am working on a problem to minimize sum of dot product. The problem can be stated as following. Given a matrix where each element is either 0 or 1. $$ \ A_{ij} = \{0,1\}; $$ with the constraint ...
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2answers
21 views

How to mathematically prove the optimality conditions for a univariate function?

Consider a univariate function $f(x)$. I know the graphical intuition behind why $f'(x)=0$ at the extrema of $f$. But how do you prove it mathematically? I start with the assumption of $x^*$ being a ...
2
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0answers
29 views

Constrained non-linear Optimization using Newton's method - Portfolio optimization

I want to solve following constrained optimization problem from portfolio optimization: The solution is supposed to be a modified risk parity portfolio: The optimization problem is: \begin{align} ...
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0answers
11 views

Nonlinear programming-separable programming

i have this function: $$5x_1x_2+4x_2x_3$$ and i need yo know if is separable or not I guees is not separable, because i can´t write the function in form: $$f_1(x_1)=x_1$$ and $$f_2(x_2)=x_2$$ ...
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0answers
90 views

How can I find the unit vector that minimizes the number of nonzero projections that a set of points has on it?

$\underset{\mathbf{w}}{\min} ~ \|\mathbf{X}^T\mathbf{w}\|_1~~~\text{subject to:}~ \|\mathbf{w}\|_2^2=1$ where $\mathbf{X}\in\mathbb{R}^{d\times m}$ is a set of $d$-dimensional points and $m>d$. ...
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1answer
133 views

How to solve this optimization problem with equality constraints?

I want to find $\delta_j$ in the following optimization problem. My variables are $\gamma_i$ and $\delta_j$ (all other symbols are known parameters). Assume $i\in\{1,\ldots,9\}$ and ...
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0answers
119 views

Finding optimal velocity profile using Dynamic Programming

This question has been asked on scicomp but I thought maybe it's more a mathematical problem of how Bellman's idea is to be applied here. The main problem for me is: How to introduce the time ...
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1answer
731 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
0
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1answer
31 views

Minimizing a non-convex rational function of two variables

I need to minimize the following function $$f(x,y)= \frac{a}{x}+\frac{bx}{y}+\frac{cy}{x}+dy+\frac{e}{y}$$ where $a,b,c,d$, and $e$ are positive constants, and $x$ and $y$ are both strictly positive. ...
2
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0answers
26 views

Constrained Quadratic Optimization(Reproducing Kernel)

I am attempting to use a constrained quadratic optimization to find the coefficients of a reproducing kernel. The problem is as follows: $y(t)=\sum_{i=0}^J\alpha_iK(t, t_i)$ $Q(\alpha)= ...
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1answer
68 views

Convex optimization approximation

Consider the optimization problem $\mathcal{P}_0$ $$ \min_{x \in \mathbb{R}^2} \left\| x-p \right\|^2 $$ $$ \text{sub. to: } \ A x \leq b, \ \ x_1^2 + x_2^2 = 1 $$ where $p \in \mathbb{R}^2$ is a ...
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2answers
18 views

Solving for gradient of Frobenius norm term

Let's first define a couple of variables: $A,B,C \in \mathbb{R}^{m \times n}, D \in \mathbb{R}^{n \times n}$, and $\mu$ is a scalar. Say I have an ADMM sub-problem that looks like this: $\arg ...
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0answers
14 views

Non-linear optimization with unknown derivative and real numbers only

Background I'm trying to optimize a set of 7 parameters which are the core configuration of some external engine (specifically Solr parameters). I already have an optimization function which grades ...
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1answer
44 views

Optimization problems on the circle

Consider the optimization problem $$ \min_{x \in \mathbb{R}^2} x^{\top} P x + q^{\top} x$$ subject to: $$ A x = b, \ x \in X, \ x_1^2 + x_2^2 = 1$$ where $X$ is compact and convex. Then ...
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0answers
23 views

Non-convex constraint made cost

Consider the non-convex optimization problem $$ \min_{x \in X} \ f(x) \quad \text{s.t.:} \ \ g(x) \leq 0, \ h(x) = 0 $$ where $X \subset \mathbb{R}^{2n}$ is compact and convex, $f$ and $g$ are ...
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0answers
31 views

Concave Quadratic Program

Let $X \subset \mathbb{R}^n$ be compact and convex. Consider $$ x^* := \arg\min_{x \in X} x^\top Q x + c^\top x $$ where $Q \prec 0$. I am wondering if there are cases where $x^*$ can be written as ...
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2answers
132 views

What is a convex optimisation problem? Objective function convex, domain convex or codomain convex?

My teacher in the course Mat-2.3139 did not want to answer this question because it would take too much time. So what does a convex optimisation problem actually mean? Convex objective function? ...
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0answers
19 views

Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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0answers
25 views

Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
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1answer
50 views

Model $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty $ into standard form

I need to solve the following convex optimization problem: $\min \frac{1}{2} \parallel Ax-B \parallel_2 + \lambda_1 \parallel Cx \parallel_1 + \lambda_2 \parallel Dx \parallel_\infty$ s.t $x ...
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1answer
199 views

Scaling a decimal number to have leading digit 5 or more

Suppose we are given two real numbers: $a, b \in \mathbb{R}$, $b > a$. Find $m \in \{1, 2, 2.5, 5\}$ and $k \in \mathbb{Z}$, satisfying the following condition: $\frac{b-a}{m10^k} \in [5,10)$. How ...
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1answer
32 views

Minimize $\ell_1$ norm subject to $\ell_2$ constraint

I am trying to solve the following optimization problem: $$\min_{\|Px\|_2=1} \|x\|_1$$ I know it is non-convex. But some non-convex problems are still solvable. Update $P$ is 2x3. $x$ is a ...
3
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2answers
166 views

Maximizing “log det + log sum exp” function

I'm trying to find a numerical solution to the following optimization problem $$ \text{maximize } f(M) = \frac{1}{2} \log \det(M) + \log \sum_{i=1}^n \exp \left\{ - \frac{1}{2} x_i^T M x_i + a_i ...
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0answers
38 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 ...
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0answers
25 views

Constrained optimization using a cutting plane on a tetrahedron

Consider the figure below where $(a,b,c,d)$ is a tetrahedron and $p=(1-t)a+tb$ is a point on the $ab$ segment. If $n_a$ and $n_b$ are two unit vectors associated with $a$ and $b$, respectively, then ...
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1answer
36 views

Evolutionary algorithm

Can someone provide me a good reference for the CMA-ES algorithm? I'm new in the world of optimization and just reading the author reference doesn't help me a lot. I know the basic idea of a genetic ...
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0answers
16 views

A non-linear optimization problem involving logarithms

I have a log likelihood function that I'm trying to minimize: $-\log(L) = -(\sum_i^n{c_i \log(a_i x + b_i y)} - \sum_i^n{\log(c_i!)} - x \sum_i^n{a_i} - y \sum_i^n{b_i})$ The function is only ...
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1answer
31 views

Analysing/Visualising shape of multi-variate function.

I have an unknown function $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ for which I'm determining a first order Taylor approximation through a non-linear optimization process in six variables (the ...
3
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2answers
51 views

Max of $3$-Variable Function

I'm trying the find the maximum of the function $$f(a,b,c)=\frac{a+b+c-\sqrt{a^2+b^2+c^2}}{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}$$ for all nonnegative real numbers $a, b, c$ with $ab + bc + ca > 0$. I ...
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2answers
623 views

Lasso with linear constraints

I want to efficiently solve the following optimization problem: \begin{align} \min &\quad \left\|\mathbf{x}-\mathbf{x}_0\right\|_2^2 + \lambda\left\|\mathbf{x}\right\|_1\\ \text{Subject to}& ...
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1answer
53 views

project a point onto the intersection of surfaces

I have several non linear equations $g_i$ that represent surfaces $s_i$. Their intersection form the surface $S$. For example $s_1 : g_1(x_1,x_2,...,x_n)=c_1$ ... $s_n : g_m(x_1,x_2,...,x_n)=c_m$ ...
0
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1answer
14 views

Armijo conditions vs Reduction Conditions in Non-Linear Line Search

Overview Line search typically consists of four stages: Direction: Search direction Initial Step Size: length to search along the line on the first sub-iteration Bracket: find an interval along the ...
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1answer
69 views

Finding Shortest distance between a Sphere and Ellipsoid?

Suppose that ,I have a Sphere and an ellipsoid as Sphere: $(x-x_1)^2 + (y-y_1)^2 + (z-z_1)^2 = R_1^2$ Ellipsoid: $\large\frac{(x-x_2)^2}{a^2} + \frac{(y-y_2)^2}{b^2} + \frac{(z-z_2)^2}{c^2} = 1$ ...
3
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0answers
28 views

Solve non-linear equation with Matrices

I'm looking more for hints than specific answers, although I would be extremely grateful if provided with one. The problem I have is as follows: $$ -\Sigma (A+\Lambda_1)+I=0 $$ Here A is a constant, ...