1
vote
2answers
28 views

Minimizing a convex cost function

I'm reviewing basic techniques in optimization and I'm stuck on the following. We aim to minimize the cost function $$f(x_1,x_2) = \frac{1}{2n} \sum_{k=1}^n \left(\cos\left(\frac{\pi k}{n}\right) x_1 ...
0
votes
1answer
28 views

Can SVD help to solve (inequality) constrained least squares problem?

Consider the following minimization problem: $$ ||Q u - h^{o} ||^{2} \to min \;\;\; s.t. \; u \geq 0 $$ where $Q$ is $m \times n$ matrix and $u$ is $n$-dimensional vector and $h^{0}$ is ...
1
vote
0answers
23 views

Effect of approximating a non-differentiable function on optimisation of minimisation

I am looking at a problem of constrained minimization, where the function to be minimized contains the Heaviside function, and as such is not twice continuously differentiable. My question is what ...
1
vote
2answers
195 views

How to find roots of a non linear multivariable equation using numerical methods

I started a course in linear algebra and numerical methods but I couldn't understand how can we numerically find roots of a nonlinear multivariable equation. f: Rn -> R Find f(x)=0 where x is ...
0
votes
0answers
51 views

This system is contractive?

I have a system which has a form of find point problem, described as following $$p_i=h_i(\mathbf{p})$$ where $$p_i\in[0,1]$$ is the $i$-th components of the $n$-dimensional column vector ...
0
votes
1answer
55 views

Approximating the optimal value of a function involving a Gaussian integral

Consider the following function $$ f(\lambda) = \alpha (1+\lambda^2) + (1-\alpha)2\int_\lambda^\infty (x-\lambda)^2 \phi(x) dx $$ where $\alpha \in (0,1)$ and $\phi$ is the standard normal probability ...
0
votes
1answer
127 views

optimization of a non-differentiable, component-wise step function

I would like to estimate the (local) minimum of a function $c:R^N \mapsto R^+$ where: $c$ is only differentiable almost everywhere, there exists a component $j$, such that $\frac{\partial ...
2
votes
1answer
188 views

Optimizing trigonometric and nonlinear functions

First, Please, keep in mind that I'm a programmer not mathematician, and I have a fair mathematical background. I used optimization in Java to fit some observations to a trigonometric function, I ...
1
vote
0answers
232 views

iterative method to solve nonlinear equations

I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations. For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear ...
0
votes
0answers
97 views

Minimization/maximization of Sum-Product cost functions

I am a little rusty in optimization. Typically in digital detection/decoding one encounters a sum-product cost function. Mathematically, this may represented by \begin{align} J(X)=\sum_i\prod_j ...
2
votes
0answers
58 views

System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)

I have a large system (N>100) of equations $\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$ where $\vec{P}$ is a vector of N functions of the variable t. What is the correct ...
2
votes
1answer
575 views

Python numerical solution for a nonlinear second order ODE with two boundary conditions

I want to solve numerical the next equation, in Python $$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$ it is a nonlinear second order $ODE$ with two $B.C$. ...
1
vote
1answer
234 views

Numerical/artifical damping in forward Euler?

I'm testing a code to find periodic solutions of nonlinear structural vibrating systems by solving a global time-discretized periodic system of equations. I am using a forward Euler (first order ...
3
votes
1answer
202 views

Optimization problem with ratio objective

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

Some type of Mixed Integer Nonlinear Programming Problem

This is a minimisation problem, to minimise the integral over possible $0\leq t \leq T$, $T$ is free, $$J = \text{min} \int_0^T (\alpha + \beta_1\cdot v \cdot R_T \cdot q+ \beta_2 \cdot ...
2
votes
2answers
573 views

how to compute the gradient of a function at an extremal point

I am writing a computer program that searches for the minimum of a multivariate function $f: \mathbb{R}^n \to \mathbb{R}$. This function is in fact the sum of many functions: $$f(x) = \sum_{i=1}^m ...
0
votes
1answer
216 views

Numerical optimization with nonlinear equality constraints

A problem that often comes up is minimizing a function $f(x_1,\ldots,x_n)$ under a constraint $g(x_1\ldots,x_n)=0$. In general this problem is very hard. When $f$ is convex and $g$ is affine, there ...
2
votes
3answers
552 views

Constrained optimization: equality constraint

I have this very general problem (for $n>2$): $$ \begin{align} & \max Z = f(x_1,\ldots ,x_n) \\[10pt] \text{s.t. } & \sum_{i=1}^{n} x_i = B \\[10pt] & x_i \geq 0 \end{align} $$ Assume ...
0
votes
1answer
593 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 ...
2
votes
2answers
768 views

Modified Cholesky factorization and retrieving the usual LT matrix

I have been looking at the modified Cholesky decomposition suggested by the following paper: Schnabel and Eskow, A Revised Modified Cholesky Factorization Algorithm, SIAM J. Optim. 9, pp. 1135-1148 ...
2
votes
1answer
231 views

Finding a function that satisfies constraints numerically

I have the following system of equations for function $p(y)$ and I need help debugging my solution: $$\begin{align} 0&=\log(p(y))+1-\lambda-\gamma y^2-\eta ...
4
votes
1answer
461 views

Solving a set of 3 nonlinear equations with constraints

Problem statement: I am given 3 sets of equations that govern the force $P$, and also the neutral axis, defined by two variables, the radius from the center $r$ and also the rotation degree in ...
1
vote
2answers
153 views

Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x) $ can be any function ( Updated)

I have a series of inequalities: $$y_{1min} \leq y_{1}(x) \leq y_{1max}$$ $$y_{2min} \leq y_{2}(x) \leq y_{2max}$$ $$..$$ $$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$ Note that $x\in\mathbb{R}$ The ...