0
votes
0answers
11 views

Get number equation using specific set of values for get given answer

I have do it for AI assignment. Need a logic for finding solution ..Here is the explanation of problem . I have answer ( any number like for example 10 ). And have some set of numbers (like for ...
2
votes
1answer
42 views

QR transformation with Householder transformation

It's a task i do to understand minimizing the error including the QR transformation with the help of Householder transformation. I think i really do something wrong but i dont get it running i hope ...
0
votes
0answers
38 views

How to characterise this non-linear optimisation (linear objective function, non-linear constraints)

I was wondering if someone may be able to help me characterise this optimisation problem as I am struggling to find a numerical library that will solve it and I suspect it is because I am using the ...
1
vote
0answers
30 views

Optimal numerial method for optimization of “Rosenbrock Banana”-like function

Which numerical methods would be optimal to find an extremum of a function with an almost flat "valley" (but a single minimum in the middle of the valley)? In this context optimal means the least ...
0
votes
0answers
19 views

Constraint approximation in non-linear optimization

In given non-linear optimization problem \begin{equation*} \begin{aligned} & \underset{x \in\mathbb R^n}{\text{maximize}} & & f(x) = \alpha^2 \\ & \text{subject to} & c(p(x)) \le ...
0
votes
1answer
31 views

Estimating rates of convergence

If I have a set of data points obtained from a numerical approximation say 15.3828 15.2458 15.2095 15.2003 how can I estimate the rate of convergence?
1
vote
0answers
25 views

Simultaneous iteration of Symmetric Matrices

Given a Matrix $A$ we can use Simultaneous iteration(Using power iteration on all columns simultaneously) to compute the d biggest eigenvalues. Now this method will give you the biggest eigenvalues, ...
0
votes
1answer
37 views

Reference request: nonlinear systems, optimization, ode/pde

Could someone suggest me one or more good books on the following topics: Nonlinear systems: fixed point and Newton's method Optimization: steepest descent and Newton's-quasi newton methods ODE ...
0
votes
0answers
29 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ...
1
vote
1answer
126 views

calculate Jacobian matrix without closed form or analytical form

The question is probably clear in the title. In many of my applications mostly computer vision, I might not have the closed-form or analytical form of f (a multivariable function). It's calculated ...
0
votes
1answer
25 views

Lowest norm solution to a system of polynomial equations

I have a system of cubic equations: $$0=A_0+A_1 x+A_2 ( x \otimes x ) + A_3( x \otimes x \otimes x )$$ where $\dim A_0 = \dim x$ (so there are as many equations as unknowns). You may assume that the ...
0
votes
0answers
35 views

Linear least squares with sparse inequality constraints for support function estimation

The initial problem is the following: $$ ||h - h^{0}|| \to min \; \; s.t. Qh \leq 0 $$ where $h^{0} \in \mathbb{R}^{n}$ is known vector and $Q$ is a $m \times n$ matrix. The problem arises in specific ...
0
votes
1answer
60 views

Can a 6-arm star be convex

Please help me with the following question. Suppose that the constant level contours of some function $V:\mathbb{R}^{2} \rightarrow \mathbb{R}$ have the shape of a symmetric 6-arm star. Can such a ...
1
vote
1answer
92 views

Does a Convex Function need to be Continuous

I have been trying the following problem and I am very confused. If possible the problem should be solved with derivatives. If the derivative exists for all the points on the graph then it is ...
0
votes
2answers
166 views

Quadratic Function must be positive definite to have a unique minimum

I have attempted the following question multiple times and I am very confused about the proof please help me solve it. Let $V(x)=a+b^{T}x+\frac{1}{2}x^{T}Cx$ for some $a \in \mathbb{R}, b \in ...
2
votes
2answers
75 views

Scale-invariance of Simpson's rule approximations to log

If I was trapped on a desert island and needed to compute $\log(2)$, the natural logaritm of $2$, one thing I could do is use the equality $$\log(2) = \int_1^2 \frac{1}{x} \ dx$$ and approximate the ...
1
vote
0answers
34 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 ...
0
votes
0answers
37 views

constrained minimization in N dimensions

I am looking to create an algorithm to minimize an N dimensional problem. I am unsure how to write it in its generic form, so I will show it in 1, 2 and 3 dimensions Minimize $ \sum_{i} x_i\left [ ...
1
vote
1answer
71 views

a conjugate gradients result for eigenvalue estimates

Consider the not preconditioned CG-method for a linear system $Ax=b$. Define $\beta_j = \frac{(r_{j+1},r_{j+1})}{(r_j,r_j)}$ and $\alpha_j=\frac{(r_j,r_j)}{(Ap_j,p_j)}$, where $(x,y) = y^Tx$ denotes ...
1
vote
1answer
26 views

How to determine coefficients of $p(x) = x^6$ with the Chebyshev processing

I want to calculate the coefficients of $p(x) = x^6$ with the Chebyshev processing. How to do that? Following question would be, how to estimate the error in $[-1,1]$, if i only use terms until ...
0
votes
0answers
69 views

What numerical methods could I use for this argmin problem?

I wish to solve the following using Numerical Methods: $$ \bar{m} = \underset{m \geq 0}{\text{argmin}} \left( \int_a^b \left( \frac{1}{\left(\sum_{i=1}^M \left(c_i^\alpha \cdot n^2 y^{-m-1} \cdot ...
0
votes
1answer
116 views

Calculate the interpolation polynom with Neville scheme

i have the following: $P \in \Pi_3$ is the interpolation polynom with $P(x_i)=f_i$, for $$x_i \quad -1 \quad 0 \quad 1 \quad 3$$ $$f_i \quad 5 \quad -6 \quad -9 \quad 33$$ (table) I want to ...
0
votes
1answer
24 views

How to show, that a relative mistake of a special function can be estimated in a given way

how to show, that if you have a function like this $$ y = f(x_1,...,x_m) := c \frac{x_1 *...*x_r}{x_{r+1},...,x_m}, \quad 1 < r \leq m,$$ the relative mistake in first order can be estimated ...
0
votes
1answer
39 views

Constrained non-linear optimisation algorithm making use of problem structure

I have a problem that in some ways is quite simple and in other ways is quite hard. I feel that there is probably an algorithm out there that is better suited to solving my problem than the one I am ...
0
votes
3answers
93 views

Does $\sqrt{\frac{2n}{n + 1}}$ have a global minimum, for large $n$?

Does the quantity $$\sqrt{\frac{2n}{n + 1}}$$ have a global minimum, for large $n$? Successive tries at WolframAlpha yield the following results: Minimize $\sqrt{\frac{2n}{n + 1}}$ for $n > ...
2
votes
1answer
155 views

Ideas on matrix factorizations and/or transformations for $\ell_1$ minimization

I am starting with a typical $\ell_1$ basis pursuit problem: $$ \min_{\mathbf{x}} \Vert \mathbf{x} \Vert_1 \quad \mathrm{s.t.} \quad \Vert \mathbf{ERx} - \mathbf{y} \Vert_2 \leq \epsilon, $$ where ...
4
votes
2answers
84 views

What numerical optimization method to use for this function?

In order to solve this over-determined system of equations numerically: $$ f_l(\mathbf x) = \displaystyle \left \lvert \sum_{k=1}^Kx_k^2e^{-j\frac{2\pi}Np_kl} \right \rvert , \qquad P = ...
0
votes
1answer
61 views

Numerical Optimization methods?

What kind of functions are suitable for numerical optimization methods such as Newton, Gradient Descent, ... ? Any conditions?
2
votes
4answers
140 views

Speeding up the convergence for Steepest Descent Method

How would one speed up the convergence of the method of steepest descent when the minimum is in a very long, narrow structure? I know the fact that is a steep minimum covers more iterations to go ...
0
votes
1answer
88 views

Better than Runge-Kutta-Fehlberg 4(5) at high order?

I wonder what are currently the best numerical solvers of ODE for high-accuracy computations. I need an efficient and accurate method to solve ODE that are not pathological (all is smooth) using ...
1
vote
0answers
86 views

Divergence of Gradient Method

Is there any example of a continuous differentiable function out there, in which the gradient method with Armijo's stepsize-rule doesn't converge? I found it pretty hard to create one myself because ...
1
vote
0answers
127 views

Qualities of Projected Gradient Methods

Consider the following constrained minimization problem: $ min_{x \in X} \ f(x) $ where $ X \subset \Bbb{R}^{n} $ is a nonempty closed convex set and f is continuously diferentiable. I'm ...
0
votes
1answer
45 views

surface approximation using least squares

I am studying the following problem. Soppose you have two Bezièr patches with a common curve; suppose that the control points of the two patches are given by some initial guess (they are all known). ...
2
votes
1answer
100 views

Formulate optimization problem

My research area has "nothing to do with mathematics" but I still find it full of optimization problems. Therefore, I would like to learn to formulate and solve such problems, even though I am not ...
0
votes
0answers
40 views

Computation of gradient of FE-function

I have a vectorfield $f_h \in \mathbb{R}^3$, which is a Finite Element function, i.e. $f_h(x)= \sum_i f^i \phi_i$. Now, i want to compute $\nabla (I_h[f_h])$ on a triangulation, where $I_h$ is the ...
1
vote
0answers
97 views

Maxima with equality constraints

I am trying to create an algorithm that finds the global maximum to a function with (in)equality constraints numerically. However, I am trying to fit that into a webpage (for example, via javascript. ...
0
votes
1answer
307 views

Finding a Fixed Point Solution

I'm trying to solve the following where $A,B >0$ in the most general case (don't assume they are equal). Wolfram Alpha cannot compute this in the allowed time, but I feel some fixed point ...
2
votes
1answer
106 views

Graphically, what is positive semidefinite-ness?

Suppose that we are trying to minimize a function $f$ on $\mathbb{R}^n$ and we apply Newton's method, updating: \begin{align} \mathbf{x}_{n+1} = \mathbf{x}_n - [\nabla^2 f(\mathbf{x}_n)]^{-1} \nabla ...
3
votes
2answers
91 views

Numerical optimization question

I need to solve the following optimization problem. \begin{aligned} \min_{\lambda_0,\lambda_1} & \sum_{t=1}^T\sum_{n\in\{4,8,12,16\}} \left(\frac{1}{n}A_n + \frac{1}{n}B_n^\top X_t + ...
0
votes
1answer
144 views

Optimize material in a can

A can in the shape of a right circular cylinder is to be constructed to contain 1000 cm$^3$. The circular top and bottom of the can must have a radius of 0.25 cm more than the radius of the can so ...
3
votes
1answer
349 views

Root Finding Algorithm for Discrete Functions

I was recently working with functions of the form $$N - \sqrt{\frac{N}{x}}\cdot\left\lfloor \frac{N}{\sqrt{N/x}}\right\rfloor + \sqrt{\frac{N}{x}} - \left\lfloor \sqrt{\frac{N}{x}}\right\rfloor$$ ...
5
votes
2answers
159 views

Non-convex optimization: $\min ||y-Ax||_p$ for very small $p$ given that $||x||_2=1$

I need to find $x$ that minimizes the cost function $\|y-Ax\|_p$ when $p$ is close to $0$, subject to the constraint $\|x\|_2=1$ where $x$ and $y$ are vectors in $\mathbb{R}^n$ and $A$ is an $n\times ...
2
votes
1answer
711 views

Optimizing integral functionals using Matlab

I am looking for some bibliography regarding solving integral optimization problems numerically (preferably using Matlab). I want to solve problems of the type $$ \min_{r \in A} \int_a^b ...
3
votes
1answer
93 views

least square problem

Let $1<p<\infty $.We define the space: $L_{V}^{p}(-1,1)=\left \{ f:(-1,1)\rightarrow \mathbb{R}:\int_{-1}^{1}\left | f(x) \right |^{p}V(x)dx<\infty \right \}$ We define the norm: $\left \| ...
2
votes
1answer
212 views

Calculating the gradient without knowing the function

I have to develop an optimizer for a simulation. There are 7 reference values $$ r_1, r_2,\ldots,r_7 $$ (certain values which are expected to show up) and 7 corresponding actual values $$ ...
0
votes
2answers
404 views

What Stopping Criteria to Use in Projected Gradient Descent

Suppose we want to solve a convex constrained minimization problem. We want to use projected gradient descent. If there was no constraint the stopping condition for a gradient descent algorithm would ...
1
vote
0answers
91 views

Ritz method for the buckling of a plate using calculus of variation

I need to find buckling of plate. And i have got the constant in analytical solution which can be founded from variation methods. How can i find polynomial for the same function for Ritz method? I ...
3
votes
1answer
111 views

What does “overdetermined” mean

When we say a problem is an overdetermined system, what do we mean by that in a rigorous fashion? Thanks.
2
votes
0answers
335 views

Logistic regression algorithm in Casio and Texas Instruments calculators

When using logistic regression on a Casio or Texas Instruments calculator, the output is of the form $$f(x) = \frac{c}{1+ae^{-bx}} $$ The problem I have (when teaching in a class where both types of ...
2
votes
3answers
618 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 ...