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9 views

what is the role of adding numerical dissipation to solve partial differential equations

I usually solve the partial differential equations (PDE), but I have never used the numerical dissipation to have a optimal results in terms on accuracy and stability of PDE's solution in generals. ...
0
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1answer
14 views

Conjugate gradient projection

Let $V$ be a collectino of the search direction for the conjugate gradient applied on a quadractic minimisation problem. As a proof of orthogonality in conjugate gradient: $$ V^T V = I $$ Now ...
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0answers
16 views

One-dimensional deblurring

I just begun studying image deblurring on my own, and I have a question. Most books I found say that I can see the images as arrays, and that I can "vectorize" the arrays of the images by stacking the ...
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2answers
32 views

Newton's method for optimization

I have been reading about Newton's method and know that you can use it for optimization problems. However, does Newton's method only guarantee convergence to a local minimum or maximum, or can it be ...
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0answers
10 views

Are there any optimization strategy suiting this framework?

For optimization problem: $min \quad f(x_1, x_2)$ Are there some strategies that are doing this sequentially, i.e. first solve $min_{x_1} \quad f(x_1, x_2^{k})$ to get $x_1^{k+1}$ then solve ...
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0answers
19 views

Quantization threshold selection

I have the $256$-bin histogram representing a distribution of the values taken by a certain descriptor element. This descriptor element takes the values in $0-255$ range, hence $256$ bins. I want to ...
1
vote
1answer
42 views

Do standard gradient descent methods work on complex variables

I am currently whishing to optimize a function numerically $f(z)$ where $z \in \mathbb{C} $ ($f(z) \in \mathbb{R}$) . I am doing this via numerical packages (specifically scipy in python) and I have ...
0
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0answers
22 views

Matrix approximation

How to solve numerically for non-negative full-rank matrices $P$ and $E$ with the following constraints? $Y$ is a known non-negative matrix with $G$ rows and $N$ columns, $G > N$ 1) ...
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0answers
27 views

Optimization on a grid

I worked a lot on defining the problem so I will be grateful to get input if i'm not clear enouth and I will fix the question. We have a grid made out of uniform points on $[x,y],$ $x,y\in[0,1],$ ...
0
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0answers
26 views

Obtaining zeroes in a matrix

I have a large matrix , with elements somewhere between 0 and 0.4 . I want to apply local unitaries, that is, unitary matrices of the form: $$U_{L} = U \otimes U \otimes U$$ in order to make my ...
2
votes
1answer
57 views

How to “separate” a matrix into two vectors?

I have a matrix $M$ and I would like to find two vectors $u$ and $v$, that minimize $$ \sum_{i,j} (M_{i,j}-u_iv_j)^2 $$ How can I do this (numerically)? Actually this is very simplified ...
1
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0answers
18 views

Find max min with linear programming

I need to solve $$ \max_x \min_y x^T M y $$ subject to $$ \sum_{i=1}^n y_i = 1, \sum_{j=1}^m x_j = 1,\\ x \geq 0, y \geq 0 $$ where $ M \in \mathbb{R}^{m\times n} $, $ x \in ...
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0answers
37 views

is this function an ill-shape convex function?

I have a function with parameter $\vec{{\alpha}}$ where it is formulated by the formula: $$ f(D|\alpha)=n_1{\alpha}_{1}+...+n_m \alpha_m -Nlog \sum_{i=1:m} exp(\alpha_{i}+g_i(D)) $$ where $g_i{(D)}$ ...
0
votes
1answer
32 views

Linear Optimization Problem with exponential variable

Hey Folks I've encountered an optimization problem which has a linear programming structure but it's coefficients are nonlinear function of another variable. here is the problem: $$\max ...
1
vote
2answers
39 views

linear least squares with equality constraints

I am looking for iterative procedures for solution of the linear least squares problems with equality constraints. That is, my problem is to solve $$\min_{x} \lVert{Ax-b} \rVert _2, \ ...
4
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1answer
179 views

Monotonic Function Optimization on Convex Constraint Region

So I have the following function, which I want to maximize: $$f(x_1,...,x_n) = \sum_{i=1}^n\alpha_i\sqrt{x_i}$$ (where all $\alpha_i$ are positive), subjected to the following equality and inequality ...
3
votes
1answer
23 views

How to create an example with exponential running time for a given implementation of the simplex algorithm?

Say I have a black box implementation of the simplex algorithm given. Even though the worst case complexity is exponential, the implementation is fast for all cases I have tried. Is there a ...
0
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0answers
15 views

Optimization problem for a piecewise constant function

I am considering the following optimization problem: $\underset{u}{\text{minimize }}\quad \underset{v}{\max} f(u,v)$ where for a fixed $u$, the function $f(u, .)$ is piecewise constant. Do you ...
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0answers
32 views

Find a symmetric matrix of minimal Frobenius norm

Let $A\in \mathbb{R}^{n\times n}$ be a symmetric matrix, And let $$x\in \mathbb{R}^n$$ be such that $\lVert Ax-b\rVert_2 = \min_{z\in \mathbb{R}^n} \lVert Az-b\rVert_2$. Show how to calculate a ...
1
vote
1answer
41 views

Number of points needed for linear interpolation of sine in $[0,\frac{\pi}{2}]$ with given error bound

I want to get a set of equispaced points in $[0,\pi/2]$ and use piecewise linear interpolation generated by those points to fit the sine function. And I want to determine how many points do I need to ...
2
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0answers
24 views

What is the logic behind the given optimization problem?

I am following a book which has a part on numerical optimization techniques. In order to elaborate Karush-Kuhn-Tucker theorem, they gave the following example: When the unconstrained solution $x=A^+ ...
0
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1answer
30 views

Consistency Error for Runge-Kutta-Methods

I'm thinking about an old exercise concerning the Friedrich Scheme $$u_{j}^{n+1} = \frac{1}{2}\left(u_{j+1}^n + u_{j-1}^n\right) - \frac{r}{2}\left(u_{j+1}^n - u_{j-1}^n\right)$$ where $u_j^n ...
3
votes
2answers
65 views

How does one choose the step size for steepest descent?

Consider finding the minimal value for any function $g$ from $\mathbb{R}^n$ to $\mathbb{R}$. The method of steepest descent for finding a local minimum for an arbitrary function $g$ from from ...
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0answers
19 views

(Convex) reformulation of a nonlinear program

Consider the following program: \begin{eqnarray*} \min_{\mathrm x}\sum_{i=1}^{n}{\sum_{j=1}^{n}{\big(x_i(Sx)_i-x_j(Sx)_j\big)^2}}\\ \mathrm{subject\; to}\quad \sum_{i=1}^{n}{x_i}=1 \\ x_i\geq 0 ...
1
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1answer
48 views

Maximization problem on an ellipsoid [closed]

for three variables, $$\max f(x,y,z)= xyz \\ \text{s.t.} \ \ (\frac{x}{a})^2+(\frac{y}{b})^2+(\frac{z}{c})^2=1$$ where $a,b,c$ are constant how to solve the maximization optimization problem? ...
0
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0answers
12 views

Finite Elements - Motivation and use: Almost affine families

Some finite elements $(K,P_k,\sigma_k)$ used for higher order problems, e.g. fourth order problems, can not be embedded into a family of affine families. Especially I'm interested in the Argyris FE. ...
0
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1answer
45 views

How would you find the profit maximising level of output of these 2 products?

Suppose a company produces two products A and B which have demand functions \begin{gather*} D_{A}=30-P_{A} \\ D_{B}=25-P_{B} \end{gather*} With $P_{A}$ and $P_{B}$ being their prices. If the combined ...
2
votes
1answer
51 views

How to replace piecewise objective function in convex optimization problem?

Suppose I have a minimization problem \begin{equation} \begin{aligned} & \min\limits_{x} & & g(x)+f(x) \end{aligned} \end{equation} \begin{equation} f(x)= \begin{cases} 1, ...
0
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0answers
25 views

Function approximation without metric, is it possible?

As I learn't from analysis/functional analysis the approximation technique are based on the use of a norm in a metric space, even the interpolation is actually a particular case of these approximation ...
1
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0answers
38 views

How to solve a system of nonlinear Hamilton-Jacobi PDE's numerically in MATLAB/Maple/other?

I've been trying recently to solve the following system of Hamilton-Jacobi PDE's, which are of the hyperbolic, first-order type: $ V_1,_t - 0.5 V_1,_x^2 + ...
2
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0answers
48 views

Optimization Problem Involving an Integral Equation

I have an integral equation for a probability current $j(t)$ given by: $$ \large\frac{1}{\sqrt{t}} e^{-\frac{\bar{x}(t)^2}{4 t}} = \int_0^t \frac{j(t')}{\sqrt{t-t'}} ...
1
vote
1answer
34 views

Minimization optimization - where have I gone wrong?

Following @littleO's advice, I've set about to minimize $\sum_n ((x-x_n)^2+(y-y_n)^2+(z-z_n)^2-d^2)^2$. Going using an exact Hessian (because the function is smooth definite) as follows: $\textbf{H} ...
0
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0answers
16 views

Convex, differentiable approximation of optimization objective function

I have defined an objective function $$f(E,B)=\sum_i \max\{ \text{abs}\{\epsilon_i\}-b_i,0 \}$$ for $\epsilon_i \in \text{E}$, deviations of a function from some target observations, and $b_i \in B$, ...
0
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0answers
17 views

Optimization of Inputs to Monte Carlo Simulation Based on Outputs

I have an optimization process that seems to work, but I want to better understand why it works and whether there's a better way to do what I'm trying to achieve. Basically I am optimizing two (or ...
1
vote
1answer
28 views

Separable linear programs

Assume, we have two distinct LPs: \begin{equation*} \begin{aligned} & \text{min}_{x_1} & & c_1^Tx_1 \\ & \text{subject to} & & A_1x_1 = b \\ & & & x_1 \geq 0 \\ ...
3
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1answer
35 views

Convex optimization where both the region and function are ugly

I am trying to build a gradient descent algorithm for a convex function over a convex region in high dimension with no closed form. All I can do is: Check whether a point is in the region Evaluate ...
0
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0answers
19 views

If weighted-sum scalarization is ok with concave function?

I want to ask a basic question related to optimization. I read that weighted-sum scalarization of multi-objective optimization problem cannot explore Pareto-front on a part of function where it is ...
0
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0answers
18 views

Holding the constraints of a constrained optimization when transformed into unconstrained optimization

Suppose there is a constrained convex optimization problem as shown below \begin{equation} \begin{aligned} & \min\limits_{\mathbf{x}} & & f(\mathbf{x}) \\ & \text{s.t.} & & ...
0
votes
0answers
9 views

Projecting on the constraint set $X = UU^T$

I'd like to explore projecting $(\hat X, \hat U)$ on the nonconvex constraint set $\{(X,U) \mid X = UU^T, U \text{ has $r$ columns}\}$ where $\hat X$ is a symmetric matrix and $\hat U$ is some ...
0
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0answers
38 views

references: L-BFGS rate of convergence

I was trying to find results about the rate of convergence for the L-BFGS algorithm (in the nonlinear case). What I end up with so far is that the BFGS-Algorithm converges Q-superlinearly this 50 ...
1
vote
0answers
25 views

Choosing a non-convex global optimization algorithm based on the number of permitted steps

Can anyone comment on the most suitable approach for the following optimization problem: We are given finite bounds for a set of $n$ real-valued parameters of an unknown deterministic function. The ...
0
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2answers
39 views

Show that n(n+1)/2 multiplications are required

$a_{11}x_1$+$a_{12}x_2$+$a_{13}x_3$+ ...+ $a_{1,n-1}x_{n-1}$+$a_{1n}x_n$ =$b_1$ $a_{22}x_2$+$a_{23}x_3$+ ...+ $a_{2,n-1}x_{n-1}$+$a_{2n}x_n$ =$b_2$ $a_{33}x_3$+ ...+ $a_{3,n-1}x_{n-1}$+$a_{3n}x_n$ ...
3
votes
1answer
44 views

How to show the existence of positive root for this equation?

$f(x)=x-(x+k)^\epsilon$ where $k>1$ and $0<\epsilon<1$. How do I show that $f(x)=0$ has a root at $x>0$? The only one root part is obvious, but I am asking this to rigorously show that ...
0
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0answers
30 views

How to obtain the best fit?

I have a complex function say $f(x,a,b,c)$ where $x$ is variable and $a,b,c$ are the parameters. Parameters $a,b$ are linked as $d = (1/a^2) - (i*pi/b)$. The limits of x is very small say -0.02 to ...
1
vote
1answer
139 views

Optimization problem involving, $L_2$, $L_1$ norm and constraints.

Can somebody suggest me how to solve the following optimization problem? \begin{equation*} F(\mathbf{w},\xi)= \begin{aligned} & \underset{\mathbf{w,\xi}}{\text{minimize}} & & ...
0
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0answers
39 views

Gradient descent and penalty method

I am seeking a minimum of a function under an inequality constraint. How can I set stop condition? The problem is that $\nabla f_p$ never goes to zero. The function: $$f(x_1, x_2)=\left(x_1 - ...
1
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0answers
37 views

Computational methods to minimizing the norm of a matrix monomial.

Linear optimization solves the problem $$\min_{\bf x}\{\|{\bf Ax - b}\|_2^2\}$$ Edit: Some clarification Doing the derivation of the optimum, first expand the norm: $$\|{\bf Ax - b}\|_2^2 = ({\bf ...
0
votes
1answer
26 views

Optimizing multivariable functions by sections

Let $f(x_1,x_2,\ldots,x_n)$ be function of several real variables $x_1, x_2, \ldots, x_n$. Suppose that every local maximum of $f(x_1,x_2,\ldots,x_n)$ is actually a global maximum. The question is to ...
0
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0answers
54 views

Formula reduction to some physical interpretation

Problem Given N 3-dimensional points which are {$p_1,p_2,..,p_n$} where $p_i = (x_i,y_i,z_i) $ . I have to find the value of the formula $$ \sum \limits_{i \neq j} \frac{ \mid ...
0
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0answers
31 views

Global optimization methods where constraints are lipschitz functions

Is there any global optimization methods where objective function is nonlinear (not lipschitz) but constraints are lipschitz functions?