Tagged Questions
1
vote
1answer
36 views
Quadratic Forms in Non-Linear Optimization
This is a rather trivial question but I am having a great deal of trouble:
Let $f(x) = (1/2)xQx-xb$
and $E(x) = (1/2)(x-x^*)Q(x-x^*)$
then $E(x) = f(x) + (1/2)x^*Qx^*$
where $x,x^*,b$ are vectors ...
3
votes
1answer
28 views
existence of solution of $Ax= \max(b-x,0) $
How do you prove the existence of a solution to the linear system:
\begin{equation}
Ax= \max(b-x,0)
\end{equation}
A is an $n\times n$ matrix and $b$ is a vector in $\mathbb{R}^n$. $x$ is the ...
2
votes
1answer
233 views
Prove every local minimum is a global minimum
Let $Q\in\mathbb{R^{dxd}}$ and $A\in\mathbb{R^{d'xd}}$ be two matrixes and $b\in\mathbb{R^d}$, $c\in\mathbb{R^{d'}}$. Suppose $d'\lt d $. For $x\in\mathbb{R^d}$.
Minimize
$$f(x)= ...
3
votes
1answer
70 views
$P_{1c} = AP$ , $P_{2c} = BP$. How to find $P$? (being that $A$ and $B$ are $3\times 4$ matrices and $P$ is a $4\times 1$ vector)
This problem arose in my stereo vision project.
$$
P_{1c} = A*P
$$
$$
P_{2c} = B*P
$$
where: $P_{1c}$ and $P_{2c}$ are $3\times1$ vectors, $A$ and $B$ are $3 \times 4$ matrices and $P$ is a ...
4
votes
4answers
252 views
Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).
To approximate a function $G$ over the interval $[0,1]$ by a polynomial $P$ of degree $n$ (or less), we minimize the function $f:R^{n+1} \to R$ given by $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$, where ...
0
votes
1answer
62 views
Minimization to Maximization doubt in SVM
I came across a lecture on Support Vector Machines and in the lecture they converted a maximization problem into a minimization problem. I am wondering how it was done...
$
Max \frac {1}{||x||}
$
...
1
vote
1answer
101 views
Max function on a closed compact convex set.
Consider a closed convex compact subset $\mathbb{S}$ of $\mathbb{R}^N$ while we denote any of its point by $x=[x_1,x_2,\ldots,x_N]^T$. Define the function
\begin{align}
f(x)=max(x_1,x_2,\ldots,x_N)
...
2
votes
2answers
75 views
A Quadratic Problem (which looks very simple)
This arises as a part of my work.
\begin{align}
\min_{x^{H}x=1}~&x^{H}A_1x \\
subject~to~&x^{H}A_2x=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ hermitian matrices and $x$ is a unit norm ...
0
votes
1answer
78 views
Lagrange method - non-linear system of equations
I have to compute optimal parametres of truncated cone so that its Volume is fixed (lets say it is 1) and its surface is minimal using Lagrange method
These are equations desribing my object:
...
0
votes
0answers
240 views
Calculation/Estimate of Lipschitz Constant for Strictly Convex Function
I have a strictly convex function
$ f(\bf{x}) = \dfrac{1}{2}\bf{x'Ax + b'x} $
where $ \bf{f} : \mathbb{R^n} \rightarrow \mathbb{R} $
and I was wondering how I can find/estimate the Lipschitz ...
0
votes
1answer
55 views
Finding the value which minimises all residuals
I have a series of observations, measurements made at various times $t$. I now need to determine the most likely value of $R$ (distance) using the model below. The guide says I should find the value ...
2
votes
1answer
86 views
Positive values for a set of quadratic forms of Hermitian Matrices. (To find a set of vectors in which a hermitian matrix is positive definite)
Assume all matrices I discuss about are $N \times N$ and the vectors conform with dimensions. Consider the following set of Quadratic inequalities where all the matrices $A_i$ are hermitian.
...
2
votes
1answer
128 views
Nonlinear optimization with rotation matrix constraint
I'm trying to optimize the equation
|| R - W || = minimum
where W is a predetermined 3x3 matrix and R is the 3x3 matrix that I'm trying to optimize, with the ...
1
vote
5answers
124 views
Independence of Rotation Matrix Definitions
I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix
$$
A = \left(
\begin{matrix}
a_{11}& a_{12}& a_{13}\\
a_{21}& ...
1
vote
1answer
61 views
Least squares and (non-)linearity of parameters
I have a question about least squares and about what happens, if the function that we minimize, $E(P)$, is not linear in its parameters $P$.
Assume we want to minimize a function (the exact terms are ...
1
vote
0answers
90 views
Nonlinear least squares and polygon area
I found this paper that describes preserving the global area of a polygon given some deformation (section 5): http://www.kunzhou.net/publications/2DShape.pdf
I'm trying to do something very similar. ...
0
votes
0answers
88 views
Linear Algebra Simplification Query
Apologies in advance; my linear algebra is not exactly up to scratch, but a program optimisation problem I've come across just feels like theres a better way mathematically rather than ...
