Approximate function by stacking building blocks
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I need some help with a 'generalised Lego problem':
Given a function $f(x)geq 0$ on an interval $[a,b]$, and a 'building block' function $p(x)geq 0$ with compact support. The maximum of f shall be larger than the maximum of p. Think of $[a,b]=[-5, 5]$ and the support of $p$ shall be $[-.2, .2]$.
My goal is to approximate $f$ by stacking translations of $p$ (i.e. local copies of the form $p(x-x^{ast})$ for given grid points $x^{ast}$).
My approach so far: Given a grid ${x_i}_{i=1,dots,n}$, I am looking for non-negative integer coefficients ${a_i}_{i=1,dots,n}$, such that the following is minimal:
$$
|f-sum_{i}a_i ,p(cdot-x_i)|_{L^2}^2 := int_a^b left|f(x)-sum_{i}a_ip(x-x_i)right|^2,dx;.
$$
It is straightforward to reformulate this problem as
$$
frac{1}{2}cdot mathbf{a}^topmathbf{Q}mathbf{a} + mathbf{b}^{top}mathbf{a} to min;,
$$
with a symmetric matrix $mathbf{Q}$ and a vector $mathbf{b}$.
(Up to constant factors, $mathbf{Q}$ and $mathbf{f}$ have entries $left(p(cdot-x_i), p(cdot-x_jright)_{L^2}$ and $left(p(cdot-x_i), fright)_{L^2}$, respectively. )
I am aware that if the coefficients were to be allowed to be arbitrarily real, this is standard. How do I tackle this problem with integer and non-negative solutions without having to buy a huge and expensive black-box optimisation software? Or am I following the wrong approach at all? Would it be even possible to optimise for the grid as well?
approximation integer-programming discrete-optimization quadratic-programming
asked Jan 10 at 9:03
Marc MingoulisMarc Mingoulis
437
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add a comment |
$begingroup$
I need some help with a 'generalised Lego problem':
Given a function $f(x)geq 0$ on an interval $[a,b]$, and a 'building block' function $p(x)geq 0$ with compact support. The maximum of f shall be larger than the maximum of p. Think of $[a,b]=[-5, 5]$ and the support of $p$ shall be $[-.2, .2]$.
My goal is to approximate $f$ by stacking translations of $p$ (i.e. local copies of the form $p(x-x^{ast})$ for given grid points $x^{ast}$).
My approach so far: Given a grid ${x_i}_{i=1,dots,n}$, I am looking for non-negative integer coefficients ${a_i}_{i=1,dots,n}$, such that the following is minimal:
$$
|f-sum_{i}a_i ,p(cdot-x_i)|_{L^2}^2 := int_a^b left|f(x)-sum_{i}a_ip(x-x_i)right|^2,dx;.
$$
It is straightforward to reformulate this problem as
$$
frac{1}{2}cdot mathbf{a}^topmathbf{Q}mathbf{a} + mathbf{b}^{top}mathbf{a} to min;,
$$
with a symmetric matrix $mathbf{Q}$ and a vector $mathbf{b}$.
(Up to constant factors, $mathbf{Q}$ and $mathbf{f}$ have entries $left(p(cdot-x_i), p(cdot-x_jright)_{L^2}$ and $left(p(cdot-x_i), fright)_{L^2}$, respectively. )
I am aware that if the coefficients were to be allowed to be arbitrarily real, this is standard. How do I tackle this problem with integer and non-negative solutions without having to buy a huge and expensive black-box optimisation software? Or am I following the wrong approach at all? Would it be even possible to optimise for the grid as well?
approximation integer-programming discrete-optimization quadratic-programming
asked Jan 10 at 9:03
Marc MingoulisMarc Mingoulis
437
$endgroup$
$begingroup$
What about using a free/open-source solver?
$endgroup$
– prubin
Jan 12 at 0:48
$begingroup$
Open for suggestions. Octave has one in principle, but it is buggy and does not work.
$endgroup$
– Marc Mingoulis
Jan 14 at 10:21
add a comment |
$begingroup$
I need some help with a 'generalised Lego problem':
Given a function $f(x)geq 0$ on an interval $[a,b]$, and a 'building block' function $p(x)geq 0$ with compact support. The maximum of f shall be larger than the maximum of p. Think of $[a,b]=[-5, 5]$ and the support of $p$ shall be $[-.2, .2]$.
My goal is to approximate $f$ by stacking translations of $p$ (i.e. local copies of the form $p(x-x^{ast})$ for given grid points $x^{ast}$).
My approach so far: Given a grid ${x_i}_{i=1,dots,n}$, I am looking for non-negative integer coefficients ${a_i}_{i=1,dots,n}$, such that the following is minimal:
$$
|f-sum_{i}a_i ,p(cdot-x_i)|_{L^2}^2 := int_a^b left|f(x)-sum_{i}a_ip(x-x_i)right|^2,dx;.
$$
It is straightforward to reformulate this problem as
$$
frac{1}{2}cdot mathbf{a}^topmathbf{Q}mathbf{a} + mathbf{b}^{top}mathbf{a} to min;,
$$
with a symmetric matrix $mathbf{Q}$ and a vector $mathbf{b}$.
(Up to constant factors, $mathbf{Q}$ and $mathbf{f}$ have entries $left(p(cdot-x_i), p(cdot-x_jright)_{L^2}$ and $left(p(cdot-x_i), fright)_{L^2}$, respectively. )
I am aware that if the coefficients were to be allowed to be arbitrarily real, this is standard. How do I tackle this problem with integer and non-negative solutions without having to buy a huge and expensive black-box optimisation software? Or am I following the wrong approach at all? Would it be even possible to optimise for the grid as well?
approximation integer-programming discrete-optimization quadratic-programming
asked Jan 10 at 9:03
Marc MingoulisMarc Mingoulis
437
$endgroup$
I need some help with a 'generalised Lego problem':
Given a function $f(x)geq 0$ on an interval $[a,b]$, and a 'building block' function $p(x)geq 0$ with compact support. The maximum of f shall be larger than the maximum of p. Think of $[a,b]=[-5, 5]$ and the support of $p$ shall be $[-.2, .2]$.
My goal is to approximate $f$ by stacking translations of $p$ (i.e. local copies of the form $p(x-x^{ast})$ for given grid points $x^{ast}$).
My approach so far: Given a grid ${x_i}_{i=1,dots,n}$, I am looking for non-negative integer coefficients ${a_i}_{i=1,dots,n}$, such that the following is minimal:
$$
|f-sum_{i}a_i ,p(cdot-x_i)|_{L^2}^2 := int_a^b left|f(x)-sum_{i}a_ip(x-x_i)right|^2,dx;.
$$
It is straightforward to reformulate this problem as
$$
frac{1}{2}cdot mathbf{a}^topmathbf{Q}mathbf{a} + mathbf{b}^{top}mathbf{a} to min;,
$$
with a symmetric matrix $mathbf{Q}$ and a vector $mathbf{b}$.
(Up to constant factors, $mathbf{Q}$ and $mathbf{f}$ have entries $left(p(cdot-x_i), p(cdot-x_jright)_{L^2}$ and $left(p(cdot-x_i), fright)_{L^2}$, respectively. )
I am aware that if the coefficients were to be allowed to be arbitrarily real, this is standard. How do I tackle this problem with integer and non-negative solutions without having to buy a huge and expensive black-box optimisation software? Or am I following the wrong approach at all? Would it be even possible to optimise for the grid as well?
approximation integer-programming discrete-optimization quadratic-programming
approximation integer-programming discrete-optimization quadratic-programming
asked Jan 10 at 9:03
Marc MingoulisMarc Mingoulis
437
asked Jan 10 at 9:03
Marc MingoulisMarc Mingoulis
437
edited Jan 16 at 14:48
Marc Mingoulis
asked Jan 10 at 9:03
Marc MingoulisMarc Mingoulis
437
asked Jan 10 at 9:03
Marc MingoulisMarc Mingoulis
437
asked Jan 10 at 9:03
Marc MingoulisMarc Mingoulis
437
437
$begingroup$
What about using a free/open-source solver?
$endgroup$
– prubin
Jan 12 at 0:48
$begingroup$
Open for suggestions. Octave has one in principle, but it is buggy and does not work.
$endgroup$
– Marc Mingoulis
Jan 14 at 10:21
add a comment |
$begingroup$
What about using a free/open-source solver?
$endgroup$
– prubin
Jan 12 at 0:48
$begingroup$
Open for suggestions. Octave has one in principle, but it is buggy and does not work.
$endgroup$
– Marc Mingoulis
Jan 14 at 10:21
$begingroup$
What about using a free/open-source solver?
$endgroup$
– prubin
Jan 12 at 0:48
$begingroup$
What about using a free/open-source solver?
$endgroup$
– prubin
Jan 12 at 0:48
$begingroup$
Open for suggestions. Octave has one in principle, but it is buggy and does not work.
$endgroup$
– Marc Mingoulis
Jan 14 at 10:21
$begingroup$
Open for suggestions. Octave has one in principle, but it is buggy and does not work.
$endgroup$
– Marc Mingoulis
Jan 14 at 10:21
add a comment |
1 Answer
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There are a number of open-source solvers available from COIN-OR. I'm not a MATLAB user, but a quick search turned up the OPTI Toolbox for MATLAB, which includes several of the COIN-OR projects.
answered Jan 15 at 18:14
prubinprubin
1,385125
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add a comment |
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$begingroup$
There are a number of open-source solvers available from COIN-OR. I'm not a MATLAB user, but a quick search turned up the OPTI Toolbox for MATLAB, which includes several of the COIN-OR projects.
answered Jan 15 at 18:14
prubinprubin
1,385125
$endgroup$
add a comment |
$begingroup$
There are a number of open-source solvers available from COIN-OR. I'm not a MATLAB user, but a quick search turned up the OPTI Toolbox for MATLAB, which includes several of the COIN-OR projects.
answered Jan 15 at 18:14
prubinprubin
1,385125
$endgroup$
add a comment |
$begingroup$
There are a number of open-source solvers available from COIN-OR. I'm not a MATLAB user, but a quick search turned up the OPTI Toolbox for MATLAB, which includes several of the COIN-OR projects.
answered Jan 15 at 18:14
prubinprubin
1,385125
$endgroup$
There are a number of open-source solvers available from COIN-OR. I'm not a MATLAB user, but a quick search turned up the OPTI Toolbox for MATLAB, which includes several of the COIN-OR projects.
answered Jan 15 at 18:14
prubinprubin
1,385125
answered Jan 15 at 18:14
prubinprubin
1,385125
answered Jan 15 at 18:14
prubinprubin
1,385125
answered Jan 15 at 18:14
prubinprubin
1,385125
1,385125
add a comment |
add a comment |
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$begingroup$
What about using a free/open-source solver?
$endgroup$
– prubin
Jan 12 at 0:48
$begingroup$
Open for suggestions. Octave has one in principle, but it is buggy and does not work.
$endgroup$
– Marc Mingoulis
Jan 14 at 10:21