I have the following constrained (global) optimization problem:
For a user defined sorted real values vector:
xi = [xi(1), ... , xi(N+1)]
I need to find unknown vector:
x = [x(1), ..., x(L)]
where the integer L is the unknown length of the vector x (L>=0, L = 0 is the trivial case)
and the unknown vector x = [x(1), ..., x(L)] must satisfy the following specific conditions:
===============================================================
0. The new "refined" sorted vector xo = union(xi,x), where length(xo) = L+N+1
should to fulfil the following set of conditions:
1. min(xo) = xo(1) = xi(1), max(xo) = xo(L+N+1) = xi(N+1)
2. q = max(z(j)/z(j+1),z(j+1)/z(j)) < q_max, for j = 1, 2, ... L+N
where z(j) = xo(j+1) - xo(j) , z = diff(xo), length(z) = L+N
and q_max is user defined max ratio, where q_max > 1 (typically q_max ~ 1.05 - 1.2)
3. min(z) -> maximal, minimal distance between xo vector elements should be maximized.
It is obvious that for small L the constraint conditions (2) is not possible to satisfy.
===============================================================
The motivation of this problem is the creation of the so called, "homogenized" 1-D grid, where consecutive distances between elements of vector xo are relatively "slowly" changing.
I will be very happy for any recommendation how to effectively solve this problem using MATLAB + (global) optimization toolbox.
See additional function to generate "realistic" (in my case) test input data xi:
function [x,q,x_base,x_aux] = nodegridtest(shift,show)
if ~(0 <= shift && shift <= 125.0)
error('shift must be in the interval [0 , 125]')
7.25,7.25,7.25,7.25,7.25,7.25,7.25,7.25, ...
7.25,7.25,7.25,7.25,7.25,7.25,7.25,7.25, ...
7.25,7.25,7.25,7.25,7.25,7.25,7.25,7.25, ...
7.25,7.25,7.25,7.25,7.25,7.25,7.25,7.25];
xmax = sum(z_base) + sum(z_aux(1:6));
x_base = [0, cumsum(z_base)];
x_aux = x_base(end)+ [0, cumsum(z_aux)];
x_aux_actual = x_aux - shift;
x_full = union(x_base,x_aux_actual);
x = [x_full(x_full < xmax),xmax];
x_aux = [x_aux_actual(x_aux_actual < xmax),xmax];
q = max([z(1:end-1)./z(2:end);z(2:end)./z(1:end-1)],[],1);
plot(x(2:end-1),q,'bo-','LineWidth',3)
title(['q distribution for shift = ' num2str(shift)])
xlabel('position');ylabel('q ratio');
then simple generate test data for any integer 0<=shift<=125 like:
to get
and
