# Math Help - [MATLAB] Optimization - Reduce Step size?

1. ## [MATLAB] Optimization - Reduce Step size?

Hi All,
I have made a number of post lately relating to this problem so I thought I would make a decent post and highlight exactly what my problem is so that hopefully I can get some feedback. Basically what I am doing is using the optimisation toolbox in matlab to find optimize the geometric potential of a system of nodes (put simply trying to get the nodes to disperse to find a minimum energy state).

The geometric potential of a system of uniformly weighted nodes is expressed by:

$G = \sum_{i=1}^{n}\frac{1}{d_{ij}^q}$

Where:
$n$ is the number of nodes in the system
$q$ is the power index
$d_{ij}$ is the distance between node $i$ and node $j$

Two forms of geometric potential are considered:
Global Geometric Potential $\left( G_G \right)$ and Relative Geometric Potential $\left( G_R \right)$

In the case of $G_G$, each node is considered to be connected to every other node. $G_R$ only considers the contributions from adjacent nodes that are connected directly. $G_R$ is the form used by the objective function in this example.

The figure below shows a set of 9 nodes placed at random inside a unit square. The black lines represent the connections between nodes and the blue lines represent the voronoi diagram of the points.

The problem is that the optimization routine is changing the values of $x_i$ and $y_i$ by too large a value per iteration. The connectivity as determined by the Delaunay triangulation is only valid inside each nodes Voronoi cell (in blue) so the maximum amount any node should move after only 1 iteration is to the edge of its respective Voronoi cell.

My understanding is that the optimization routine recognizes that moving node $1$ and node $4$ together to the point $\left(1,0\right)$ (for example) is a valid since the two nodes are not connected. The problem then arises that the Delaunay triangulation can no longer be recomputed since two nodes would now occupy the same position so there are duplicate nodes in the system.

I have tried writing a constraint function to limit how close 2 nodes can get to each other, but the step size taken by the routine appears to be the biggest issue that needs addressing. my problem in a nutshell is:

Is there a way to reduce the step size of the problem limit
$x_i$ and $y_i$ to move by a maximum absolute value of say $\delta = 0.05$ per iteration?

The matlab code used is given here. Running the code without any parameters will use the default values shown in the figure above. I am useing MATLAB 2008b with the optimisation toolbox version 4.1.
Code:
function [xy,fval,eflag] = Relative_Geometric_Potential(xy_start,q)
% Take the nodes xy_start and optimize their position inside a
% unit sqaure in order to minimize the relative geometric
% potential of the system subject to the power index q.
% > xy_start is a 1x2n vector where there are n nodes in the system.
% > q is a possitive integer.

if nargin < 2;q = 2;end%set q=2 by default.(Not that important for this example)
%Start position of nodes [x,y]
if nargin < 1 % use default
xy_start = [0.196 0.992 0.802 0.729 0.498 0.809 0.357 0.073 0.591,...%xvals
0.910 0.194 0.432 0.749 0.946 0.764 0.559 0.184 0.498];%yvals
end

n = length(xy_start)/2;%number of nodes
LBounds = zeros(1,2*n);%lower Bounds
UBounds = ones(1,2*n);%upper bounds

options = optimset('Display','iter','Algorithm','active-set');
f = @(xy)Gr(xy,q);%use power index of 2 for Geometric Potential Function

[xy,fval,eflag] = fmincon(f,xy_start,...
[],[],[],[],LBounds,UBounds,[],options);

xy = reshape(xy,length(xy)/2,2);%reshape to nx2 matrix
end

function fval = Gr(xy,q)
%xy is row vector = [x y]
%q is power index

xy = reshape(xy,length(xy)/2,2);%reshape to nx2 matrix
x = xy(:,1);y = xy(:,2);%pull column vectors

TRI = delaunay(x,y,{'QJ' 'Pp'});%determine connectivity between local nodes
tmp = [TRI(:,[1,2]);TRI(:,[2,3])];%find nodes i,j of individual element
tmp = sort(tmp,2);%sort so node i < node j
mem_ij = unique(tmp,'rows');%take unique rows to get connectivity

k = xy(mem_ij(:,1),:)-xy(mem_ij(:,2),:);%vector delta
dij = sqrt(sum(k.^2,2));%normal of k
foo = sum(dij.^(-q));%sum for final ans
fval = 2*foo;%account for dij -> dji connections  (ie symetric dij matrix)
end
Any advice on how to solve this issue would be greatly appreciated.

Kind Regards,
Elbarto