C++ Monte Carlo Code

Intro

An adaptive monte carlo algorithm to minimize a function. A space or volume is randomly sampled. From the best samples another bounding box/volume is constructed. Repeating the process defines the algorithm.

The algorithm implemented assumes a local minimum. This is in line with say mimimizing on the line with the fibonacci search where a minimum is assumed. So this algorithm is implemented when the solution is within the bounding volume.

Source

Files

Makefile
main.cpp
minmonte.h

projcompile.txt
unittestsreport.txt

Doxygen

main.cpp
Makefile
minmonte
valueindex

Issues

As the algorithm is addaptive there are essentially two variable paramters, the number of sample point ns and the sample size s to form the next bounding box. Actually the third would by the number of iterations.

Let $w^{[k]} (s_{i}, ns_{i})$ be notation for the algorithm iterated $k$ times.

Let $m = \sum_{i = 0}^{k - 1} ns_{i}$ be a measure of the number of function evaluations.
How are $k$ , $ns_{i}$ and $s_{i}$ choosen to mimimize $m$ ?

TODO Remember the best solution and put it into the next sample space.

TODO Overshoot - often variables approach the true solution in a limiting sense. It is quite real that by taking the best results the true solution bound is cut away. A way to address this problem may be to take the best s results and form a bounding box and increase the box's size by say 5%(in truth it would be made into another variable supplied by the user as there are always situations where this approach wouldn't make sense - say when your bounding volume isn't going to zero), thereby always capturing the minimum. Convergence is a little slower but the algorithm would be more adaptive - much less likely to deteriorate with overshoot.

Adding Point Rejection

TODO; Two classes, one to start the algorithm when the bounding box is greater than the problems space. Reject points outside the space and accept others. Iterate until the bounding box is within the problems space.

To test if the bounding box is in the problem space, possibilities include testing all combinations with the bounding box points(for small dim). Alternatively when all samples land in the correct space accept that the box is within the problems bounds.

Derive minmonteinit which adds a conditional template argument that tests if a point is within the problem region, accepting or rejecting the point. s good points are sampled.

Example

$f (x_{0}, x_{1}) = (x_{0} - 3)^{2} + 9 (x_{1} - 5)^{2}$ has a minimum at $[3, 5]$

80 function evaluations per iteration with the best 5 forming a new bounding volume.

ns=80; s=5;
a=[1,1]  b=[8,8]  lenab=[7,7]
a=[1.99122,4.59381]  b=[4.34178,5.4021]  lenab=[2.35056,0.80829]
a=[2.64947,5.03522]  b=[3.16214,5.07755]  lenab=[0.512662,0.0423337]
a=[2.93421,5.03547]  b=[3.06427,5.04618]  lenab=[0.130061,0.0107063]
a=[2.99541,5.03583]  b=[3.01512,5.03752]  lenab=[0.0197176,0.00169228]
a=[2.99627,5.03585]  b=[3.00706,5.03596]  lenab=[0.0107894,0.000104257]

TODO

I believe this is an adaptive random search algorithm.
The monte carlo boxing is subseptible to convergence problem where points converge to outer level curves. This keeps a large region open. Rework and clear this theory up. Find a solution.
Direct search on random interval hybrid.
Implement the pattern searches exploritory move with instead a random interval. If no better position is found reduce the interval about the optimal point. Let each exploration step have k random point tests. Implement this in Pattern Search as it is a variation of that algorithm.
Rewrite the class so that the dimension is not a templated parameter. Rewite the whole class with a formal review. Document with Doxygen comments.
Test with the Rosenbrock function.