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https://github.com/thchang/delaunaysearch

Delaunay Search Black-Box Optimizer in Python 3
https://github.com/thchang/delaunaysearch

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Delaunay Search Black-Box Optimizer in Python 3

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# Delaunay Search Black-Box Optimizer



This repo contains a Python 3 implementation of an experimental Delaunay

mesh refinedment based black-box optimizer.

The Delaunay search technique is intended for usage with expensive black-box

functions in situations where the user already has a significant database of

pre-evaluated cost values.

To refine the current database toward the true minima, a Delaunay triangulation

(unstructured mesh) is implemented over the dataset.

In each iteration:

 - The objective value at the circumcenter of each element in the mesh is

   estimated based on the average linearly extrapolated value predicted by

   its d+1 neighboring cells.

 - The candidate cell with the lowest predicted objective value is subdivided

   by way of a function evaluation at its circumcenter.



For objective functions with Lipschitz continuous first derivatives, the

linear fit in each Delaunay cell will converge to the average gradient in that

cell almost surely as more data is sampled.

Therefore, the Delaunay search will converge to a local minima.



This is experimental research code, and is not yet suitable for all use

cases.

In particular, if the global minimum is not in the convex hull of the provided

data set, then the Delaunay search can never reach it.



## Contents



Note that all test functions have been shifted so that their global minima

has value f(x) = 0.



 - delsearch.py contains the DelaunaySearch subroutine and an optional main

   program which uses a quadratic function and latin hypercube design for

   testing.

 - Rosenbrock.py contains a 2D test function with a single minima at the

   bottom of a shallow "banana shaped" Rosenbrock valley at (0,0).

 - Griewank.py contains a 2D test function with many local minima, and

   a single global minima at (0,0).

 - Michaelwicz.py contains a 2D test function with a single global minima

   at approximately (2.2, 1.57), amid large flat ridges and sharp narrow

   valleys.



## Prerequisites



The DelaunaySearch function is written for Python 3 and depends on several

other python modules that can be installed via pip3.

 - numpy for numerical linear algebra

 - scipy.spatial for their Delaunay qhull wrapper

 - matplotlib.pyplot for visualizing results

 - pyDOE for their lhs (latin hypercube design) function



### User Interface



To use DelaunaySearch, call

``

[minind, x, f] = DelaunaySearch(data, obj_func, budget)

``

where

 - data is a python list containing the input data set (the first d entries

in each row are the coordinates of the points, and the last entry is the

objective value),

 - obj\_func is the objective function,

 - budget (optional) is the function evaluation budget

 - minind is the index of the minimum objective value observed

 - x is a list of design points evaluated (including those provided in data)

 - f is a list of corresponding objective values



### Test cases



To optimize a 2 dimensional quadratic using a latin hypercube design and

visualize the results in matplotlib:



``

python3 delsearch.py

``



To do the same for the Rosenbrock, Griewank, and Michaelwicz functions,

execute ``python3 Rosenbrock.py``, ``python3 Griewank.py``, and

``python3 Michaelwicz.py`` respectively.



## Author



* **Tyler H. Chang** - *Primary author*