COORDINATE_SEARCH is a MATLAB program which seeks the minimizer of a scalar function of several variables, by Jeff Borggaard.
The algorithm, which goes back to Fermi and Metropolis, is easy to describe. As originally designed, the algorithm begins with a central point, as well as 2*M "test" points, where M is the number of spatial dimensions. The I-th pair of test points differs from the central point only in the I-th coordinate. One test point increases, and one decreases, this coordinate by a fixed amount H.
The function value of all 2*M+1 points is computed; if the lowest value occurs at the center point, then H is decreased. If the lowest value occurs at a test point, then that test point becomes the new center point.
On the next step, using the current center point and size H, a new set of 2*M test points is computed, and the process is repeated.
Under certain simple conditions, the process will converge to a local minimizer of the function.
Note that fminsearch is a MATLAB built in command which minimizes a scalar function of several variables using the Nelder-Mead algorithm.
x_opt = coordinate_search ( x_center, f, flag )where
Very simple functions can be input as a quoted string. Thus, one could specify the f argument as '(x(1)-2*x(2)+7)^2'; However, for more complicated functions it makes sense to prepare an M-file that defines the function. For this same example, a suitable M-file would be:
function f = example ( x )
f = ( x(1) - 2 * x(2) + 7 )^2;
If this information was stored in an M-file called example.m, then one might invoke the optimization program with a command like
x_opt = coordinate_search ( x_init, @example, 0 )
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
COORDINATE_SEARCH is available in a MATLAB version.
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Jeff Borggaard,
Mathematics Department,
Virginia Tech.
COORDINATE_SEARCH_TEST calls all the tests.
BEALE is the Beale function, for which N=2.
BOHACH1 is the Bohachevsky function #1, for which N=2.
BOHACH2 is the Bohachevsky function #2, for which N=2.
EXTENDED_ROSENBROCK is the "extended" Rosenbrock function. This version of the Rosenbrock function allows the spatial dimension N to be arbitrary, except that it must be even.
GOLDSTEIN_PRICE is the Goldstein-Price polynomial, for which N=2.
HIMMELBLAU is the Himmelblau function, for which N = 2, and which has four global minima.
LOCAL is a badly scaled function with a local minimum, for which N=2.
MCKINNON is the McKinnon function, for which N=2. This function can cause problems for the Nelder-Mead optimization algorithm.
POWELL is the Powell singular quartic function, for which N = 4.
ROSENBROCK is the Rosenbrock "banana" function.
You can go up one level to the MATLAB source codes.