The Coordinate Search Optimization Algorithm

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 )

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.

Related Data and Programs:

ASA047, a MATLAB library which minimizes a scalar function of several variables using the Nelder-Mead algorithm.

COMPASS_SEARCH, is a MATLAB library which seeks the minimizer of a scalar function of several variables using compass search, a direct search algorithm that does not use derivatives.

ENTRUST, a MATLAB program which minimizes a scalar function of several variables using trust-region methods.

NELDER_MEAD, a MATLAB program which minimizes a scalar function of several variables using the Nelder-Mead algorithm.

PRAXIS, a FORTRAN90 library which implements the principal axis method of Richard Brent for minimization of a function without the use of derivatives.

TEST_OPT, a MATLAB library which defines test problems requiring the minimization of a scalar function of several variables.

TOMS178, a MATLAB library which optimizes a scalar functional of multiple variables using the Hooke-Jeeves method.


Jeff Borggaard,
Mathematics Department,
Virginia Tech.


  1. Evelyn Beale,
    On an Iterative Method for Finding a Local Minimum of a Function of More than One Variable,
    Technical Report 25,
    Statistical Techniques Research Group,
    Princeton University, 1958.
  2. Richard Brent,
    Algorithms for Minimization without Derivatives,
    Dover, 2002,
    ISBN: 0-486-41998-3,
    LC: QA402.5.B74.
  3. David Himmelblau,
    Applied Nonlinear Programming,
    McGraw Hill, 1972,
    ISBN13: 978-0070289215,
    LC: T57.8.H55.
  4. Ken McKinnon,
    Convergence of the Nelder-Mead simplex method to a nonstationary point,
    SIAM Journal on Optimization,
    Volume 9, Number 1, 1998, pages 148-158.
  5. Zbigniew Michalewicz,
    Genetic Algorithms + Data Structures = Evolution Programs,
    Third Edition,
    Springer, 1996,
    ISBN: 3-540-60676-9,
    LC: QA76.618.M53.
  6. Michael Powell,
    An Iterative Method for Finding Stationary Values of a Function of Several Variables,
    Computer Journal,
    Volume 5, 1962, pages 147-151.
  7. Howard Rosenbrock,
    An Automatic Method for Finding the Greatest or Least Value of a Function,
    Computer Journal,
    Volume 3, 1960, pages 175-184.

Source Code:

Examples and Tests:

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.

Last revised on 01 January 2012.