The Compass Search Optimization 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.

The algorithm, which goes back to Fermi and Metropolis, is easy to describe. The algorithm begins with a starting point X, and a step size DELTA.

For each dimension I, the algorithm considers perturbing X(I) by adding or subtracting DELTA.

If a perturbation is found which decreases the function, this becomes the new X. Otherwise DELTA is halved.

The iteration halts when DELTA reaches a minimal value.

The algorithm is not guaranteed to find a global minimum. It can, for instance, easily be attracted to a local minimum. Moreover, the algorithm can diverge if, for instance, the function decreases as the argument goes to infinity.


[ x, fx, k ] = compass_search ( @f, m, x, delta_tol, delta, k_max )


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


COMPASS_SEARCH is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

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

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 MATLAB library which minimizes a scalar function of several variables, without requiring derivative information, by Richard Brent.

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.


John Burkardt


  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. Charles Broyden,
    A class of methods for solving nonlinear simultaneous equations,
    Mathematics of Computation,
    Volume 19, 1965, pages 577-593.
  4. David Himmelblau,
    Applied Nonlinear Programming,
    McGraw Hill, 1972,
    ISBN13: 978-0070289215,
    LC: T57.8.H55.
  5. Tamara Kolda, Robert Michael Lewis, Virginia Torczon,
    Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods,
    SIAM Review,
    Volume 45, Number 3, 2003, pages 385-482.
  6. 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.
  7. Zbigniew Michalewicz,
    Genetic Algorithms + Data Structures = Evolution Programs,
    Third Edition,
    Springer, 1996,
    ISBN: 3-540-60676-9,
    LC: QA76.618.M53.
  8. Jorge More, Burton Garbow, Kenneth Hillstrom,
    Testing unconstrained optimization software,
    ACM Transactions on Mathematical Software,
    Volume 7, Number 1, March 1981, pages 17-41.
  9. Michael Powell,
    An Iterative Method for Finding Stationary Values of a Function of Several Variables,
    Computer Journal,
    Volume 5, 1962, pages 147-151.
  10. 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:

COMPASS_SEARCH_TEST calls all the tests.

BEALE is the Beale function, for which M=2.

BOHACH1 is the Bohachevsky function #1, for which M=2.

BOHACH2 is the Bohachevsky function #2, for which M=2.

BROYDEN is the two dimensional modified Broyden function, for which M=2.

EXTENDED_ROSENBROCK is the "extended" Rosenbrock function. This version of the Rosenbrock function allows the spatial dimension M to be arbitrary, except that it must be even.

GOLDSTEIN_PRICE is the Goldstein-Price polynomial, for which M=2.

HIMMELBLAU is the Himmelblau function, for which M = 2, and which has four global minima.

LOCAL is a badly scaled function with a local minimum, for which M=2.

MCKINNON is the McKinnon function, for which M=2. This function can cause problems for the Nelder-Mead optimization algorithm.

POWELL is the Powell singular quartic function, for which M = 4.

ROSENBROCK is the Rosenbrock "banana" function, for which M = 2.

You can go up one level to the MATLAB source codes.

Last revised on 04 January 2012.