compass_search
compass_search,
a MATLAB code 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.
Usage:
[ x, fx, k ] = compass_search ( @f, m, x,
delta_tol, delta, k_max )
where

@f is the "function handle"; that is, either a quoted
expression for the function, or the name of an Mfile that defines
the function, preceded by an "@" sign. The function should have the
form function value = f(m,x).

m is the number of variables.

x is an Mvector containing a starting point for the iteration.

delta_tol is the minimum allowed size of DELTA, and must be positive.

delta is the starting stepsize.

k_max is the maximum number of steps allowed. This is the only
way to avoid an infinite loop if the function decreases as it goes to infinity.

x is the program's estimate for the minimizer of the
function;

fx is the function value at x.

k is the number of steps taken.
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the MIT license
Languages:
compass_search is available in
a C version and
a C++ version and
a FORTRAN90 version and
a MATLAB version and
a Python version.
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compass_search_test
nelder_mead,
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minimizes a scalar function of several variables using the NelderMead algorithm.
praxis,
a MATLAB code which
minimizes a scalar function of several variables, without
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test_opt,
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Author:
John Burkardt
Reference:

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.

Richard Brent,
Algorithms for Minimization without Derivatives,
Dover, 2002,
ISBN: 0486419983,
LC: QA402.5.B74.

Charles Broyden,
A class of methods for solving nonlinear simultaneous equations,
Mathematics of Computation,
Volume 19, 1965, pages 577593.

David Himmelblau,
Applied Nonlinear Programming,
McGraw Hill, 1972,
ISBN13: 9780070289215,
LC: T57.8.H55.

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 385482.

Ken McKinnon,
Convergence of the NelderMead simplex method to a nonstationary point,
SIAM Journal on Optimization,
Volume 9, Number 1, 1998, pages 148158.

Zbigniew Michalewicz,
Genetic Algorithms + Data Structures = Evolution Programs,
Third Edition,
Springer, 1996,
ISBN: 3540606769,
LC: QA76.618.M53.

Jorge More, Burton Garbow, Kenneth Hillstrom,
Testing unconstrained optimization software,
ACM Transactions on Mathematical Software,
Volume 7, Number 1, March 1981, pages 1741.

Michael Powell,
An Iterative Method for Finding Stationary Values of a Function
of Several Variables,
Computer Journal,
Volume 5, 1962, pages 147151.

Howard Rosenbrock,
An Automatic Method for Finding the Greatest or Least Value of a Function,
Computer Journal,
Volume 3, 1960, pages 175184.
Source Code:
Last revised on 08 December 2018.