MATLAB Function Reference |
Minimize a function of several variables
Syntax
x = fminsearch(fun,x0) x = fminsearch(fun,x0,options) [x,fval] = fminsearch(...) [x,fval,exitflag] = fminsearch(...) [x,fval,exitflag,output] = fminsearch(...)
Description
fminsearch finds the minimum of a scalar function of several variables, starting at an initial estimate. This is generally referred to as unconstrained nonlinear optimization.
starts at the point x =
fminsearch(fun,x0)
x0
and finds a local minimum x
of the function described in fun
. x0
can be a scalar, vector, or matrix. fun
is a function handle. See Function Handles in the MATLAB Programming documentation for more information.
Parameterizing Functions Called by Function Functions, in the MATLAB mathematics documentation, explains how to provide additional parameters to the function fun
, if necessary.
minimizes with the optimization parameters specified in the structure x =
fminsearch(fun,x0,options)
options
. You can define these parameters using the optimset
function. fminsearch
uses these options
structure fields:
[x,fval] = fminsearch(...)
returns in fval
the value of the objective function fun
at the solution x
.
[x,fval,exitflag] = fminsearch(...)
returns a value exitflag
that describes the exit condition of fminsearch:
1 |
fminsearch converged to a solution x . |
0 |
Maximum number of function evaluations or iterations was reached. |
-1 |
Algorithm was terminated by the output function. |
[x,fval,exitflag,output] = fminsearch(...)
returns a structure output
that contains information about the optimization:
output.algorithm |
Algorithm used |
output.funcCount |
Number of function evaluations |
output.iterations |
Number of iterations |
output.message |
Exit message |
Arguments
fun
is the function to be minimized. It accepts an input x
and returns a scalar f
, the objective function evaluated at x
. The function fun
can be specified as a function handle for an M-file function
where myfun
is an M-file function such as
or as a function handle for an anonymous function, such as
Other arguments are described in the syntax descriptions above.
Examples
Example 1. A classic test example for multidimensional minimization is the Rosenbrock banana function
The minimum is at (1,1)
and has the value 0
. The traditional starting point is (-1.2,1)
. The anonymous function shown here defines the function and returns a function handle called banana
:
Pass the function handle to fminsearch
:
This indicates that the minimizer was found to at least four decimal places with a value near zero.
Example 2. If fun
is parameterized, you can use anonymous functions to capture the problem-dependent parameters. For example, suppose you want to minimize the objective function myfun
defined by the following M-file function.
Note that myfun
has an extra parameter a
, so you cannot pass it directly to fminsearch
. To optimize for a specific value of a
, such as a = 1.5
.
a
.
fminsearch
with a one-argument anonymous function that captures that value of a
and calls myfun
with two arguments:
Example 3. You can modify the first example by adding a parameter a to the second term of the banana function:
This changes the location of the minimum to the point [a,a^2]
. To minimize this function for a specific value of a
, for example a = sqrt(2)
, create a one-argument anonymous function that captures the value of a
.
seeks the minimum [sqrt(2), 2]
to an accuracy higher than the default on x
.
Algorithm
fminsearch uses the simplex search method of [1]. This is a direct search method that does not use numerical or analytic gradients.
If n
is the length of x
, a simplex in n
-dimensional space is characterized by the n+1
distinct vectors that are its vertices. In two-space, a simplex is a triangle; in three-space, it is a pyramid. At each step of the search, a new point in or near the current simplex is generated. The function value at the new point is compared with the function's values at the vertices of the simplex and, usually, one of the vertices is replaced by the new point, giving a new simplex. This step is repeated until the diameter of the simplex is less than the specified tolerance.
Limitations
fminsearch can often handle discontinuity, particularly if it does not occur near the solution. fminsearch may only give local solutions.
fminsearch only minimizes over the real numbers, that is, must only consist of real numbers and must only return real numbers. When has complex variables, they must be split into real and imaginary parts.
See Also
fminbnd
, optimset
, function_handle
(@), anonymous functions
References
[1] Lagarias, J.C., J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions," SIAM Journal of Optimization, Vol. 9 Number 1, pp. 112-147, 1998.
fminbnd | fopen |
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