Find minimum of unconstrained multivariable function using derivativefree method
Finds the minimum of a problem specified by
$$\underset{x}{\mathrm{min}}f(x)$$
where f(x) is a function that returns a scalar.
x is a vector or a matrix; see Matrix Arguments.
x = fminsearch(fun,x0)
x = fminsearch(fun,x0,options)
x = fminsearch(problem)
[x,fval] = fminsearch(...)
[x,fval,exitflag] = fminsearch(...)
[x,fval,exitflag,output] = fminsearch(...)
fminsearch
attempts to find a minimum of
a scalar function of several variables, starting at an initial estimate.
This is generally referred to as unconstrained nonlinear
optimization.
Note: Passing Extra Parameters explains how to pass extra parameters to the objective function, if necessary. 
x = fminsearch(fun,x0)
starts
at the point x0
and returns a value x
that
is a local minimizer of the function described in fun
. fun
is
either a function handle to a file or is an anonymous function. x0
can
be a scalar, vector, or matrix.
x = fminsearch(fun,x0,options)
minimizes
with the optimization options specified in the structure options
.
Use optimset
to set these options.
x = fminsearch(problem)
finds the minimum
for problem
, where problem
is
a structure described in Input Arguments.
Create the structure problem
by exporting
a problem from Optimization app, as described in Exporting Your Work.
[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.
[x,fval,exitflag,output] = fminsearch(...)
returns
a structure output
that contains information about
the optimization.
Function Arguments contains
general descriptions of arguments passed into fminsearch
.
This section provides functionspecific details for fun
, options
,
and problem
:
 The function to be
minimized. x = fminsearch(@myfun,x0) where function f = myfun(x) f = ... % Compute function value at x
x = fminsearch(@(x)norm(x)^2,x0,A,b);  
 Options provides the functionspecific details for the  
problem 
 Objective function 
 Initial point for x  
 'fminsearch'  
 Options structure created using optimset 
Function Arguments contains
general descriptions of arguments returned by fminsearch
.
This section provides functionspecific details for exitflag
and output
:
 Integer identifying the
reason the algorithm terminated. The following lists the values of  
 The function converged to a solution  
 Number of iterations exceeded  
 The algorithm was terminated by the output function.  
 Structure containing information about the optimization. The fields of the structure are  
iterations  Number of iterations  
funcCount  Number of function evaluations  
algorithm 
 
message  Exit message 
Optimization options used by fminsearch
.
You can use optimset
to set or
change the values of these fields in the options structure options
.
See Optimization Options Reference for detailed
information.
 Level of display:

FunValCheck  Check whether objective function values are valid. 
 Maximum number of function evaluations allowed, a positive
integer. The default is 
 Maximum number of iterations allowed, a positive integer.
The default value is 
OutputFcn  Specify one or more userdefined functions that an optimization
function calls at each iteration, either as a function handle or as
a cell array of function handles. The default is none ( 
 Plots various measures of progress while the algorithm
executes, select from predefined plots or write your own. Pass a function
handle or a cell array of function handles. The default is none (
For information on writing a custom plot function, see Plot Functions. 
 Termination tolerance on the function value, a positive
scalar. The default is 
 Termination tolerance on 
A classic test example for multidimensional minimization is
the Rosenbrock banana
function:
$$f(x)=100{\left({x}_{2}{x}_{1}^{2}\right)}^{2}+{(1{x}_{1})}^{2}.$$
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
:
banana = @(x)100*(x(2)x(1)^2)^2+(1x(1))^2;
Pass the function handle to fminsearch
:
[x,fval,exitflag] = fminsearch(banana,[1.2, 1])
This produces
x = 1.0000 1.0000 fval = 8.1777e010 exitflag = 1
This indicates that the minimizer was found at [1 1] with a value near zero.
You can modify the first example by adding a parameter a to the second term of the banana function:
$$f(x)=100{\left({x}_{2}{x}_{1}^{2}\right)}^{2}+{(a{x}_{1})}^{2}.$$
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 oneargument anonymous function that captures the value of a
.
a = sqrt(2); banana = @(x)100*(x(2)x(1)^2)^2+(ax(1))^2;
Then the statement
[x,fval,exitflag] = fminsearch(banana, [1.2, 1], ... optimset('TolX',1e8))
seeks the minimum [sqrt(2), 2]
to an accuracy
higher than the default on x
. The result is
x = 1.4142 2.0000 fval = 4.2065e018 exitflag = 1
fminsearch
solves nondifferentiable
problems and can often handle discontinuity, particularly if it does
not occur near the solution. fminsearch
might only
give local solutions.
fminsearch
only minimizes over the real numbers,
that is, x must only consist of real numbers and f(x)
must only return real numbers. When x has complex
variables, they must be split into real and imaginary parts.
fminsearch
is not the preferred choice for
solving problems that are sums of squares, that is, of the form
$$\underset{x}{\mathrm{min}}{\Vert f(x)\Vert}_{2}^{2}=\underset{x}{\mathrm{min}}\left({f}_{1}{(x)}^{2}+{f}_{2}{(x)}^{2}+\mathrm{...}+{f}_{n}{(x)}^{2}\right)$$
Instead use the lsqnonlin
function,
which has been optimized for problems of this form.
[1] Lagarias, J. C., J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence Properties of the NelderMead Simplex Method in Low Dimensions," SIAM Journal of Optimization, Vol. 9, Number 1, pp. 112–147, 1998.