Documentation 
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:
fun  The function to be minimized. fun is a function handle for a function that accepts a vector x and returns a scalar f, the objective function evaluated at x. The function fun can be specified as a function handle for a file: x = fminsearch(@myfun,x0) where myfun is a MATLAB^{®} function such as function f = myfun(x) f = ... % Compute function value at x fun can also be a function handle for an anonymous function, such as x = fminsearch(@(x)norm(x)^2,x0,A,b);  
options  Options provides the functionspecific details for the options values.  
problem  objective  Objective function 
x0  Initial point for x  
solver  'fminsearch'  
options  Options structure created using optimset 
Function Arguments contains general descriptions of arguments returned by fminsearch. This section provides functionspecific details for exitflag and output:
exitflag  Integer identifying the reason the algorithm terminated. The following lists the values of exitflag and the corresponding reasons the algorithm terminated.  
1  The function converged to a solution x.  
0  Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.MaxFunEvals.  
1  The algorithm was terminated by the output function.  
output  Structure containing information about the optimization. The fields of the structure are  
iterations  Number of iterations  
funcCount  Number of function evaluations  
algorithm  'NelderMead simplex direct search'  
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.
Display  Level of display:

FunValCheck  Check whether objective function values are valid. 'on' displays an error when the objective function returns a value that is complex, Inf, or NaN. The default 'off' displays no error. 
MaxFunEvals  Maximum number of function evaluations allowed, a positive integer. The default is 200*numberOfVariables. 
MaxIter  Maximum number of iterations allowed, a positive integer. The default value is 200*numberOfVariables. 
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 ([]). See Output Function. 
PlotFcns  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. 
TolFun  Termination tolerance on the function value, a positive scalar. The default is 1e4. 
TolX  Termination tolerance on x, a positive scalar. The default value is 1e4. 
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.