fminsearch (MATLAB Function Reference)

Minimize a function of several variables

Syntax

x = fminsearch(fun,x0)
x = fminsearch(fun,x0,options)
x = fminsearch(fun,x0,options,P1,P2,...)
[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.

x = fminsearch(fun,x0)

returns a vector x that is a local minimizer of the function described in fun (usually an M-file, built-in function or an inline object) near the starting vector x0. fun should return a scalar function value f evaluated at x when called with feval: f=feval(fun,x).

x = fminsearch(fun,x0,options)

minimizes with the optimization parameters specified in the structure options. You can define these parameters using the optimset function. fminsearch uses these options structure fields:

Display - Level of display. off displays no output; iter displays output at each iteration; final displays just the final output.
MaxFunEvals - Maximum number of function evaluations allowed.
MaxIter - Maximum number of iterations allowed.
TolFun - Termination tolerance on the function value.
TolX - Termination tolerance on x.

x = fminsearch(fun,x0,options,P1,P2,...)

passes the problem-dependent parameters P1, P2, etc., directly to the function fun: feval(fun,x,P1,P2,...). Pass an empty matrix for options to use the default values.

[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:

> 0 indicates that the function converged to a solution x.
0 indicates that the maximum number of function evaluations was reached.
< 0 indicates that the function did not converge to a solution.

[x,fval,exitflag,output] = fminsearch(...)

returns a structure output that contains information about the optimization:

output.algorithm - The algorithm used.
output.funcCount - The number of function evaluations.
output.iterations - The number of iterations taken.

Arguments

fun is a string containing the name of the function that computes the objective function to be minimized at the point x. The function returns one argument, a scalar valued function f to be minimized, given a vector x. For example, if fun='fun', the first line of the M-file fun.m is

f = fun(x)

fun can also be the name of a built-in function such as fun='norm'.(Note that norm takes a vector and returns a scalar.)

Alternatively, you can specify an inline object. For example,

fun = inline('sin(x''*x)');

Other arguments are described in the syntax descriptions above.

Examples

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 M-file banana.m defines the function.

function f = banana(x)
f = 100*(x(2)-x(1)^2)^2+(1-x(1))^2;

The statement

[x,fval] = fminsearch('banana',[-1.2, 1])

produces

x =

    1.0000    1.0000

fval =

    8.1777e-010

This indicates that the minimizer was found to at least four decimal places with a value near zero.

Move the location of the minimum to the point [a,a^2] by adding a second parameter to banana.m.

function f = banana(x,a)
if nargin < 2, a = 1; end
f = 100*(x(2)-x(1)^2)^2+(a-x(1))^2;

Then the statement

[x,fval] = fminsearch('banana', [-1.2, 1], ...
   optimset('TolX',1e-8), sqrt(2));

sets the new parameter to sqrt(2) and seeks the minimum 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, 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.