Equation

Finds the minimum of a problem specified by

$$\underset{x}{\min }f(x)$$

where f(x) is a function that returns a scalar.

x is a vector or a matrix; see Matrix Arguments.

Syntax

Description

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.

Input Arguments

Function Arguments contains general descriptions of arguments passed into fminsearch. This section provides function-specific details for fun, options, and problem:

x = fminsearch(@myfun,x0)

function f = myfun(x)
f = ...            % Compute function value at x

x = fminsearch(@(x)norm(x)^2,x0,A,b);

Output Arguments

Function Arguments contains general descriptions of arguments returned by fminsearch. This section provides function-specific details for exitflag and output:

Options

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.

Examples

Example 1: Minimizing Rosenbrock's Function with fminsearch

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+(1-x(1))^2;

Pass the function handle to fminsearch:

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

This produces

x =
    1.0000    1.0000

fval =
    8.1777e-010

exitflag =
     1

This indicates that the minimizer was found at [1 1] with a value near zero.

Example 2

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 one-argument anonymous function that captures the value of a.

a = sqrt(2);
banana = @(x)100*(x(2)-x(1)^2)^2+(a-x(1))^2;

Then the statement

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

seeks the minimum [sqrt(2), 2] to an accuracy higher than the default on x. The result is

x =
    1.4142    2.0000

fval =
  4.2065e-018

exitflag =
     1

Limitations

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.

Notes

fminsearch is not the preferred choice for solving problems that are sums of squares, that is, of the form

$$\underset{x}{\min }{\Vert f(x)\Vert }_2^2=\underset{x}{\min }\left(f_1{(x)}^2+f_2{(x)}^2+\mathrm{...}+f_n{(x)}^2\right)$$

Instead use the lsqnonlin function, which has been optimized for problems of this form.

References

See Also

fminbnd | fminunc | optimset | optimtool