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# fminsearch

Find minimum of unconstrained multivariable function using derivative-free method

## Equation

Finds the minimum of a problem specified by

$\underset{x}{\mathrm{min}}f\left(x\right)$

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

## Syntax

x = fminsearch(fun,x0)
x = fminsearch(fun,x0,options)
x = fminsearch(problem)
[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) starts at the point x0 and returns a value x that is a local minimizer of the function described in fun. x0 can be a scalar, vector, or matrix. fun is a function handle; see Function Handles.

Parameterizing Functions in the MATLAB® Mathematics documentation explains how to pass additional parameters to your objective function fun. See also Example 2 and Example 3 below.

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; 'notify' (default) displays output only if the function does not converge. See Iterative Display in MATLAB Mathematics for more information. FunValCheck Check whether objective function values are valid. 'on' displays an error when the objective function returns a value that is complex or NaN. 'off' (the default) 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 user-defined 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 Functions in MATLAB Mathematics for more information. 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 ([]). @optimplotx plots the current point@optimplotfval plots the function value@optimplotfunccount plots the function countSee Plot Functions in MATLAB Mathematics for more information. TolFun Termination tolerance on the function value, a positive scalar. The default is 1e-4. See Tolerances and Stopping Criteria. Unlike other solvers, fminsearch stops when it satisfies both TolFun and TolX. TolX Termination tolerance on x, a positive scalar. The default value is 1e-4. See Tolerances and Stopping Criteria. Unlike other solvers, fminsearch stops when it satisfies both TolFun and TolX.

x = fminsearch(problem) finds the minimum for problem, where problem is a structure with the following fields:

 objective Objective function x0 Initial point for x solver 'fminsearch' options Options structure created using optimset

[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 in the following fields:

 algorithm 'Nelder-Mead simplex direct search' funcCount Number of function evaluations iterations Number of iterations 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 a function file

x = fminsearch(@myfun, x0)

where myfun is a function file such as

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

or as a function handle for an anonymous function, such as

x = fminsearch(@(x)sin(x^2), x0);

Other arguments are described in the syntax descriptions above.

## Examples

### Example 1

The Rosenbrock banana function is a classic test example for multidimensional minimization:

$f\left(x\right)=100{\left({x}_{2}-{x}_{1}^{2}\right)}^{2}+{\left(1-{x}_{1}\right)}^{2}.$

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] = fminsearch(banana,[-1.2, 1])

This 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.

### 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 function file:

function f = myfun(x,a)
f = x(1)^2 + a*x(2)^2;

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.

1. Assign the value to a.

a = 1.5; % define parameter first
2. Call fminsearch with a one-argument anonymous function that captures that value of a and calls myfun with two arguments:

x = fminsearch(@(x) myfun(x,a),[0,1])

### Example 3

You can modify the first example by adding a parameter a to the second term of the banana function:

$f\left(x\right)=100{\left({x}_{2}-{x}_{1}^{2}\right)}^{2}+{\left(a-{x}_{1}\right)}^{2}.$

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

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

## 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.

## More About

collapse all

### Algorithms

fminsearch uses the simplex search method of Lagarias et al. [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.

For more information, see fminsearch Algorithm.

## 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.

## See Also

#### Introduced before R2006a

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