MATLAB Function Reference |

Minimize a function of several variables

**Syntax**

x = fminsearch(fun,x0) x = fminsearch(fun,x0,options) [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*.

starts at the point `x = `

fminsearch```
(fun,x0)
```

`x0`

and finds a local minimum `x`

of the function described in `fun`

. `x0`

can be a scalar, vector, or matrix. `fun`

is a function handle. See Function Handles in the MATLAB Programming documentation for more information.

Parameterizing Functions Called by Function Functions, in the MATLAB mathematics documentation, explains how to provide additional parameters to the function `fun`

, if necessary.

minimizes with the optimization parameters specified in the structure `x = `

fminsearch```
(fun,x0,options)
```

`options`

. You can define these parameters using the `optimset`

function. `fminsearch`

uses these `options`

structure fields:

```
[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:

`output.algorithm` |
Algorithm used |

`output.funcCount` |
Number of function evaluations |

`output.iterations` |
Number of iterations |

`output.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 an M-file function

where `myfun`

is an M-file function such as

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

Other arguments are described in the syntax descriptions above.

**Examples**

**Example 1.** 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 anonymous function shown here defines the function and returns a function handle called `banana`

:

Pass the function handle to `fminsearch`

:

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 M-file function.

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`

.

- Assign the value to
`a`

. - Call
`fminsearch`

with a one-argument anonymous function that captures that value of`a`

and calls`myfun`

with two arguments:

**Example 3.** You can modify the first example by adding a parameter *a* to the second term of the banana function:

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`

.

seeks the minimum `[sqrt(2), 2]`

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, must only consist of real numbers and must only return real numbers. When* * has complex variables, they must be split into real and imaginary parts.

**See Also**

`fminbnd`

, `optimset`

, `function_handle`

(@), anonymous functions

**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.

fminbnd | fopen |

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