# 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; see Matrix Arguments.

## 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` 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`:

 `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 function-specific details for the `options` values. `problem` `objective` Objective function `x0` Initial point for `x` `solver` `'fminsearch'` `options` Options structure created using `optimset`

## Output Arguments

Function Arguments contains general descriptions of arguments returned by `fminsearch`. This section provides function-specific 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` `'Nelder-Mead simplex direct search'` `message` Exit message

## 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.

 `Display` Level of display:`'off'` or `'none'` displays no output.`'iter'` displays output at each iteration.`'notify'` displays output only if the function does not converge.`'final'` (default) displays just the final output. `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 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 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 (`[]`):`@optimplotx` plots the current point.`@optimplotfunccount` plots the function count.`@optimplotfval` plots the function value.For information on writing a custom plot function, see Plot Functions. `TolFun` Termination tolerance on the function value, a positive scalar. The default is `1e-4`. `TolX` Termination tolerance on `x`, a positive scalar. The default value is `1e-4`.

## Examples

### Example 1: Minimizing Rosenbrock's Function with fminsearch

A classic test example for multidimensional minimization is the Rosenbrock `banana` function:

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

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,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:

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

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,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

$\underset{x}{\mathrm{min}}{‖f\left(x\right)‖}_{2}^{2}=\underset{x}{\mathrm{min}}\left({f}_{1}{\left(x\right)}^{2}+{f}_{2}{\left(x\right)}^{2}+...+{f}_{n}{\left(x\right)}^{2}\right)$

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

## More About

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### Algorithms

`fminsearch` uses the simplex search method of [1]. This is a direct search method that does not use numerical or analytic gradients as in `fminunc`. The algorithm is described in detail in fminsearch Algorithm.

`fminsearch` is generally less efficient than `fminunc` for problems of order greater than two. However, when the problem is highly discontinuous, `fminsearch` might be more robust.

## 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.

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