fminsearch (Optimization Toolbox)

Find a minimum of an unconstrained multivariable function

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

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 a 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 finds a local minimum x of the function described in fun. x0 can be a scalar, vector, or matrix.

x = fminsearch(fun,x0,options) minimizes with the optimization parameters specified in the structure options. Use optimset to set these parameters.

x = fminsearch(fun,x0,options,P1,P2,...) passes the problem-dependent parameters P1, P2, etc., directly to the function fun. Use options = [] as a placeholder if no options are set.

[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 in to fminsearch. This section provides function-specific details for fun and options:

fun
The function to be minimized. fun is 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.
x = fminsearch(@myfun,x0,A,b)
where myfun is a MATLAB function such as
function f = myfun(x) f = ... % Compute function value at x
fun can also be an inline object.
x = fminsearch(inline('norm(x)^2'),x0,A,b);

options
Options provides the function-specific details for the options parameters.

`fun`	The function to be minimized. `fun` is 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. x = fminsearch(@myfun,x0,A,b) where `myfun` is a MATLAB function such as function f = myfun(x) f = ... % Compute function value at x `fun` can also be an inline object. x = fminsearch(inline('norm(x)^2'),x0,A,b);
`options`	Options provides the function-specific details for the `options` parameters.

Output Arguments

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

exitflag
Describes the exit condition:

> 0
The function converged to a solution x.

0
The maximum number of function evaluations or iterations was exceeded.

< 0
The function did not converge to a solution.

output
Structure containing information about the optimization. The fields of the structure are:

iterations
Number of iterations taken.

funcCount
Number of function evaluations.

algorithm
Algorithm used.

`exitflag`	Describes the exit condition:
	`> 0`	The function converged to a solution `x`.
	`0`	The maximum number of function evaluations or iterations was exceeded.
	`< 0`	The function did not converge to a solution.
`output`	Structure containing information about the optimization. The fields of the structure are:
	`iterations`	Number of iterations taken.
	`funcCount`	Number of function evaluations.
	`algorithm`	Algorithm used.

Options

Optimization options parameters used by fminsearch. You can use optimset to set or change the values of these fields in the parameters structure, options. See Optimization Parameters, for detailed information:

Display
Level of display. 'off' displays no output; 'iter' displays output at each iteration; 'final' displays just the final output; 'notify' (default) dislays output only if the function does not converge.

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.

`Display`	Level of display. `'off'` displays no output; `'iter'` displays output at each iteration; `'final'` displays just the final output; `'notify'` (default) dislays output only if the function does not converge.
`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`.

Examples

Minimize the one-dimensional function f(x) = sin(x) + 3.

To use an M-file, i.e., fun = 'myfun', create a file myfun.m.

```
function f = myfun(x)
f = sin(x) + 3;
```

Then call fminsearch to find a minimum of fun near 2.

```
x = fminsearch(@myfun,2)
```

To minimize the function f(x) = sin(x) + 3 using an inline object

f = inline('sin(x)+3');
x = fminsearch(f,2);

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.

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.

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

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.

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

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

See Also

@ (function_handle), fminbnd, fminunc, inline, optimset

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.

fminimax fminunc