fminsearch

Find minimum of unconstrained multivariable function using derivative-free method

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

Nonlinear programming solver. Searches for the minimum of a problem specified by

$\min_{x} f (x)$

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

example

x = fminsearch(fun,x0) starts at the point x0 and attempts to find a local minimum x of the function described in fun.

example

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.

example

[x,fval] = fminsearch(___), for any previous input syntax, returns in fval the value of the objective function fun at the solution x.

[x,fval,exitflag] = fminsearch(___) additionally returns a value exitflag that describes the exit condition.

example

[x,fval,exitflag,output] = fminsearch(___) additionally returns a structure output with information about the optimization process.

Examples

collapse all

Minimize Rosenbrock's Function

Open Live Script

Minimize Rosenbrock's function, a notoriously difficult optimization problem for many algorithms:

$f (x) = 100 (x_{2} - x_{1}^{2})^{2} + (1 - x_{1})^{2} .$

The function is minimized at the point x = [1,1] with minimum value 0.

Set the start point to x0 = [-1.2,1] and minimize Rosenbrock's function using fminsearch.

fun = @(x)100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;
x0 = [-1.2,1];
x = fminsearch(fun,x0)

x = 1×2

    1.0000    1.0000

Monitor Optimization Process

Open Live Script

Set options to monitor the process as fminsearch attempts to locate a minimum.

Set options to plot the objective function at each iteration.

options = optimset('PlotFcns',@optimplotfval);

Set the objective function to Rosenbrock's function,

$f (x) = 100 (x_{2} - x_{1}^{2})^{2} + (1 - x_{1})^{2} .$

The function is minimized at the point x = [1,1] with minimum value 0.

Set the start point to x0 = [-1.2,1] and minimize Rosenbrock's function using fminsearch.

fun = @(x)100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;
x0 = [-1.2,1];
x = fminsearch(fun,x0,options)

x = 1×2

    1.0000    1.0000

Minimize a Function Specified by a File

Open Script

Minimize an objective function whose values are given by executing a file. A function file must accept a real vector x and return a real scalar that is the value of the objective function.

Copy the following code and include it as a file named objectivefcn1.m on your MATLAB® path.

function f = objectivefcn1(x)
f = 0;
for k = -10:10
    f = f + exp(-(x(1)-x(2))^2 - 2*x(1)^2)*cos(x(2))*sin(2*x(2));
end

Start at x0 = [0.25,-0.25] and search for a minimum of objectivefcn.

x0 = [0.25,-0.25];
x = fminsearch(@objectivefcn1,x0)

x =

   -0.1696   -0.5086

Minimize with Extra Parameters

Open Live Script

Sometimes your objective function has extra parameters. These parameters are not variables to optimize, they are fixed values during the optimization. For example, suppose that you have a parameter a in the Rosenbrock-type function

$f (x, a) = 100 (x_{2} - x_{1}^{2})^{2} + (a - x_{1})^{2} .$

This function has a minimum value of 0 at $x_{1} = a$ , $x_{2} = a^{2}$ . If, for example, $a = 3$ , you can include the parameter in your objective function by creating an anonymous function.

Create the objective function with its extra parameters as extra arguments.

f = @(x,a)100*(x(2) - x(1)^2)^2 + (a-x(1))^2;

Put the parameter in your MATLAB® workspace.

a = 3;

Create an anonymous function of x alone that includes the workspace value of the parameter.

fun = @(x)f(x,a);

Solve the problem starting at x0 = [-1,1.9].

x0 = [-1,1.9];
x = fminsearch(fun,x0)

x = 1×2

    3.0000    9.0000

For more information about using extra parameters in your objective function, see Parameterizing Functions.

Find Minimum Location and Value

Open Live Script

Find both the location and value of a minimum of an objective function using fminsearch.

Write an anonymous objective function for a three-variable problem.

x0 = [1,2,3];
fun = @(x)-norm(x+x0)^2*exp(-norm(x-x0)^2 + sum(x));

Find the minimum of fun starting at x0. Find the value of the minimum as well.

[x,fval] = fminsearch(fun,x0)

x = 1×3

    1.5359    2.5645    3.5932

fval = -5.9565e+04

Inspect Optimization Process

Open Script

Inspect the results of an optimization, both while it is running and after it finishes.

Set options to provide iterative display, which gives information on the optimization as the solver runs. Also, set a plot function to show the objective function value as the solver runs.

options = optimset('Display','iter','PlotFcns',@optimplotfval);

Set an objective function and start point.

function f = objectivefcn1(x)
f = 0;
for k = -10:10
    f = f + exp(-(x(1)-x(2))^2 - 2*x(1)^2)*cos(x(2))*sin(2*x(2));
end

Include the code for objectivefcn1 as a file on your MATLAB® path.

x0 = [0.25,-0.25];
fun = @objectivefcn1;

Obtain all solver outputs. Use these outputs to inspect the results after the solver finishes.

[x,fval,exitflag,output] = fminsearch(fun,x0,options)

 
 Iteration   Func-count     min f(x)         Procedure
     0            1         -6.70447         
     1            3         -6.89837         initial simplex
     2            5         -7.34101         expand
     3            7         -7.91894         expand
     4            9         -9.07939         expand
     5           11         -10.5047         expand
     6           13         -12.4957         expand
     7           15         -12.6957         reflect
     8           17         -12.8052         contract outside
     9           19         -12.8052         contract inside
    10           21         -13.0189         expand
    11           23         -13.0189         contract inside
    12           25         -13.0374         reflect
    13           27          -13.122         reflect
    14           28          -13.122         reflect
    15           29          -13.122         reflect
    16           31          -13.122         contract outside
    17           33         -13.1279         contract inside
    18           35         -13.1279         contract inside
    19           37         -13.1296         contract inside
    20           39         -13.1301         contract inside
    21           41         -13.1305         reflect
    22           43         -13.1306         contract inside
    23           45         -13.1309         contract inside
    24           47         -13.1309         contract inside
    25           49          -13.131         reflect
    26           51          -13.131         contract inside
    27           53          -13.131         contract inside
    28           55          -13.131         contract inside
    29           57          -13.131         contract outside
    30           59          -13.131         contract inside
    31           61          -13.131         contract inside
    32           63          -13.131         contract inside
    33           65          -13.131         contract outside
    34           67          -13.131         contract inside
    35           69          -13.131         contract inside
 
Optimization terminated:
 the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-04 
 and F(X) satisfies the convergence criteria using OPTIONS.TolFun of 1.000000e-04 


x =

   -0.1696   -0.5086


fval =

  -13.1310


exitflag =

     1


output = 

  struct with fields:

    iterations: 35
     funcCount: 69
     algorithm: 'Nelder-Mead simplex direct search'
       message: 'Optimization terminated:...'

The value of exitflag is 1, meaning fminsearch likely converged to a local minimum.

The output structure shows the number of iterations. The iterative display and the plot show this information as well. The output structure also shows the number of function evaluations, which the iterative display shows, but the chosen plot function does not.

Input Arguments

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`fun` — Function to minimize
function handle | function name

Function to minimize, specified as a function handle or function name. fun is a function that accepts a vector or array x and returns a real scalar f (the objective function evaluated at x).

Specify fun 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

You can also specify fun as a function handle for an anonymous function:

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

Example: fun = @(x)-x*exp(-3*x)

Data Types: char | function_handle | string

`x0` — Initial point
real vector | real array

Initial point, specified as a real vector or real array. Solvers use the number of elements in, and size of, x0 to determine the number and size of variables that fun accepts.

Example: x0 = [1,2,3,4]

Data Types: double

`options` — Optimization options
structure such as `optimset` returns

Optimization options, specified as a structure such as optimset returns. You can use optimset to set or change the values of these fields in the options structure. See Set Optimization Options for detailed information.

`Display`	Level of display (see Optimization Solver Iterative Display): `'notify'` (default) displays output only if the function does not converge. `'final'` displays just the final output. `'off'` or `'none'` displays no output. `'iter'` displays output at each iteration.
`FunValCheck`	Check whether objective function values are valid. `'on'` displays an error when the objective function returns a value that is `complex` or `NaN`. The default `'off'` displays no error.
`MaxFunEvals`	Maximum number of function evaluations allowed, a positive integer. The default is `200*numberOfVariables`. See Tolerances and Stopping Criteria.
`MaxIter`	Maximum number of iterations allowed, a positive integer. The default value is `200*numberOfVariables`. See Tolerances and Stopping Criteria.
`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 Optimization Solver Output Functions.
`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 Optimization Solver Plot Functions.
`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`.

Example: options = optimset('Display','iter')

Data Types: struct

`problem` — Problem structure
structure

Problem structure, specified as a structure with the following fields.

Field Name	Entry
`objective`	Objective function
`x0`	Initial point for `x`
`solver`	`'fminsearch'`
`options`	Options structure such as returned by `optimset`

Data Types: struct

Output Arguments

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`x` — Solution
real vector | real array

Solution, returned as a real vector or real array. The size of x is the same as the size of x0. Typically, x is a local solution to the problem when exitflag is positive.

`fval` — Objective function value at solution
real number

Objective function value at the solution, returned as a real number. Generally, fval = fun(x).

`exitflag` — Reason `fminsearch` stopped
integer

Reason fminsearch stopped, returned as an integer.

`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` — Information about the optimization process
structure

Information about the optimization process, returned as a structure with fields:

`iterations`	Number of iterations
`funcCount`	Number of function evaluations
`algorithm`	`'Nelder-Mead simplex direct search'`
`message`	Exit message

Tips

fminsearch only minimizes over the real numbers, that is, the vector or array x must only consist of real numbers and f(x) must only return real numbers. When x has complex values, split x into real and imaginary parts.
Use fminsearch to solve nondifferentiable problems or problems with discontinuities, particularly if no discontinuity occurs near the solution.

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 as in fminunc (Optimization Toolbox). The algorithm is described in detail in fminsearch Algorithm. The algorithm is not guaranteed to converge to a local minimum.

Alternative Functionality

App

The Optimize Live Editor task provides a visual interface for fminsearch.

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, 1998, pp. 112–147.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

For C/C++ code generation:

fminsearch ignores the Display option and does not give iterative display or an exit message. To check solution quality, examine the exit flag.
The output structure does not include the algorithm or message fields.
fminsearch ignores the OutputFcn and PlotFcns options.

Documentation

fminsearch

Syntax

Description

Examples

Minimize Rosenbrock's Function

Monitor Optimization Process

Minimize a Function Specified by a File

Minimize with Extra Parameters

Find Minimum Location and Value

Inspect Optimization Process

Input Arguments

`fun` — Function to minimize
function handle | function name

`x0` — Initial point
real vector | real array

`options` — Optimization options
structure such as `optimset` returns

`problem` — Problem structure
structure

Output Arguments

`x` — Solution
real vector | real array

`fval` — Objective function value at solution
real number

`exitflag` — Reason `fminsearch` stopped
integer

`output` — Information about the optimization process
structure

Tips

Algorithms

Alternative Functionality

App

References

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

See Also

Topics

MATLAB Documentation

Support

Documentation

fminsearch

Syntax

Description

Examples

Minimize Rosenbrock's Function

Monitor Optimization Process

Minimize a Function Specified by a File

Minimize with Extra Parameters

Find Minimum Location and Value

Inspect Optimization Process

Input Arguments

fun — Function to minimize function handle | function name

x0 — Initial point real vector | real array

options — Optimization options structure such as optimset returns

problem — Problem structure structure

Output Arguments

x — Solution real vector | real array

fval — Objective function value at solution real number

exitflag — Reason fminsearch stopped integer

output — Information about the optimization process structure

Tips

Algorithms

Alternative Functionality

App

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

See Also

Topics

MATLAB Documentation

Support

`fun` — Function to minimize
function handle | function name

`x0` — Initial point
real vector | real array

`options` — Optimization options
structure such as `optimset` returns

`problem` — Problem structure
structure

`x` — Solution
real vector | real array

`fval` — Objective function value at solution
real number

`exitflag` — Reason `fminsearch` stopped
integer

`output` — Information about the optimization process
structure

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.