fminsearch

Search for local 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 a local minimum of a problem specified by

$$\underset{x}{\min }f(x)$$

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

For details, see Local vs. Global Minimum.

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

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

x = fminsearch(problem) searches for a local minimum for problem, where problem is a structure.

[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.

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

Examples

Minimize Rosenbrock's Function

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

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

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

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

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

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         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

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).

fminsearch passes x to your objective function in the shape of the x0 argument. For example, if x0 is a 5-by-3 array, then fminsearch passes x to fun as a 5-by-3 array.

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 name, function handle, or a cell array of function names or 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

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 an approximate local solution to the problem when exitflag is positive. See Local vs. Global Minimum. However, as stated in Algorithms, the solution x is not guaranteed to be a local minimum.

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

More About

Local vs. Global Minimum

In general, optimization solvers return a local minimum (or optimum). The result might be a global minimum (or optimum), but might not.

• A local minimum of a function is a point where the function value is smaller than at nearby points, but possibly greater than at a distant point.

• A global minimum is a point where the function value is smaller than at all other feasible points.

MATLAB and Optimization Toolbox™ optimization solvers typically return a local minimum. Global Optimization Toolbox solvers can search for a global minimum, but do not guarantee that their solutions are global. For an example of global search, see Find Global or Multiple Local Minima (Global Optimization Toolbox).

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.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a