fminsearch - Search for local minimum of unconstrained multivariable function using derivative-free

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Search for local minimum of unconstrained multivariable function using derivative-free method

Syntax

Description

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

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

For details, see Local vs. Global Minimum.

[x](#bvadxhn-1-x) = fminsearch([fun](#bvadxhn-1-fun),[x0](#bvadxhn-1-x0)) starts at the point x0 and searches for a local minimumx of the function described in fun.

example

[x](#bvadxhn-1-x) = fminsearch([fun](#bvadxhn-1-fun),[x0](#bvadxhn-1-x0),[options](#bvadxhn-1-options)) searches with the optimization options specified in the structureoptions. Use optimset to set these options.

example

[x](#bvadxhn-1-x) = fminsearch([problem](#bvadxhn-1-problem)) searches for a local minimum for problem, whereproblem is a structure.

[[x](#bvadxhn-1-x),[fval](#bvadxhn-1-fval)] = fminsearch(___), for any previous input syntax, returns in fval the value of the objective function fun at the solution x.

example

[[x](#bvadxhn-1-x),[fval](#bvadxhn-1-fval),[exitflag](#bvadxhn-1-exitflag)] = fminsearch(___) additionally returns a value exitflag that describes the exit condition.

[[x](#bvadxhn-1-x),[fval](#bvadxhn-1-fval),[exitflag](#bvadxhn-1-exitflag),[output](#bvadxhn-1-output)] = fminsearch(___) additionally returns a structure output with information about the optimization process.

example

Examples

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

f(x)=100(x2-x12)2+(1-x1)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)

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(x2-x12)2+(1-x1)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)

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(2x(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)

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(x2-x12)2+(a-x1)2.

This function has a minimum value of 0 at x1=a, x2=a2. 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.

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

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

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

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

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

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(2x(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 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 ↵'

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

fminsearch passesx to your objective function in the shape of thex0 argument. For example, if x0 is a 5-by-3 array, then fminsearch passesx 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

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

Optimization options, specified as a structure such asoptimset 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.

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

Data Types: struct

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

Data Types: struct

Output Arguments

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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.

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

Reason fminsearch stopped, returned as an integer.

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

More About

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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

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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.

Version History

Introduced before R2006a