fminbnd

Find local minimum of single-variable function on fixed interval

Syntax

x = fminbnd(fun,x1,x2)

x = fminbnd(fun,x1,x2,options)

x = fminbnd(problem)

[x,fval] = fminbnd(___)

[x,fval,exitflag] = fminbnd(___)

[x,fval,exitflag,output] = fminbnd(___)

Description

fminbnd is a one-dimensional minimizer that finds a local minimum for a problem specified by

$$\underset{x}{\min }f(x)\text{ such that }{x}_1<x<x_2.$$

x, x₁, and x₂ are finite scalars, and f(x) is a function that returns a scalar.

If multiple local minima exist on the interval (x1,x2), fminbnd returns only one, which is not guaranteed to be the global minimum. For details, see Local vs. Global Minimum.

x = fminbnd(fun,x1,x2) returns a value x that is a local minimizer of the scalar valued function that is described in fun in the interval x1 < x < x2.

x = fminbnd(fun,x1,x2,options) minimizes with the optimization options specified in options. Use optimset to set these options.

x = fminbnd(problem) minimizes problem, where problem is a structure.

[x,fval] = fminbnd(___), for any input arguments, returns the value of the objective function computed in fun at the solution x.

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

[x,fval,exitflag,output] = fminbnd(___) additionally returns a structure output that contains information about the optimization.

Examples

Minimum of sin

Find the point where the $$\sin (x)$$ function takes its minimum in the range $$0<x<2\pi$$.

fun = @sin;
x1 = 0;
x2 = 2*pi;
x = fminbnd(fun,x1,x2)

x = 
4.7124

To display precision, this is the same as the correct value $$x=3\pi /2$$.

3*pi/2

ans = 
4.7124

Minimize a Function Specified by a File

Minimize a function that is specified by a separate function file. A function accepts a point x and returns a real scalar representing the value of the objective function at x.

Write the following function as a file, and save the file as scalarobjective.m on your MATLAB® path.

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

Find the x that minimizes scalarobjective on the interval 1 <= x <= 3.

x = fminbnd(@scalarobjective,1,3)

x =

    2.0061

Minimize with Extra Parameter

Minimize a function when there is an extra parameter. The function $$\sin (x-a)$$ has a minimum that depends on the value of the parameter $$a$$. Create an anonymous function of $$x$$ that includes the value of the parameter $$a$$. Minimize this function over the interval $$0<x<2\pi$$.

a = 9/7;
fun = @(x)sin(x-a);
x = fminbnd(fun,1,2*pi)

x = 
5.9981

This answer is correct; the theoretical value is

3*pi/2 + 9/7

ans = 
5.9981

For more information about including extra parameters, see Parameterizing Functions.

Monitor Iterations

Monitor the steps fminbnd takes to minimize the $$\sin (x)$$ function for $$0<x<2\pi$$.

fun = @sin;
x1 = 0;
x2 = 2*pi;
options = optimset('Display','iter');
x = fminbnd(fun,x1,x2,options)

 
 Func-count     x          f(x)         Procedure
    1        2.39996      0.67549        initial
    2        3.88322     -0.67549        golden
    3        4.79993    -0.996171        golden
    4        5.08984    -0.929607        parabolic
    5        4.70582    -0.999978        parabolic
    6         4.7118           -1        parabolic
    7        4.71239           -1        parabolic
    8        4.71236           -1        parabolic
    9        4.71242           -1        parabolic
 
Optimization terminated:
 the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-04 

x = 
4.7124

Find Minimum Location and Function Value

Find the location of the minimum of $$\sin (x)$$ and the value of the minimum for $$0<x<2\pi$$.

fun = @sin;
[x,fval] = fminbnd(fun,1,2*pi)

x = 
4.7124

fval = 
-1.0000

Obtain All Information

Return all information about the fminbnd solution process by requesting all outputs. Also, monitor the solution process using a plot function.

fun = @sin;
x1 = 0;
x2 = 2*pi;
options = optimset('PlotFcns',@optimplotfval);
[x,fval,exitflag,output] = fminbnd(fun,x1,x2,options)

x = 
4.7124

fval = 
-1.0000

exitflag = 
1

output = struct with fields:
    iterations: 8
     funcCount: 9
     algorithm: 'golden section search, parabolic interpolation'
       message: 'Optimization terminated:...'

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 real scalar x and returns a real scalar f (the objective function evaluated at x).

Specify fun as a function handle for a file:

x = fminbnd(@myfun,x1,x2)

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 = fminbnd(@(x)norm(x)^2,x1,x2);

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

Data Types: char | function_handle | string

x1 — Lower bound

finite real scalar

Lower bound, specified as a finite real scalar.

Example: x1 = -3

Data Types: double

x2 — Upper bound

finite real scalar

Upper bound, specified as a finite real scalar.

Example: x2 = 5

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. 'off' or 'none' displays no output. 'iter' displays output at each iteration. 'final' displays just the final output.
FunValCheck	Check whether objective function values are valid. The default 'off' allows fminbnd to proceed when the objective function returns a value that is complex or NaN. The 'on' setting throws an error when the objective function returns a value that is complex or NaN.
MaxFunEvals	Maximum number of function evaluations allowed, a positive integer. The default is 500. See Tolerances and Stopping Criteria.
MaxIter	Maximum number of iterations allowed, a positive integer. The default is 500. 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.
TolX	Termination tolerance on x, a positive scalar. The default is 1e-4. See Tolerances and Stopping Criteria.

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
x1	Left endpoint
x2	Right endpoint
solver	'fminbnd'
options	Options structure such as returned by optimset

Data Types: struct

Output Arguments

x — Solution

real scalar

Solution, returned as a real scalar. Typically, x is a local solution to the problem when exitflag is positive. See Local vs. Global 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 fminbnd stopped

integer

Reason fminbnd stopped, returned as an integer.

1	Function converged to a solution x.
0	Number of iterations exceeded options.MaxIter or number of function evaluations exceeded options.MaxFunEvals.
-1	Stopped by an output function or plot function.
-2	The bounds are inconsistent, meaning x1 > x2.

output — Information about the optimization process

structure

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

iterations	Number of iterations taken
funcCount	Number of function evaluations
algorithm	'golden section search, parabolic interpolation'
message	Exit message

Limitations

• The function to be minimized must be continuous.

• fminbnd might only give local solutions.

• fminbnd can exhibit slow convergence when the solution is on a boundary of the interval.

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).

Algorithms

fminbnd is a function file. The algorithm is based on golden section search and parabolic interpolation. Unless the left endpoint x₁ is very close to the right endpoint x₂, fminbnd never evaluates fun at the endpoints, so fun need only be defined for x in the interval x₁ < x < x₂.

If the minimum actually occurs at x₁ or x₂, fminbnd returns a point x in the interior of the interval (x₁,x₂) that is close to the minimizer. In this case, the distance of x from the minimizer is no more than 2*(TolX + 3*abs(x)*sqrt(eps)). See [1] or [2] for details about the algorithm.

Alternative Functionality

App

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

References

[1] Forsythe, G. E., M. A. Malcolm, and C. B. Moler. Computer Methods for Mathematical Computations. Englewood Cliffs, NJ: Prentice Hall, 1976.

[2] Brent, Richard. P. Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ: Prentice-Hall, 1973.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

For C/C++ code generation:

• fminbnd does not support the problem structure argument.

• fminbnd 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.

• fminbnd 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