simulannealbnd

Find minimum of function using simulated annealing algorithm

Syntax

x = simulannealbnd(fun,x0)

x = simulannealbnd(fun,x0,lb,ub)

x = simulannealbnd(fun,x0,lb,ub,options)

x = simulannealbnd(problem)

[x,fval] = simulannealbnd(___)

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

Description

x = simulannealbnd(fun,x0) finds a local minimum, x, to the function handle fun that computes the values of the objective function. x0 is an initial point for the simulated annealing algorithm, a real vector.

Note

Passing Extra Parameters (Optimization Toolbox) explains how to pass extra parameters to the objective function, if necessary.

x = simulannealbnd(fun,x0,lb,ub) defines a set of lower and upper bounds on the design variables in x, so that the solution is always in the range lb ≤ x ≤ ub. If x(i) is unbounded below, set lb(i) = -Inf, and if x(i) is unbounded above, set ub(i) = Inf.

x = simulannealbnd(fun,x0,lb,ub,options) minimizes with the optimization options specified in options. Create options using optimoptions. If no bounds exist, set lb = [] and/or ub = [].

x = simulannealbnd(problem) finds the minimum for problem, where problem is a structure described in . Create the problem structure by exporting a problem from Optimization app, as described in Exporting Your Work (Optimization Toolbox).

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

[x,fval,exitflag,output] = simulannealbnd(___) additionally returns a value exitflag that describes the exit condition of simulannealbnd, and a structure output with information about the optimization process.

Examples

Minimize a Function with Many Local Minima

Minimize De Jong's fifth function, a two-dimensional function with many local minima.

Plot De Jong's fifth function.

dejong5fcn

Minimize De Jong's fifth function using simulannealbnd starting from the point [0,0].

fun = @dejong5fcn;
x0 = [0 0];
x = simulannealbnd(fun,x0)

Optimization terminated: change in best function value less than options.FunctionTolerance.

x = 1×2

  -32.0285   -0.1280

The simulannealbnd algorithm uses the MATLAB® random number stream, so you might obtain a different result.

Minimize Subject to Bounds

Minimize De Jong’s fifth function within a bounded region.

Plot De Jong's fifth function.

dejong5fcn

Start simulannealbnd starting at the point [0,0], and set lower bounds of -64 and upper bounds of 64 on each component.

fun = @dejong5fcn;
x0 = [0 0];
lb = [-64 -64];
ub = [64 64];
x = simulannealbnd(fun,x0,lb,ub)

Optimization terminated: change in best function value less than options.FunctionTolerance.

x = 1×2

  -15.9790  -31.9593

The simulannealbnd algorithm uses the MATLAB® random number stream, so you might obtain a different result.

Minimize Using Nondefault Options

Observe the progress of simulannealbnd by setting options to use some plot functions.

Set simulated annealing options to use several plot functions.

options = optimoptions('simulannealbnd','PlotFcns',...
          {@saplotbestx,@saplotbestf,@saplotx,@saplotf});

Start simulannealbnd starting at the point [0,0], and set lower bounds of -64 and upper bounds of 64 on each component.

rng default % For reproducibility
fun = @dejong5fcn;
x0 = [0,0];
lb = [-64,-64];
ub = [64,64];
x = simulannealbnd(fun,x0,lb,ub,options)

Optimization terminated: change in best function value less than options.FunctionTolerance.

x = 1×2

  -15.9790  -31.9593

Obtain All Outputs

Obtain all the outputs of a simulated annealing minimization.

Plot De Jong's fifth function.

dejong5fcn

Start simulannealbnd starting at the point [0,0], and set lower bounds of -64 and upper bounds of 64 on each component.

fun = @dejong5fcn;
x0 = [0,0];
lb = [-64,-64];
ub = [64,64];
[x,fval,exitflag,output] = simulannealbnd(fun,x0,lb,ub)

Optimization terminated: change in best function value less than options.FunctionTolerance.

x = 1×2

  -15.9790  -31.9593

fval = 1.9920

exitflag = 1

output = struct with fields:
     iterations: 1762
      funccount: 1779
        message: 'Optimization terminated: change in best function value less than options.FunctionTolerance.'
       rngstate: [1x1 struct]
    problemtype: 'boundconstraints'
    temperature: [2x1 double]
      totaltime: 1.3331

The simulannealbnd algorithm uses the MATLAB® random number stream, so you might obtain a different result.

Input Arguments

fun — Function to be minimized

function handle | function name

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

fun can be specified as a function handle for a file:

x = simulannealbnd(@myfun,x0)

where myfun is a MATLAB^® function such as

function f = myfun(x)
f = ...            % Compute function value at x

fun can also be a function handle for an anonymous function:

x = simulannealbnd(@(x)norm(x)^2,x0,lb,ub);

Example: fun = @(x)sin(x(1))*cos(x(2))

Data Types: char | function_handle | string

x0 — Initial point

real vector

Initial point, specified as a real vector. simulannealbnd uses the number of elements in x0 to determine the number of variables that fun accepts.

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

Data Types: double

lb — Lower bounds

real vector | real array

Lower bounds, specified as a real vector or real array. If the number of elements in x0 is equal to that of lb, then lb specifies that

x(i) >= lb(i) for all i.

If numel(lb) < numel(x0), then lb specifies that

x(i) >= lb(i) for 1 <= i <= numel(lb).

In this case, solvers issue a warning.

Example: To specify that all control variables are positive, lb = zeros(size(x0))

Data Types: double

ub — Upper bounds

real vector | real array

Upper bounds, specified as a real vector or real array. If the number of elements in x0 is equal to that of ub, then ub specifies that

x(i) <= ub(i) for all i.

If numel(ub) < numel(x0), then ub specifies that

x(i) <= ub(i) for 1 <= i <= numel(ub).

In this case, solvers issue a warning.

Example: To specify that all control variables are less than one, ub = ones(size(x0))

Data Types: double

options — Optimization options

object returned by optimoptions | structure

Optimization options, specified as an object returned by optimoptions or a structure. For details, see Simulated Annealing Options.

optimoptions hides the options listed in italics; see Options that optimoptions Hides.

{} denotes the default value. See option details in Simulated Annealing Options.

Option	Description	Values
AcceptanceFcn	Function the algorithm uses to determine if a new point is accepted. Specify as 'acceptancesa' or a function handle.	Function handle | {'acceptancesa'}
AnnealingFcn	Function the algorithm uses to generate new points. Specify as a name of a built-in annealing function or a function handle.	Function handle | function name | 'annealingboltz' | {'annealingfast'}
DataType	Type of decision variable	'custom' | {'double'}
Display	Level of display	'off' | 'iter' | 'diagnose' | {'final'}
DisplayInterval	Interval for iterative display	Positive integer | {10}
FunctionTolerance	Termination tolerance on function value For an options structure, use TolFun.	Positive scalar | {1e-6}
HybridFcn	Automatically run HybridFcn (another optimization function) during or at the end of iterations of the solver. Specify as a name or a function handle. See When to Use a Hybrid Function.	'fminsearch' | 'patternsearch' | 'fminunc' | 'fmincon' | {[]} or 1-by-2 cell array | {@solver, hybridoptions}, where solver = fminsearch, patternsearch, fminunc, or fmincon {[]}
HybridInterval	Interval (if not 'end' or 'never') at which HybridFcn is called	Positive integer | 'never' | {'end'}
InitialTemperature	Initial value of temperature	Positive scalar | positive vector | {100}
MaxFunctionEvaluations	Maximum number of objective function evaluations allowed For an options structure, use MaxFunEvals.	Positive integer | {3000*numberOfVariables}
MaxIterations	Maximum number of iterations allowed For an options structure, use MaxIter.	Positive integer | {Inf}
MaxStallIterations	Number of iterations over which average change in fitness function value at current point is less than options.FunctionTolerance For an options structure, use StallIterLimit.	Positive integer | {500*numberOfVariables}
MaxTime	The algorithm stops after running for MaxTime seconds For an options structure, use TimeLimit.	Positive scalar | {Inf}
ObjectiveLimit	Minimum objective function value desired	Scalar | {-Inf}
OutputFcn	Function(s) get(s) iterative data and can change options at run time For an options structure, use OutputFcns.	Function handle | cell array of function handles | {[]}
PlotFcn	Plot function(s) called during iterations For an options structure, use PlotFcns.	Function handle | built-in plot function name | cell array of function handles | cell array of built-in plot function names | 'saplotbestf' | 'saplotbestx' | 'saplotf' | 'saplotstopping' | 'saplottemperature' | {[]}
PlotInterval	Plot functions are called at every interval	Positive integer | {1}
ReannealInterval	Reannealing interval	Positive integer | {100}
TemperatureFcn	Function used to update temperature schedule	Function handle | built-in temperature function name | 'temperatureboltz' | 'temperaturefast' | {'temperatureexp'}

Example: options = optimoptions(@simulannealbnd,'MaxIterations',150)

Data Types: struct

problem — Problem structure

structure

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

• objective — Objective function

• x0 — Starting point

• lb — Lower bound for x

• ub — Upper bound for x

• solver — 'simulannealbnd'

• options — Options created with optimoptions or an options structure

• rngstate — Optional field to reset the state of the random number generator

Create the structure problem by exporting a problem from the Optimization app, as described in Importing and Exporting Your Work (Optimization Toolbox).

Note

problem must have all the fields as specified above.

Data Types: struct

Output Arguments

x — Solution

real vector

Solution, returned as a real vector. 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 the solution

real number

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

exitflag — Reason simulannealbnd stopped

integer

Reason simulannealbnd stopped, returned as an integer.

Exit Flag	Meaning
1	Average change in the value of the objective function over options.MaxStallIterations iterations is less than options.FunctionTolerance.
5	options.ObjectiveLimit limit reached.
0	Maximum number of function evaluations or iterations reached.
-1	Optimization terminated by an output function or plot function.
-2	No feasible point found.
-5	Time limit exceeded.

output — Information about the optimization process

structure

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

• problemtype — Type of problem: unconstrained or bound constrained.

• iterations — The number of iterations computed.

• funccount — The number of evaluations of the objective function.

• message — The reason the algorithm terminated.

• temperature — Temperature when the solver terminated.

• totaltime — Total time for the solver to run.

• rngstate — State of the MATLAB random number generator, just before the algorithm started. You can use the values in rngstate to reproduce the output of simulannealbnd. See Reproduce Your Results.