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# Documentation

## Simulated Annealing Options

### saoptimset At The Command Line

Specify options by creating an options structure using the saoptimset function as follows:

`options = saoptimset('Param1',value1,'Param2',value2, ...);`

See Set Options for simulannealbnd at the Command Line for examples.

Each option in this section is listed by its field name in the options structure. For example, InitialTemperature refers to the corresponding field of the options structure.

### Plot Options

Plot options enable you to plot data from the simulated annealing solver while it is running.

PlotInterval specifies the number of iterations between consecutive calls to the plot function.

To display a plot when calling simulannealbnd from the command line, set the PlotFcns field of options to be a function handle to the plot function. You can specify any of the following plots:

• @saplotbestf plots the best objective function value.

• @saplotbestx plots the current best point.

• @saplotf plots the current function value.

• @saplotx plots the current point.

• @saplotstopping plots stopping criteria levels.

• @saplottemperature plots the temperature at each iteration.

• @myfun plots a custom plot function, where myfun is the name of your function. See Structure of the Plot Functions for a description of the syntax.

For example, to display the best objective plot, set options as follows

`options = saoptimset('PlotFcns',@saplotbestf);`

To display multiple plots, use the cell array syntax

`options = saoptimset('PlotFcns',{@plotfun1,@plotfun2, ...});`

where @plotfun1, @plotfun2, and so on are function handles to the plot functions.

If you specify more than one plot function, all plots appear as subplots in the same window. Right-click any subplot to obtain a larger version in a separate figure window.

#### Structure of the Plot Functions

The first line of a plot function has the form

`function stop = plotfun(options,optimvalues,flag)`

The input arguments to the function are

• options — Options structure created using saoptimset.

• optimvalues — Structure containing information about the current state of the solver. The structure contains the following fields:

• x — Current point

• fval — Objective function value at x

• bestx — Best point found so far

• bestfval — Objective function value at best point

• temperature — Current temperature

• iteration — Current iteration

• funccount — Number of function evaluations

• t0 — Start time for algorithm

• k — Annealing parameter

• flag — Current state in which the plot function is called. The possible values for flag are

• 'init' — Initialization state

• 'iter' — Iteration state

• 'done' — Final state

The output argument stop provides a way to stop the algorithm at the current iteration. stop can have the following values:

• false — The algorithm continues to the next iteration.

• true — The algorithm terminates at the current iteration.

### Temperature Options

Temperature options specify how the temperature will be lowered at each iteration over the course of the algorithm.

• InitialTemperature — Initial temperature at the start of the algorithm. The default is 100. The initial temperature can be a vector with the same length as x, the vector of unknowns. simulannealbnd expands a scalar initial temperature into a vector.

• TemperatureFcn — Function used to update the temperature schedule. Let k denote the annealing parameter. (The annealing parameter is the same as the iteration number until reannealing.) The options are:

• @temperatureexp — The temperature is equal to InitialTemperature * 0.95^k. This is the default.

• @temperaturefast — The temperature is equal to InitialTemperature / k.

• @temperatureboltz — The temperature is equal to InitialTemperature / ln(k).

• @myfun — Uses a custom function, myfun, to update temperature. The syntax is:

`temperature = myfun(optimValues,options)`

where optimValues is a structure described in Structure of the Plot Functions. options is either the structure created with saoptimset, or the structure of default options, if you did not create an options structure. Both the annealing parameter optimValues.k and the temperature optimValues.temperature are vectors with length equal to the number of elements of the current point x. For example, the function temperaturefast is:

`temperature = options.InitialTemperature./optimValues.k;`

### Algorithm Settings

Algorithm settings define algorithmic specific parameters used in generating new points at each iteration.

Parameters that can be specified for simulannealbnd are:

• DataType — Type of data to use in the objective function. Choices:

• 'double' (default) — A vector of type double.

• 'custom' — Any other data type. You must provide a 'custom' annealing function. You cannot use a hybrid function.

• AnnealingFcn — Function used to generate new points for the next iteration. The choices are:

• @annealingfast — The step has length temperature, with direction uniformly at random. This is the default.

• @annealingboltz — The step has length square root of temperature, with direction uniformly at random.

• @myfun — Uses a custom annealing algorithm, myfun. The syntax is:

`newx = myfun(optimValues,problem)`

where optimValues is a structure described in Structure of the Output Function, and problem is a structure containing the following information:

• objective: function handle to the objective function

• x0: the start point

• nvar: number of decision variables

• lb: lower bound on decision variables

• ub: upper bound on decision variables

For example, the current position is optimValues.x, and the current objective function value is problem.objective(optimValues.x).

• ReannealInterval — Number of points accepted before reannealing. The default value is 100.

• AcceptanceFcn — Function used to determine whether a new point is accepted or not. The choices are:

• @acceptancesa — Simulated annealing acceptance function, the default. If the new objective function value is less than the old, the new point is always accepted. Otherwise, the new point is accepted at random with a probability depending on the difference in objective function values and on the current temperature. The acceptance probability is

$\frac{1}{1+\mathrm{exp}\left(\frac{\Delta }{\mathrm{max}\left(T\right)}\right)}\text{\hspace{0.17em}},$

where Δ = new objective – old objective, and T is the current temperature. Since both Δ and T are positive, the probability of acceptance is between 0 and 1/2. Smaller temperature leads to smaller acceptance probability. Also, larger Δ leads to smaller acceptance probability.

• @myfun — A custom acceptance function, myfun. The syntax is:

`acceptpoint = myfun(optimValues,newx,newfval);`

where optimValues is a structure described in Structure of the Output Function, newx is the point being evaluated for acceptance, and newfval is the objective function at newx. acceptpoint is a Boolean, with value true to accept newx, and false to reject newx.

### Hybrid Function Options

A hybrid function is another minimization function that runs during or at the end of iterations of the solver. HybridInterval specifies the interval (if not never or end) at which the hybrid function is called. You can specify a hybrid function using the HybridFcn option. The choices are:

• [] — No hybrid function.

• @fminsearch — Uses the MATLAB® function fminsearch to perform unconstrained minimization.

• @patternsearch — Uses patternsearch to perform constrained or unconstrained minimization.

• @fminunc — Uses the Optimization Toolbox™ function fminunc to perform unconstrained minimization.

• @fmincon — Uses the Optimization Toolbox function fmincon to perform constrained minimization.

You can set separate options for the hybrid function. Use optimset for fminsearch, psoptimset for patternsearch, or optimoptions for fmincon or fminunc. For example:

`hybridopts = optimoptions('fminunc','Display','iter','Algorithm','quasi-newton');`

Include the hybrid options in the simulannealbnd options structure as follows:

`options = saoptimset(options,'HybridFcn',{@fminunc,hybridopts}); `

hybridopts must exist before you set options.

See Include a Hybrid Function for an example.

### Stopping Criteria Options

Stopping criteria determine what causes the algorithm to terminate. You can specify the following options:

• TolFun — The algorithm runs until the average change in value of the objective function in StallIterLim iterations is less than TolFun. The default value is 1e-6.

• MaxIter — The algorithm stops if the number of iterations exceeds this maximum number of iterations. You can specify the maximum number of iterations as a positive integer or Inf. Inf is the default.

• MaxFunEval specifies the maximum number of evaluations of the objective function. The algorithm stops if the number of function evaluations exceeds the maximum number of function evaluations. The allowed maximum is 3000*numberofvariables.

• TimeLimit specifies the maximum time in seconds the algorithm runs before stopping.

• ObjectiveLimit — The algorithm stops if the best objective function value is less than or equal to the value of ObjectiveLimit.

### Output Function Options

Output functions are functions that the algorithm calls at each iteration. The default value is to have no output function, []. You must first create an output function using the syntax described in Structure of the Output Function.

Using the Optimization app:

• Specify Output function as @myfun, where myfun is the name of your function.

• To pass extra parameters in the output function, use Anonymous Functions.

• For multiple output functions, enter a cell array of output function handles: {@myfun1,@myfun2,...}.

At the command line:

• options = saoptimset('OutputFcns',@myfun);

• For multiple output functions, enter a cell array:

`options = saoptimset('OutputFcns',{@myfun1,@myfun2,...});`

To see a template that you can use to write your own output functions, enter

`edit saoutputfcntemplate`

at the MATLAB command line.

#### Structure of the Output Function

The output function has the following calling syntax.

`[stop,options,optchanged] = myfun(options,optimvalues,flag)`

The function has the following input arguments:

• options — Options structure created using saoptimset.

• optimvalues — Structure containing information about the current state of the solver. The structure contains the following fields:

• x — Current point

• fval — Objective function value at x

• bestx — Best point found so far

• bestfval — Objective function value at best point

• temperature — Current temperature, a vector the same length as x

• iteration — Current iteration

• funccount — Number of function evaluations

• t0 — Start time for algorithm

• k — Annealing parameter, a vector the same length as x

• flag — Current state in which the output function is called. The possible values for flag are

• 'init' — Initialization state

• 'iter' — Iteration state

• 'done' — Final state

Passing Extra Parameters in the Optimization Toolbox documentation explains how to provide additional parameters to the output function.

The output function returns the following arguments:

• stop — Provides a way to stop the algorithm at the current iteration. stop can have the following values:

• false — The algorithm continues to the next iteration.

• true — The algorithm terminates at the current iteration.

• options — Options structure modified by the output function.

• optchanged — A boolean flag indicating changes were made to options. This must be set to true if options are changed.

### Display Options

Use the Display option to specify how much information is displayed at the command line while the algorithm is running. The available options are

• off — No output is displayed. This is the default value for an options structure created using saoptimset.

• iter — Information is displayed at each iteration.

• diagnose — Information is displayed at each iteration. In addition, the diagnostic lists some problem information and the options that have been changed from the defaults.

• final — The reason for stopping is displayed. This is the default.

Both iter and diagnose display the following information:

• Iteration — Iteration number

• f-count — Cumulative number of objective function evaluations

• Best f(x) — Best objective function value

• Current f(x) — Current objective function value

• Mean Temperature — Mean temperature function value