# bayesopt

Select optimal machine learning hyperparameters using Bayesian optimization

## Description

attempts
to find values of `results`

= bayesopt(`fun`

,`vars`

)`vars`

that minimize `fun(vars)`

.

**Note**

To include extra parameters in an objective function, see Parameterizing Functions.

modifies the optimization process according to the `results`

= bayesopt(`fun`

,`vars`

,`Name,Value`

)`Name,Value`

arguments.

## Examples

### Create a `BayesianOptimization`

Object Using `bayesopt`

This example shows how to create a `BayesianOptimization`

object by using `bayesopt`

to minimize cross-validation loss.

Optimize hyperparameters of a KNN classifier for the `ionosphere`

data, that is, find KNN hyperparameters that minimize the cross-validation loss. Have `bayesopt`

minimize over the following hyperparameters:

Nearest-neighborhood sizes from 1 to 30

Distance functions

`'chebychev'`

,`'euclidean'`

, and`'minkowski'`

.

For reproducibility, set the random seed, set the partition, and set the `AcquisitionFunctionName`

option to `'expected-improvement-plus'`

. To suppress iterative display, set `'Verbose'`

to `0`

. Pass the partition `c`

and fitting data `X`

and `Y`

to the objective function `fun`

by creating `fun`

as an anonymous function that incorporates this data. See Parameterizing Functions.

load ionosphere rng default num = optimizableVariable('n',[1,30],'Type','integer'); dst = optimizableVariable('dst',{'chebychev','euclidean','minkowski'},'Type','categorical'); c = cvpartition(351,'Kfold',5); fun = @(x)kfoldLoss(fitcknn(X,Y,'CVPartition',c,'NumNeighbors',x.n,... 'Distance',char(x.dst),'NSMethod','exhaustive')); results = bayesopt(fun,[num,dst],'Verbose',0,... 'AcquisitionFunctionName','expected-improvement-plus')

results = BayesianOptimization with properties: ObjectiveFcn: [function_handle] VariableDescriptions: [1x2 optimizableVariable] Options: [1x1 struct] MinObjective: 0.1197 XAtMinObjective: [1x2 table] MinEstimatedObjective: 0.1213 XAtMinEstimatedObjective: [1x2 table] NumObjectiveEvaluations: 30 TotalElapsedTime: 29.7824 NextPoint: [1x2 table] XTrace: [30x2 table] ObjectiveTrace: [30x1 double] ConstraintsTrace: [] UserDataTrace: {30x1 cell} ObjectiveEvaluationTimeTrace: [30x1 double] IterationTimeTrace: [30x1 double] ErrorTrace: [30x1 double] FeasibilityTrace: [30x1 logical] FeasibilityProbabilityTrace: [30x1 double] IndexOfMinimumTrace: [30x1 double] ObjectiveMinimumTrace: [30x1 double] EstimatedObjectiveMinimumTrace: [30x1 double]

### Bayesian Optimization with Coupled Constraints

A coupled constraint is one that can be evaluated only by evaluating the objective function. In this case, the objective function is the cross-validated loss of an SVM model. The coupled constraint is that the number of support vectors is no more than 100. The model details are in Optimize Cross-Validated Classifier Using bayesopt.

Create the data for classification.

rng default grnpop = mvnrnd([1,0],eye(2),10); redpop = mvnrnd([0,1],eye(2),10); redpts = zeros(100,2); grnpts = redpts; for i = 1:100 grnpts(i,:) = mvnrnd(grnpop(randi(10),:),eye(2)*0.02); redpts(i,:) = mvnrnd(redpop(randi(10),:),eye(2)*0.02); end cdata = [grnpts;redpts]; grp = ones(200,1); grp(101:200) = -1; c = cvpartition(200,'KFold',10); sigma = optimizableVariable('sigma',[1e-5,1e5],'Transform','log'); box = optimizableVariable('box',[1e-5,1e5],'Transform','log');

The objective function is the cross-validation loss of the SVM model for partition `c`

. The coupled constraint is the number of support vectors minus 100.5. This ensures that 100 support vectors give a negative constraint value, but 101 support vectors give a positive value. The model has 200 data points, so the coupled constraint values range from -99.5 (there is always at least one support vector) to 99.5. Positive values mean the constraint is not satisfied.

function [objective,constraint] = mysvmfun(x,cdata,grp,c) SVMModel = fitcsvm(cdata,grp,'KernelFunction','rbf',... 'BoxConstraint',x.box,... 'KernelScale',x.sigma); cvModel = crossval(SVMModel,'CVPartition',c); objective = kfoldLoss(cvModel); constraint = sum(SVMModel.IsSupportVector)-100.5;

Pass the partition `c`

and fitting data `cdata`

and `grp`

to the objective function `fun`

by creating `fun`

as an anonymous function that incorporates this data. See Parameterizing Functions.

fun = @(x)mysvmfun(x,cdata,grp,c);

Set the `NumCoupledConstraints`

to `1`

so the optimizer knows that there is a coupled constraint. Set options to plot the constraint model.

results = bayesopt(fun,[sigma,box],'IsObjectiveDeterministic',true,... 'NumCoupledConstraints',1,'PlotFcn',... {@plotMinObjective,@plotConstraintModels},... 'AcquisitionFunctionName','expected-improvement-plus','Verbose',0);

Most points lead to an infeasible number of support vectors.

### Parallel Bayesian Optimization

Improve the speed of a Bayesian optimization by using parallel objective function evaluation.

Prepare variables and the objective function for Bayesian optimization.

The objective function is the cross-validation error rate for the ionosphere data, a binary classification problem. Use `fitcsvm`

as the classifier, with `BoxConstraint`

and `KernelScale`

as the parameters to optimize.

load ionosphere box = optimizableVariable('box',[1e-4,1e3],'Transform','log'); kern = optimizableVariable('kern',[1e-4,1e3],'Transform','log'); vars = [box,kern]; fun = @(vars)kfoldLoss(fitcsvm(X,Y,'BoxConstraint',vars.box,'KernelScale',vars.kern,... 'Kfold',5));

Search for the parameters that give the lowest cross-validation error by using parallel Bayesian optimization.

`results = bayesopt(fun,vars,'UseParallel',true);`

Copying objective function to workers... Done copying objective function to workers.

|===============================================================================================================| | Iter | Active | Eval | Objective | Objective | BestSoFar | BestSoFar | box | kern | | | workers | result | | runtime | (observed) | (estim.) | | | |===============================================================================================================| | 1 | 2 | Accept | 0.2735 | 0.56171 | 0.13105 | 0.13108 | 0.0002608 | 0.2227 | | 2 | 2 | Accept | 0.35897 | 0.4062 | 0.13105 | 0.13108 | 3.6999 | 344.01 | | 3 | 2 | Accept | 0.13675 | 0.42727 | 0.13105 | 0.13108 | 0.33594 | 0.39276 | | 4 | 2 | Accept | 0.35897 | 0.4453 | 0.13105 | 0.13108 | 0.014127 | 449.58 | | 5 | 2 | Best | 0.13105 | 0.45503 | 0.13105 | 0.13108 | 0.29713 | 1.0859 |

| 6 | 6 | Accept | 0.35897 | 0.16605 | 0.13105 | 0.13108 | 8.1878 | 256.9 |

| 7 | 5 | Best | 0.11396 | 0.51146 | 0.11396 | 0.11395 | 8.7331 | 0.7521 | | 8 | 5 | Accept | 0.14245 | 0.24943 | 0.11396 | 0.11395 | 0.0020774 | 0.022712 |

| 9 | 6 | Best | 0.10826 | 4.0711 | 0.10826 | 0.10827 | 0.0015925 | 0.0050225 |

| 10 | 6 | Accept | 0.25641 | 16.265 | 0.10826 | 0.10829 | 0.00057357 | 0.00025895 |

| 11 | 6 | Accept | 0.1339 | 15.581 | 0.10826 | 0.10829 | 1.4553 | 0.011186 |

| 12 | 6 | Accept | 0.16809 | 19.585 | 0.10826 | 0.10828 | 0.26919 | 0.00037649 |

| 13 | 6 | Accept | 0.20513 | 18.637 | 0.10826 | 0.10828 | 369.59 | 0.099122 |

| 14 | 6 | Accept | 0.12536 | 0.11382 | 0.10826 | 0.10829 | 5.7059 | 2.5642 |

| 15 | 6 | Accept | 0.13675 | 2.63 | 0.10826 | 0.10828 | 984.19 | 2.2214 |

| 16 | 6 | Accept | 0.12821 | 2.0743 | 0.10826 | 0.11144 | 0.0063411 | 0.0090242 |

| 17 | 6 | Accept | 0.1339 | 0.1939 | 0.10826 | 0.11302 | 0.00010225 | 0.0076795 |

| 18 | 6 | Accept | 0.12821 | 0.20933 | 0.10826 | 0.11376 | 7.7447 | 1.2868 |

| 19 | 4 | Accept | 0.55556 | 17.564 | 0.10826 | 0.10828 | 0.0087593 | 0.00014486 | | 20 | 4 | Accept | 0.1396 | 16.473 | 0.10826 | 0.10828 | 0.054844 | 0.004479 | |===============================================================================================================| | Iter | Active | Eval | Objective | Objective | BestSoFar | BestSoFar | box | kern | | | workers | result | | runtime | (observed) | (estim.) | | | |===============================================================================================================| | 21 | 4 | Accept | 0.1339 | 0.17127 | 0.10826 | 0.10828 | 9.2668 | 1.2171 |

| 22 | 4 | Accept | 0.12821 | 0.089065 | 0.10826 | 0.10828 | 12.265 | 8.5455 |

| 23 | 4 | Accept | 0.12536 | 0.073586 | 0.10826 | 0.10828 | 1.3355 | 2.8392 |

| 24 | 4 | Accept | 0.12821 | 0.08038 | 0.10826 | 0.10828 | 131.51 | 16.878 |

| 25 | 3 | Accept | 0.11111 | 10.687 | 0.10826 | 0.10867 | 1.4795 | 0.041452 | | 26 | 3 | Accept | 0.13675 | 0.18626 | 0.10826 | 0.10867 | 2.0513 | 0.70421 |

| 27 | 6 | Accept | 0.12821 | 0.078559 | 0.10826 | 0.10868 | 980.04 | 44.19 |

| 28 | 5 | Accept | 0.33048 | 0.089844 | 0.10826 | 0.10843 | 0.41821 | 10.208 | | 29 | 5 | Accept | 0.16239 | 0.12688 | 0.10826 | 0.10843 | 172.39 | 141.43 |

| 30 | 5 | Accept | 0.11966 | 0.14597 | 0.10826 | 0.10846 | 639.15 | 14.75 |

__________________________________________________________ Optimization completed. MaxObjectiveEvaluations of 30 reached. Total function evaluations: 30 Total elapsed time: 48.2085 seconds. Total objective function evaluation time: 128.3472 Best observed feasible point: box kern _________ _________ 0.0015925 0.0050225 Observed objective function value = 0.10826 Estimated objective function value = 0.10846 Function evaluation time = 4.0711 Best estimated feasible point (according to models): box kern _________ _________ 0.0015925 0.0050225 Estimated objective function value = 0.10846 Estimated function evaluation time = 2.8307

Return the best feasible point in the Bayesian model `results`

by using the `bestPoint`

function. Use the default criterion `min-visited-upper-confidence-interval`

, which determines the best feasible point as the visited point that minimizes an upper confidence interval on the objective function value.

zbest = bestPoint(results)

`zbest=`*1×2 table*
box kern
_________ _________
0.0015925 0.0050225

The table `zbest`

contains the optimal estimated values for the `'BoxConstraint'`

and `'KernelScale'`

name-value pair arguments. Use these values to train a new optimized classifier.

Mdl = fitcsvm(X,Y,'BoxConstraint',zbest.box,'KernelScale',zbest.kern);

Observe that the optimal parameters are in `Mdl`

.

Mdl.BoxConstraints(1)

ans = 0.0016

Mdl.KernelParameters.Scale

ans = 0.0050

## Input Arguments

`fun`

— Objective function

function handle | `parallel.pool.Constant`

whose `Value`

is a function handle

Objective function, specified as a function handle or, when the `UseParallel`

name-value pair is `true`

, a `parallel.pool.Constant`

(Parallel Computing Toolbox)
whose `Value`

is a function handle. Typically,
`fun`

returns a measure of loss (such as a
misclassification error) for a machine learning model that has tunable
hyperparameters to control its training. `fun`

has these
signatures:

objective = fun(x) % or [objective,constraints] = fun(x) % or [objective,constraints,UserData] = fun(x)

`fun`

accepts `x`

, a 1-by-`D`

table
of variable values, and returns `objective`

, a real
scalar representing the objective function value `fun(x)`

.

Optionally, `fun`

also returns:

`constraints`

, a real vector of coupled constraint violations. For a definition, see Coupled Constraints.`constraint(j) > 0`

means constraint`j`

is violated.`constraint(j) < 0`

means constraint`j`

is satisfied.`UserData`

, an entity of any type (such as a scalar, matrix, structure, or object). For an example of a custom plot function that uses`UserData`

, see Create a Custom Plot Function.

For details about using `parallel.pool.Constant`

with
`bayesopt`

, see Placing the Objective Function on Workers.

**Example: **`@objfun`

**Data Types: **`function_handle`

`vars`

— Variable descriptions

vector of `optimizableVariable`

objects defining
the hyperparameters to be tuned

Variable descriptions, specified as a vector of `optimizableVariable`

objects
defining the hyperparameters to be tuned.

**Example: **`[X1,X2]`

, where `X1`

and `X2`

are `optimizableVariable`

objects

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

**Example: **`results = bayesopt(fun,vars,'AcquisitionFunctionName','expected-improvement-plus')`

**Algorithm Control**

`AcquisitionFunctionName`

— Function to choose next evaluation point

`'expected-improvement-per-second-plus'`

(default) | `'expected-improvement'`

| `'expected-improvement-plus'`

| `'expected-improvement-per-second'`

| `'lower-confidence-bound'`

| `'probability-of-improvement'`

Function to choose next evaluation point, specified as one of the listed choices.

Acquisition functions whose names include
`per-second`

do not yield reproducible results because the optimization
depends on the runtime of the objective function. Acquisition functions whose names include
`plus`

modify their behavior when they are overexploiting an area. For more
details, see Acquisition Function Types.

**Example: **`'AcquisitionFunctionName','expected-improvement-per-second'`

`IsObjectiveDeterministic`

— Specify deterministic objective function

`false`

(default) | `true`

Specify deterministic objective function, specified as `false`

or
`true`

. If `fun`

is stochastic
(that is, `fun(x)`

can return different values for the
same `x`

), then set
`IsObjectiveDeterministic`

to
`false`

. In this case,
`bayesopt`

estimates a noise level during
optimization.

**Example: **`'IsObjectiveDeterministic',true`

**Data Types: **`logical`

`ExplorationRatio`

— Propensity to explore

`0.5`

(default) | positive real

Propensity to explore, specified as a positive real. Applies
to the `'expected-improvement-plus'`

and `'expected-improvement-per-second-plus'`

acquisition
functions. See Plus.

**Example: **`'ExplorationRatio',0.2`

**Data Types: **`double`

`GPActiveSetSize`

— Fit Gaussian Process model to `GPActiveSetSize`

or fewer points

`300`

(default) | positive integer

Fit Gaussian Process model to `GPActiveSetSize`

or
fewer points, specified as a positive integer. When
`bayesopt`

has visited more than
`GPActiveSetSize`

points, subsequent iterations
that use a GP model fit the model to `GPActiveSetSize`

points. `bayesopt`

chooses points uniformly at random
without replacement among visited points. Using fewer points leads to
faster GP model fitting, at the expense of possibly less accurate
fitting.

**Example: **`'GPActiveSetSize',80`

**Data Types: **`double`

`UseParallel`

— Compute in parallel

`false`

(default) | `true`

Compute in parallel, specified as `false`

(do not
compute in parallel) or `true`

(compute in parallel).
Computing in parallel requires Parallel Computing Toolbox™.

`bayesopt`

performs parallel objective function
evaluations concurrently on parallel workers. For algorithmic details,
see Parallel Bayesian Optimization.

**Example: **`'UseParallel',true`

**Data Types: **`logical`

`ParallelMethod`

— Imputation method for parallel worker objective function values

`'clipped-model-prediction'`

(default) | `'model-prediction'`

| `'max-observed'`

| `'min-observed'`

Imputation method for parallel worker objective function values,
specified as `'clipped-model-prediction'`

,
`'model-prediction'`

,
`'max-observed'`

, or
`'min-observed'`

. To generate a new point to
evaluate, `bayesopt`

fits a Gaussian process to all
points, including the points being evaluated on workers. To fit the
process, `bayesopt`

imputes objective function values
for the points that are currently on workers.
`ParallelMethod`

specifies the method used for
imputation.

`'clipped-model-prediction'`

— Impute the maximum of these quantities:Mean Gaussian process prediction at the point

`x`

Minimum observed objective function among feasible points visited

Minimum model prediction among all feasible points

`'model-prediction'`

— Impute the mean Gaussian process prediction at the point`x`

.`'max-observed'`

— Impute the maximum observed objective function value among feasible points.`'min-observed'`

— Impute the minimum observed objective function value among feasible points.

**Example: **`'ParallelMethod','max-observed'`

`MinWorkerUtilization`

— Tolerance on number of active parallel workers

`floor(0.8*Nworkers)`

(default) | positive integer

Tolerance on the number of active parallel workers, specified as a
positive integer. After `bayesopt`

assigns a point to
evaluate, and before it computes a new point to assign, it checks
whether fewer than `MinWorkerUtilization`

workers are
active. If so, `bayesopt`

assigns random points
within bounds to all available workers. Otherwise,
`bayesopt`

calculates the best point for one
worker. `bayesopt`

creates random points much faster
than fitted points, so this behavior leads to higher utilization of
workers, at the cost of possibly poorer points. For details, see Parallel Bayesian Optimization.

**Example: **`'MinWorkerUtilization',3`

**Data Types: **`double`

**Starting and Stopping**

`MaxObjectiveEvaluations`

— Objective function evaluation limit

`30`

(default) | positive integer

Objective function evaluation limit, specified as a positive integer.

**Example: **`'MaxObjectiveEvaluations',60`

**Data Types: **`double`

`NumSeedPoints`

— Number of initial evaluation points

`4`

(default) | positive integer

Number of initial evaluation points, specified as a positive integer.
`bayesopt`

chooses these points randomly within
the variable bounds, according to the setting of the `Transform`

setting for each
variable (uniform for `'none'`

, logarithmically spaced
for `'log'`

).

**Example: **`'NumSeedPoints',10`

**Data Types: **`double`

**Constraints**

`XConstraintFcn`

— Deterministic constraints on variables

`[]`

(default) | function handle

Deterministic constraints on variables, specified as a function handle.

For details, see Deterministic Constraints — XConstraintFcn.

**Example: **`'XConstraintFcn',@xconstraint`

**Data Types: **`function_handle`

`ConditionalVariableFcn`

— Conditional variable constraints

`[]`

(default) | function handle

Conditional variable constraints, specified as a function handle.

For details, see Conditional Constraints — ConditionalVariableFcn.

**Example: **`'ConditionalVariableFcn',@condfun`

**Data Types: **`function_handle`

`NumCoupledConstraints`

— Number of coupled constraints

`0`

(default) | positive integer

Number of coupled constraints, specified as a positive integer. For details, see Coupled Constraints.

**Note**

`NumCoupledConstraints`

is required when you
have coupled constraints.

**Example: **`'NumCoupledConstraints',3`

**Data Types: **`double`

`AreCoupledConstraintsDeterministic`

— Indication of whether coupled constraints are deterministic

`true`

for all coupled
constraints (default) | logical vector

Indication of whether coupled constraints are deterministic, specified
as a logical vector of length `NumCoupledConstraints`

.
For details, see Coupled Constraints.

**Example: **`'AreCoupledConstraintsDeterministic',[true,false,true]`

**Data Types: **`logical`

**Reports, Plots, and Halting**

`Verbose`

— Command-line display level

`1`

(default) | `0`

| `2`

Command-line display level, specified as `0`

, `1`

,
or `2`

.

`0`

— No command-line display.`1`

— At each iteration, display the iteration number, result report (see the next paragraph), objective function model, objective function evaluation time, best (lowest) observed objective function value, best (lowest) estimated objective function value, and the observed constraint values (if any). When optimizing in parallel, the display also includes a column showing the number of active workers, counted after assigning a job to the next worker.The result report for each iteration is one of the following:

`Accept`

— The objective function returns a finite value, and all constraints are satisfied.`Best`

— Constraints are satisfied, and the objective function returns the lowest value among feasible points.`Error`

— The objective function returns a value that is not a finite real scalar.`Infeas`

— At least one constraint is violated.

`2`

— Same as`1`

, adding diagnostic information such as time to select the next point, model fitting time, indication that "plus" acquisition functions declare overexploiting, and parallel workers are being assigned to random points due to low parallel utilization.

**Example: **`'Verbose',2`

**Data Types: **`double`

`OutputFcn`

— Function called after each iteration

`{}`

(default) | function handle | cell array of function handles

Function called after each iteration, specified as a function handle or cell array of function handles. An output function can halt the solver, and can perform arbitrary calculations, including creating variables or plotting. Specify several output functions using a cell array of function handles.

There are two built-in output functions:

`@assignInBase`

— Constructs a`BayesianOptimization`

instance at each iteration and assigns it to a variable in the base workspace. Choose a variable name using the`SaveVariableName`

name-value pair.`@saveToFile`

— Constructs a`BayesianOptimization`

instance at each iteration and saves it to a file in the current folder. Choose a file name using the`SaveFileName`

name-value pair.

You can write your own output functions. For details, see Bayesian Optimization Output Functions.

**Example: **`'OutputFcn',{@saveToFile @myOutputFunction}`

**Data Types: **`cell`

| `function_handle`

`SaveFileName`

— File name for the `@saveToFile`

output function

`'BayesoptResults.mat'`

(default) | character vector | string scalar

File name for the `@saveToFile`

output function, specified as a character
vector or string scalar. The file name can include a path, such as
`'../optimizations/September2.mat'`

.

**Example: **`'SaveFileName','September2.mat'`

**Data Types: **`char`

| `string`

`SaveVariableName`

— Variable name for the `@assignInBase`

output function

`'BayesoptResults'`

(default) | character vector | string scalar

Variable name for the `@assignInBase`

output function, specified as a
character vector or string scalar.

**Example: **`'SaveVariableName','September2Results'`

**Data Types: **`char`

| `string`

`PlotFcn`

— Plot function called after each iteration

`{@plotObjectiveModel,@plotMinObjective}`

(default) | `'all'`

| function handle | cell array of function handles

Plot function called after each iteration, specified as `'all'`

,
a function handle, or a cell array of function handles. A plot function
can halt the solver, and can perform arbitrary calculations, including
creating variables, in addition to plotting.

Specify no plot function as `[]`

.

`'all'`

calls all built-in plot functions.
Specify several plot functions using a cell array of function handles.

The built-in plot functions appear in the following tables.

Model Plots — Apply When D ≤ 2 | Description |
---|---|

`@plotAcquisitionFunction` | Plot the acquisition function surface. |

`@plotConstraintModels` | Plot each constraint model surface. Negative values indicate feasible points. Also plot a
Also plot the error model, if
it exists, which ranges from Plotted error = 2*Probability(error) – 1. |

`@plotObjectiveEvaluationTimeModel` | Plot the objective function evaluation time model surface. |

`@plotObjectiveModel` | Plot the |

Trace Plots — Apply to All D | Description |
---|---|

`@plotObjective` | Plot each observed function value versus the number of function evaluations. |

`@plotObjectiveEvaluationTime` | Plot each observed function evaluation run time versus the number of function evaluations. |

`@plotMinObjective` | Plot the minimum observed and estimated function values versus the number of function evaluations. |

`@plotElapsedTime` | Plot three curves: the total elapsed time of the optimization, the total function evaluation time, and the total modeling and point selection time, all versus the number of function evaluations. |

You can write your own plot functions. For details, see Bayesian Optimization Plot Functions.

**Note**

When there are coupled constraints, iterative display and plot functions can give counterintuitive results such as:

A

*minimum objective*plot can increase.The optimization can declare a problem infeasible even when it showed an earlier feasible point.

The reason for this behavior is that the decision about whether
a point is feasible can change as the optimization progresses. `bayesopt`

determines
feasibility with respect to its constraint model, and this model changes
as `bayesopt`

evaluates points. So a “minimum
objective” plot can increase when the minimal point is later
deemed infeasible, and the iterative display can show a feasible point
that is later deemed infeasible.

**Example: **`'PlotFcn','all'`

**Data Types: **`char`

| `string`

| `cell`

| `function_handle`

**Initialization**

`InitialX`

— Initial evaluation points

`NumSeedPoints`

-by-`D`

random
initial points within bounds (default) | `N`

-by-`D`

table

Initial evaluation points, specified as an `N`

-by-`D`

table,
where `N`

is the number of evaluation points, and `D`

is
the number of variables.

**Note**

If only `InitialX`

is provided, it is interpreted
as initial points to evaluate. The objective function is evaluated
at `InitialX`

.

If any other initialization parameters are also provided, `InitialX`

is
interpreted as prior function evaluation data. The objective function
is not evaluated. Any missing values are set to `NaN`

.

**Data Types: **`table`

`InitialObjective`

— Objective values corresponding to `InitialX`

`[]`

(default) | length-`N`

vector

Objective values corresponding to `InitialX`

,
specified as a length-`N`

vector, where
`N`

is the number of evaluation points.

**Example: **`'InitialObjective',[17;-3;-12.5]`

**Data Types: **`double`

`InitialConstraintViolations`

— Constraint violations of coupled constraints

`[]`

(default) | `N`

-by-`K`

matrix

Constraint violations of coupled constraints, specified as an `N`

-by-`K`

matrix,
where `N`

is the number of evaluation points and `K`

is
the number of coupled constraints. For details, see Coupled Constraints.

**Data Types: **`double`

`InitialErrorValues`

— Errors for `InitialX`

`[]`

(default) | length-`N`

vector with entries `-1`

or `1`

Errors for `InitialX`

, specified as a length-`N`

vector
with entries `-1`

or `1`

, where `N`

is
the number of evaluation points. Specify `-1`

for
no error, and `1`

for an error.

**Example: **`'InitialErrorValues',[-1,-1,-1,-1,1]`

**Data Types: **`double`

`InitialUserData`

— Initial data corresponding to `InitialX`

`[]`

(default) | length-`N`

cell vector

Initial data corresponding to `InitialX`

,
specified as a length-`N`

cell vector, where `N`

is
the number of evaluation points.

**Example: **`'InitialUserData',{2,3,-1}`

**Data Types: **`cell`

`InitialObjectiveEvaluationTimes`

— Evaluation times of objective function at `InitialX`

`[]`

(default) | length-`N`

vector

Evaluation times of objective function at `InitialX`

,
specified as a length-`N`

vector, where `N`

is
the number of evaluation points. Time is measured in seconds.

**Data Types: **`double`

`InitialIterationTimes`

— Times for the first `N`

iterations

`{}`

(default) | length-`N`

vector

Times for the first `N`

iterations, specified
as a length-`N`

vector, where `N`

is
the number of evaluation points. Time is measured in seconds.

**Data Types: **`double`

## Output Arguments

`results`

— Bayesian optimization results

`BayesianOptimization`

object

Bayesian optimization results, returned as a `BayesianOptimization`

object.

## More About

### Coupled Constraints

Coupled constraints are those constraints whose value comes from the objective function calculation. See Coupled Constraints.

## Tips

Bayesian optimization is not reproducible if one of these conditions exists:

You specify an acquisition function whose name includes

`per-second`

, such as`'expected-improvement-per-second'`

. The`per-second`

modifier indicates that optimization depends on the run time of the objective function. For more details, see Acquisition Function Types.You specify to run Bayesian optimization in parallel. Due to the nonreproducibility of parallel timing, parallel Bayesian optimization does not necessarily yield reproducible results. For more details, see Parallel Bayesian Optimization.

## Extended Capabilities

### Automatic Parallel Support

Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

To run in parallel, set the `UseParallel`

name-value argument to
`true`

in the call to this function.

For more general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

## Version History

**Introduced in R2016b**

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