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Select optimal machine learning hyperparameters using Bayesian optimization

`results = bayesopt(fun,vars)`

`results = bayesopt(fun,vars,Name,Value)`

`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'`

. Set options give no iterative display.

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: 79.7617 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]

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 a Cross-Validated SVM 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;

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.

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 | 3 | Accept | 0.35897 | 0.10852 | 0.27066 | 0.27066 | 0.40251 | 972.28 | | 2 | 3 | Best | 0.27066 | 0.10422 | 0.27066 | 0.27066 | 0.0073203 | 1.1872 | | 3 | 3 | Accept | 0.35897 | 0.11258 | 0.27066 | 0.27066 | 0.63346 | 220.31 | | 4 | 3 | Accept | 0.35897 | 0.10982 | 0.27066 | 0.27066 | 0.24185 | 444.02 | | 5 | 6 | Best | 0.15385 | 0.099978 | 0.15385 | 0.15386 | 0.00011168 | 0.093528 | | 6 | 6 | Best | 0.12821 | 0.12967 | 0.12821 | 0.12818 | 0.00063866 | 0.018135 | | 7 | 6 | Accept | 0.19373 | 0.10296 | 0.12821 | 0.12776 | 0.0001005 | 0.11991 | | 8 | 6 | Accept | 0.37322 | 20.593 | 0.12821 | 0.13249 | 8.7031 | 0.00011125 | | 9 | 6 | Accept | 0.1339 | 0.086394 | 0.12821 | 0.13216 | 0.00010824 | 0.023311 | | 10 | 6 | Accept | 0.5812 | 19.209 | 0.12821 | 0.13415 | 237.89 | 0.00010012 | | 11 | 6 | Accept | 0.14245 | 0.10978 | 0.12821 | 0.1376 | 0.00010041 | 0.022664 | | 12 | 6 | Accept | 0.17664 | 23.135 | 0.12821 | 0.1289 | 0.30037 | 0.0014962 | | 13 | 5 | Accept | 0.23647 | 22.508 | 0.12821 | 0.12809 | 876.57 | 0.012627 | | 14 | 5 | Accept | 0.35897 | 0.099327 | 0.12821 | 0.12809 | 0.0001005 | 988.9 | | 15 | 5 | Accept | 0.35897 | 0.083674 | 0.12821 | 0.12814 | 990.27 | 985.91 | | 16 | 6 | Best | 0.12251 | 0.13437 | 0.12251 | 0.12571 | 0.00093129 | 0.033266 | | 17 | 6 | Best | 0.11966 | 0.18898 | 0.11966 | 0.12289 | 0.0011252 | 0.011984 | | 18 | 6 | Accept | 0.1339 | 0.16574 | 0.11966 | 0.12306 | 0.0028325 | 0.036476 | | 19 | 6 | Best | 0.10541 | 0.29174 | 0.10541 | 0.1142 | 0.0022156 | 0.013997 | | 20 | 5 | Accept | 0.11681 | 3.4201 | 0.10541 | 0.1179 | 0.00019144 | 0.004792 | |===============================================================================================================| | Iter | Active | Eval | Objective | Objective | BestSoFar | BestSoFar | box | kern | | | workers | result | | runtime | (observed) | (estim.) | | | |===============================================================================================================| | 21 | 5 | Accept | 0.13675 | 0.21393 | 0.10541 | 0.1179 | 0.00081046 | 0.01568 | | 22 | 5 | Accept | 0.12821 | 0.49429 | 0.10541 | 0.12063 | 0.0031922 | 0.0095725 | | 23 | 6 | Accept | 0.1453 | 0.49944 | 0.10541 | 0.12027 | 0.038828 | 0.026798 | | 24 | 6 | Accept | 0.12251 | 0.44988 | 0.10541 | 0.12041 | 0.0034879 | 0.012684 | | 25 | 5 | Accept | 0.11111 | 1.7729 | 0.10256 | 0.11523 | 206.59 | 1.1184 | | 26 | 5 | Best | 0.10256 | 0.32152 | 0.10256 | 0.11523 | 0.00083077 | 0.0090062 | | 27 | 5 | Accept | 0.1339 | 0.094543 | 0.10256 | 0.1178 | 999.01 | 81.922 | | 28 | 6 | Accept | 0.11681 | 0.20018 | 0.10256 | 0.11785 | 987.2 | 13.241 | | 29 | 6 | Accept | 0.11681 | 0.14953 | 0.10256 | 0.11787 | 995.8 | 22.417 | | 30 | 6 | Accept | 0.11681 | 0.71489 | 0.10256 | 0.11786 | 962.75 | 4.0028 | __________________________________________________________ Optimization completed. MaxObjectiveEvaluations of 30 reached. Total function evaluations: 30 Total elapsed time: 38.2851 seconds. Total objective function evaluation time: 95.7048 Best observed feasible point: box kern __________ _________ 0.00083077 0.0090062 Observed objective function value = 0.10256 Estimated objective function value = 0.11786 Function evaluation time = 0.32152 Best estimated feasible point (according to models): box kern _________ ________ 0.0011252 0.011984 Estimated objective function value = 0.11786 Estimated function evaluation time = 0.28326

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
_________ ________
0.0011252 0.011984

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 = 0.0011

Mdl.KernelParameters.Scale

ans = 0.0120

`fun`

— Objective functionfunction handle |

`parallel.pool.Constant`

whose `Value`

is a function handleObjective function, specified as a function handle or, when the `UseParallel`

name-value pair is `true`

, a `parallel.pool.Constant`

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 descriptionsvector of

`optimizableVariable`

objects defining
the hyperparameters to be tunedVariable descriptions, specified as a vector of `optimizableVariable`

objects
defining the hyperparameters to be tuned.

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

, where `X1`

and `X2`

are `optimizableVariable`

objects

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside single quotes (`' '`

). You can
specify several name and value pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

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

`'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. For algorithm descriptions, see Acquisition Function Types.

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

**Data Types: **`char`

`'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 realPropensity 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 integerFit 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).
`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'`

**Data Types: **`char`

`'MinWorkerUtilization'`

— Tolerance on number of active parallel workers`floor(0.8*Nworkers)`

(default) | positive integerTolerance 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`

`'MaxObjectiveEvaluations'`

— Objective function evaluation limit`30`

(default) | positive integerObjective function evaluation limit, specified as a positive integer.

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

**Data Types: **`double`

`'NumSeedPoints'`

— Number of initial evaluation points`4`

(default) | positive integerNumber 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`

`'XConstraintFcn'`

— Deterministic constraints on variables`[]`

(default) | function handleDeterministic 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 handleConditional 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 integerNumber of coupled constraints, specified as a positive integer. For details, see Coupled Constraints.

`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 vectorIndication 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`

`'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 handlesFunction 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 vectorFile name for the `@saveToFile`

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

.

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

**Data Types: **`char`

`'SaveVariableName'`

— Variable name for the `@assignInBase`

output function`'BayesoptResults'`

(default) | character vectorVariable name for the `@assignInBase`

output
function, specified as a character vector.

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

**Data Types: **`char`

`'PlotFcn'`

— Plot function called after each iteration`{@plotObjectiveModel,@plotMinObjective}`

(default) | `'all'`

| function handle | cell array of function handlesPlot 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. Also plots 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.

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`

| `cell`

| `function_handle`

`'InitialX'`

— Initial evaluation points`NumSeedPoints`

-by-`D`

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

-by-`D`

tableInitial evaluation points, specified as an `N`

-by-`D`

table,
where `N`

is the number of evaluation points, and `D`

is
the number of variables.

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`

vectorObjective 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`

matrixConstraint 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 vectorInitial 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`

vectorEvaluation 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`

vectorTimes 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`

`results`

— Bayesian optimization results`BayesianOptimization`

objectBayesian optimization results, returned as a `BayesianOptimization`

object.

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

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