# Options¶

## Strategy¶

We provide implementations of Stochastic RBF (SRBF), DYCORS, Expected Improvement (EI), lower confidence bound (LCB) and random search (RS). EI can only be used in combination with GPRegressor since uncertainty predictions are necessary. All strategies support running in serial, batch synchronous parallel, and asynchronous parallel.

New optimization strategies can be implemented by inheriting from SurrogateBaseStrategy and implementing the abstract generate_evals method that proposes num_pts new sample points:

• Required methods
• generate_evals(num_pts): Proposes num_pts new samples.

The following strategies are currently supported:

### SRBFStrategy¶

This is an implementation of the SRBF strategy by Regis and Shoemaker:

Rommel G Regis and Christine A Shoemaker.
A stochastic radial basis function method for the global optimization of expensive functions.
INFORMS Journal on Computing, 19(4): 497-509, 2007.
Rommel G Regis and Christine A Shoemaker.
Parallel stochastic global optimization using radial basis functions.
INFORMS Journal on Computing, 21(3):411-426, 2009.

The main idea is to pick the new evaluations from a set of candidate points where each candidate point is generated as an N(0, sigma^2) distributed perturbation from the current best solution. The value of sigma is modified based on progress and follows the same logic as in many trust region methods; we increase sigma if we make a lot of progress (the surrogate is accurate) and decrease sigma when we aren’t able to make progress (the surrogate model is inaccurate). More details about how sigma is updated is given in the original papers.

After generating the candidate points we predict their objective function value and compute the minimum distance to previously evaluated point. Let the candidate points be denoted by C and let the function value predictions be s(x_i) and the distance values be d(x_i), both rescaled through a linear transformation to the interval [0,1]. This is done to put the values on the same scale. The next point selected for evaluation is the candidate point x that minimizes the weighted-distance merit function:

$\text{merit}(x) := w s(x) + (1 - w) (1 - d(x))$

where $$0 \leq w \leq 1$$. That is, we want a small function value prediction and a large minimum distance from previously evalauted points. The weight w is commonly cycled between a few values to achieve both exploitation and exploration. When w is close to zero we do pure exploration while w close to 1 corresponds to explotation.

• Parameters:
• max_evals: Evaluation budget (int)
• opt_prob: Optimization problem object, must implement OptimizationProblem
• exp_design: Experimental design object, must implement ExperimentalDesign
• surrogate: Surrogate object, must implement Surrogate
• asynchronous: Whether or not to use asynchrony (True / False).
• batch_size: Size of the batch. This value is ignored if asynchronous is True. Use 1 for serial or run with asynchronous set to True.
• extra_points: n Extra points to add to the experimental design (numpy.array of size n x dim)
• extra_vals: Values for extra_points. Set elements to np.nan if unknown (numpy.array of size n x 1)
• reset_surrogate: Specify whether or not we are resetting the surrogate model i.e., removing current points (True / False)
• weights: Weights for merit function (list or numpy.array). Default is [0.3, 0.5, 0.8, 0.95]
• num_cand: Number of candidate points (int). Default = 100*dim

### DYCORStrategy¶

This is an implementation of the DYCORS strategy by Regis and Shoemaker:

Rommel G Regis and Christine A Shoemaker.
Combining radial basis function surrogates and dynamic coordinate search in high-dimensional expensive black-box optimization.
Engineering Optimization, 45(5): 529-555, 2013.

This is an extension of the SRBF strategy that changes how the candidate points are generated. The main idea is that many objective functions depend only on a few directions so it may be advantageous to perturb only a few directions. In particular, we use a perturbation probability to perturb a given coordinate and decrease this probability after each function evaluation so fewer coordinates are perturbed later in the optimization.

The parameters are the same as in the SRBF strategy.

### SOPStrategy¶

This is an implementation of the SOP strategy by Krityakierne, Akhtar and Shoemaker:

Tipaluck Krityakierne, Taimoor Akhtar and Christine A. Shoemaker.
SOP: parallel surrogate global optimization with Pareto center selection for computationally expensive single objective problems.
Journal of Global Optimization, 66(3): 417-437, 2016.

The core idea of SOP is to maintain a ranked archive of all previously evaluated points, as per non-dominated sorting between two objectives, i.e., i) Objective function value(minimize) and ii) Minimum distance from other evaluated points(maximize). A sub-archive of center points is subsequently maintained via selection from the ranked evaluated points. The number of points in the sub-archive of centers should be equal to (or greater than) the number of parallel threads. Candidate points are generated around each ‘center point’ via the DYCORS sampling strategy, i.e., an N(0, sigma^2) distributed perturbation of a subset of decision variables. A separate value of sigma is maintained for each center point, where sigma is decreased if no progress is registered in the bi-objective objective value and distance criterion trade-off. One point is selected for expensive evaluation from each set of candidate points, based on the surrogate approximation only. Hence the merit function is s(x), where s(x) is the surrogate prediction.

Exploration and exploitation are simultaneously achieved (in parallel) via the bi-objective ranking of previously evaluated points, and subsequent selection of these points as centers of DYCORS perturbations. Exploitation is achieved when the point with best objective value is the perturbation center, and the candidate around it with best surrogate value is selected as the new evaluation point. Exploration is achieved when the point with the maximum distance (max-min) from other evaluated points is selected as the perturbation center.

Parameters are the same as in SRBF strategy, but exclude weights, and include the following:
• ncenters: Number of center points for candidate search where one point is selected for evaluation per, each center.

### EIStrategy¶

This is an implementation of Expected Improvement (EI), arguably the most popular acquisition function in Bayesian optimization. Under a Gaussian process (GP) prior, the expected value of the improvement:

\begin{split}\begin{align*} \text{I}(x) &:= \max(f_{\text{best}} - f(x), 0) \\ \text{EI}[x] &:= \mathbb{E}[I(x)] \end{align*}\end{split}

can be computed analytically, where f_best is the best observed function value.EI is one-step optimal in the sense that selecting the maximizer of EI is the optimal action if we have exactly one function value remaining and must return a solution with a known function value.

When using parallelism, we constrain each new evaluation to be a distance dtol away from previous and pending evaluations to avoid that the same point is being evaluated multiple times. We use a default value of dtol = 1e-3 * norm(ub - lb), but note that this value has not been tuned carefully and may be far from optimal.

The optimization strategy terminates when the evaluatio budget has been exceeded or when the EI of the next point falls below some threshold, where the default threshold is 1e-6 * (max(fX) - min(fX)).

• Parameters:
• max_evals: Evaluation budget (int)
• opt_prob: Optimization problem object, must implement OptimizationProblem
• exp_design: Experimental design object, must implement ExperimentalDesign
• surrogate: Surrogate object, must implement Surrogate
• asynchronous: Whether or not to use asynchrony (True / False).
• batch_size: Size of the batch. This value is ignored if asynchronous is True. Use 1 for serial or run with asynchronous set to True.
• extra_points: n Extra points to add to the experimental design (numpy.array of size n x dim)
• extra_vals: Values for extra_points. Set elements to np.nan if unknown (numpy.array of size n x 1)
• reset_surrogate: Specify whether or not we are resetting the surrogate model i.e., removing current points (True / False)
• ei_tol: Terminate if the largest EI falls below this threshold (float). Default: 1e-6 * (max(fX) - min(fX))
• dtol: Minimum distance between new and pending/finished evaluations (float). Default: 1e-3 * norm(ub - lb)

### LCBStrategy¶

This is an implementation of Lower Confidence Bound (LCB), a popular acquisition function in Bayesian optimization. The main idea is to minimize:

$\text{LCB}(x) := \mathbb{E}[x] - \kappa * \sqrt{\mathbb{V}[x]}$

where $$\mathbb{E}[x]$$ is the predicted function value, $$V[x]$$ is the predicted variance, and kappa is a constant that balances exploration and exploitation. We use a default value of kappa = 2.

When using parallelism, we constrain each new evaluation to be a distance dtol away from previous and pending evaluations to avoid that the same point is being evaluated multiple times. We use a default value of dtol = 1e-3 * norm(ub - lb), but note that this value has not been tuned carefully and may be far from optimal.

The optimization strategy terminates when the evaluatio budget has been exceeded or when the LCB of the next point falls below some threshold, where the default threshold is 1e-6 * (max(fX) - min(fX)).

• Parameters:
• max_evals: Evaluation budget (int)
• opt_prob: Optimization problem object, must implement OptimizationProblem
• exp_design: Experimental design object, must implement ExperimentalDesign
• surrogate: Surrogate object, must implement Surrogate
• asynchronous: Whether or not to use asynchrony (True / False).
• batch_size: Size of the batch. This value is ignored if asynchronous is True. Use 1 for serial or run with asynchronous set to True.
• extra_points: n Extra points to add to the experimental design (numpy.array of size n x dim)
• extra_vals: Values for extra_points. Set elements to np.nan if unknown (numpy.array of size n x 1)
• reset_surrogate: Specify whether or not we are resetting the surrogate model i.e., removing current points (True / False)
• kappa: Constant in the LCB merit function (float). Default: 2.0
• lcb_tol: Terminate if min(fX) - min(LCB(x)) < lcb_tol (float). Default: 1e-6 * (max(fX) - min(fX))
• dtol: Minimum distance between new and pending/finished evaluations (float). Default: 1e-3 * norm(ub - lb)

## Experimental design¶

The experimental design generates the initial points to be evaluated. A well-chosen experimental design is critical in order to fit a surrogate model that captures the behavior of the underlying objective function. Any implementation must have the following attributes and method:

• Attributes:
• dim: Dimensionality
• num_pts: Number of points in the design
• Required methods
• generate_points(lb, ub, int_var): Returns an experimental design of size num_pts x dim where num_pts is the number of points in the initial design, which was specified when the object was created. You can supply lb, ub, and int_var to have the design mapped before it’s scored instead of having the rounding take place in the strategy.

The following experimental designs are supported:

### LatinHypercube¶

A Latin hypercube design

• Parameters:
• dim: Number of dimensions (int).
• num_pts: Number of desired sampling points (int).
• iterations: Number of designs to generate and choose the best from (int)

Example:

from pySOT.experimental_design import LatinHypercube
lhd = LatinHypercube(dim=3, num_pts=10)


creates a Latin hypercube design with 10 points in 3 dimensions

### SymmetricLatinHypercube¶

A symmetric Latin hypercube design

• Parameters:
• dim: Number of dimensions (int).
• num_pts: Number of desired sampling points (int). Use 2*dim + 1 to make sure the design has full rank.
• iterations: Number of designs to generate and choose the best from (int)

Example:

from pySOT.experimental_design import SymmetricLatinHypercube
slhd = SymmetricLatinHypercube(dim=3, num_pts=10)


creates a symmetric Latin hypercube design with 10 points in 3 dimensions

### TwoFactorial¶

The corners of the unit hypercube

• Parameters:
• dim: Number of dimensions (int).

Example:

from pySOT.experimental_design import TwoFactorial
two_factorial = TwoFactorial(dim=3)


creates a two factorial design with 8 points in 3 dimensions

## Surrogate model¶

The surrogate model approximates the underlying objective function given all of the points that have been evaluated. Any implementation of a surrogate model must have the following attributes and methods

• Attributes:
• dim: Number of dimensions
• lb: Lower variable bounds
• ub: Upper variable bounds
• output_transformation: Transformation to apply to function values before fitting (for example median capping)
• num_pts: Number of points in the surrogate model
• X: Data points, of size num_pts x dim, currently incorporated in the model
• _X: Data points scaled to the unit hypercube. We use these internally to for conditioning reasons
• fX: Function values at the data points
• updated: True if all information is incorporated in the model, else a new fit will be triggered
• Required methods
• reset(): Resets the surrogate model
• add_points(x, fx): Adds point(s) x with value(s) fx to the surrogate model. This SHOULD NOT trigger a new fit of the model.
• predict(x): Evaluates the surrogate model at points x
• predict_deriv(x): Evaluates the derivative of surrogate model at points x
• Optional methods
• predict_std(x): Evaluates the uncertainty of the surrogate model at points x

The following surrogate models are supported:

### RBFInterpolant¶

A radial basis function (RBF) takes the form:

$s(x) = \sum_j c_j \phi(\|x-x_j\|) + \sum_j \lambda_j p_j(x)$

where the functions $$p_j(x)$$ are low-degree polynomials. The fitting equations are

$\begin{split}\begin{bmatrix} \eta I & P^T \\ P & \Phi + \eta I \end{bmatrix} \begin{bmatrix} \lambda \\ c \end{bmatrix} = \begin{bmatrix} 0 \\ f \end{bmatrix}\end{split}$

where $$P_{ij} = p_j(x_i)$$ and $$\Phi_{ij}=\phi(\|x_i-x_j\|)$$ The regularization parameter $$\eta$$ allows us to avoid problems with potential poor conditioning of the system. Consider using the SurrogateUnitBox wrapper or manually scaling the domain to the unit hypercube to avoid issues with the domain scaling.

We add k new points to the RBFInterpolant in $$O(kn^2)$$ flops by updating the LU factorization of the old RBF system. This is better than computing the RBF coefficients from scratch, which costs $$O(n^3)$$ flops.

• Parameters:
• dim: Number of dimensions (int)
• lb: Lower variable bounds (numpy.array)
• ub: Upper variable bounds (numpy.array)
• output_transformation: Transformation to apply to function values before fitting (callable)
• kernel: RBF kernel object, must implement Kernel. Default: CubicKernel()
• tail: RBF polynomial tail object, must implement Tail. Default: LinearTail(dim)
• eta: Regularization parameter. Use something small like 1e-6 if the domain is [0, 1]^dim

Example:

from pySOT.surrogate import RBFInterpolant, CubicKernel, LinearTail
lb, ub = np.zeros(5), np.ones(5)  # Domain is [0, 1]^5
rbf = RBFInterpolant(dim=5, lb=lb, ub=ub, kernel=CubicKernel(), tail=LinearTail(dim=dim))


creates a cubic RBF with a linear tail in dim dimensions.

Example:

from pySOT.surrogate import RBFInterpolant, CubicKernel, LinearTail, median_capping
lb, ub = np.zeros(5), np.ones(5)  # Domain is [0, 1]^5
rbf = RBFInterpolant(
dim=5, lb=lb, ub=ub, output_transformation=median_capping, kernel=CubicKernel(), tail=LinearTail(dim=5))


will apply median capping (replace values above median by the median) to the function values before fitting. This is useful for minimization problems where we do not want large values to influence the fit of the model.

### GPRegressor¶

Generate a Gaussian process regression object. This is just a wrapper around the GPRegressor in scikit-learn.

• Parameters:
• dim: Number of dimensions (int)
• lb: Lower variable bounds (numpy.array)
• ub: Upper variable bounds (numpy.array)
• output_transformation: Transformation to apply to function values before fitting (callable)
• gp: GPRegressor model in scikit-learn. Uses the SE/RBF/Gaussian kernel as a default if None is passed.
• n_restarts_optimizer: Number of restarts in hyperparamater fitting (int)

Example:

from pySOT.surrogate import GPRegressor
lb, ub = np.zeros(5), np.ones(5)  # Domain is [0, 1]^5
gp = GPRegressor(dim=5, lb=lb, ub=ub)


creates a GPRegressor object in dim dimensions.

### MARSInterpolant¶

Generate a Multivariate Adaptive Regression Splines (MARS) model.

$\hat{f}(x) = \sum_{i=1}^{k} c_i B_i(x).$

The model is a weighted sum of basis functions $$B_i(x)$$. Each basis function $$B_i(x)$$ takes one of the following three forms:

1. A constant 1.
2. A hinge function of the form $$\max(0, x - const)$$ or $$\max(0, const - x)$$. MARS automatically selects variables and values of those variables for knots of the hinge functions.
3. A product of two or more hinge functions. These basis functions c an model interaction between two or more variables.
• Parameters:
• dim: Number of dimensions (int)
• lb: Lower variable bounds (numpy.array)
• ub: Upper variable bounds (numpy.array)
• output_transformation: Transformation to apply to function values before fitting (callable)

Note

This implementation depends on the py-earth module (see Dependencies)

Example:

from pySOT.surrogate import MARSInterpolant
lb, ub = np.zeros(5), np.ones(5)  # Domain is [0, 1]^5
mars = MARSInterpolant(dim=5, lb=lb, ub=ub)


creates a MARS interpolant in dim dimensions.

### PolyRegressor¶

Multivariate polynomial regression with cross-terms. This is just a wrapper around PolynomialFeatures in scikit-learn.

• Parameters:
• dim: Number of dimensions (int)
• lb: Lower variable bounds (numpy.array)
• ub: Upper variable bounds (numpy.array)
• output_transformation: Transformation to apply to function values before fitting (callable)
• degree: Polynomial degree (int)

Example:

from pySOT.surrogate import PolyRegressor
lb, ub = np.zeros(5), np.ones(5)  # Domain is [0, 1]^5
poly = PolyRegressor(dim=5, lb=lb, ub=ub, degree=2)


creates a polynomial regressor of degree 2.

## Optimization problem¶

The optimization problem is its own object and must have certain attributes and methods in order to work with the framework. The following attributes and methods must always be specified in the optimization problem class:

• Attributes
• lb: Lower bounds for the variables.
• ub: Upper bounds for the variables.
• dim: Number of dimensions
• int_var: Specifies the integer variables. If no variables have discrete, set to []
• cont_var: Specifies the continuous variables. If no variables are continuous, set to []
• Required methods
• eval: Takes one input in the form of an numpy.ndarray with shape (1, dim), which corresponds to one point in dim dimensions. Returns the value (a scalar) of the objective function at this point.