cvxoptdes

Example gallery

Sun, 18 Oct 2026 00:00:00 +0000

Optimization

Volcano height dataset

The volcano dataset included in the datasets R-package reports topographic information of the Maunga Whau (Mt Eden) volcano in the Auckland volcano field. To reduce the data grid size, we first downsample the volcano data, resulting in volcano heights across a $(44 \times 31)$ -dimensional matrix with rows corresponding to grid lines running east to west and columns to grid lines running south to north.

## initialize data
data(volcano)

## subsample grid
volcano <- volcano[c(TRUE, FALSE), c(TRUE, FALSE)]

data <- expand.grid(
 x1 = seq(from = -1, to = 1, length = dim(volcano)[1]),
 x2 = seq(from = -1, to = 1, length = dim(volcano)[2])
)
## standardize response
data[["y"]] <- (c(volcano) - min(volcano)) / (max(volcano) - min(volcano))

The irregular height response surface is not easily captured by a parametric linear or nonlinear model. Instead, a nonparametric regression model, such as a generalized additive model (GAM) or a Gaussian process regression (GPR) model, is better suited, allowing the complexity of the model to grow with the number of observations. In vignette("glm_gam_design"), the use of a GAM to model the volcano height was already discussed. In this example, we consider penalized design augmentation subject to a GPR prediction model, which is expected to behave similar to (multi-proposal) Bayesian Optimization. The objective is to maximize the observed height response, where the observed height at a specific grid point is assumed to be noise-free.

Generalized linear/additive model optimal design

Sun, 18 Oct 2026 00:00:00 +0000

Introduction

In vignette("lm_design"), the use of the lm_design R6-class is demonstrated to construct optimal designs for linear models. This vignette introduces the glm_design and gam_design R6-classes to calculate approximate and exact optimal designs for generalized linear and generalized additive models.

The methods available in the glm_design and gam_design classes are identical to those of the lm_design class, and it is recommended to read vignette("lm_design") before going through this vignette. The principal difference between optimal experiment design for generalized linear/additive models and linear models is that the information matrix of a generalized linear model is usually not independent of the model parameters $\boldsymbol{\theta}$ , similar to the information matrix of a nonlinear model, as explained in vignette("nlm_design").

Linear model optimal design

Sun, 18 Oct 2026 00:00:00 +0000

Getting started

Optimal designs for linear models are calculated using the R6-class lm_design. When initializing a design object with lm_design$new(), the minimal required inputs are a (discrete) candidate design grid and a one-sided linear model formula:

library(cvxoptdes)

## 3 factors w/ 5, 5 and 3 levels
data <- expand.grid(
 x1 = -2:2,
 x2 = -2:2,
 x3 = -1:1
)

design <- lm_design$new(
 formula = ~(x1 + x2 + x3)^2 + I(x1^2) + I(x2^2) + I(x3^2), ## full quadratic model
 data = data ## design points
)

design$data
#> x1 x2 x3
#> 1 -2 -2 -1
#> 2 -1 -2 -1
#> 3 0 -2 -1
#> 4 1 -2 -1
#> 5 2 -2 -1
#> 6 -2 -1 -1
#> 7 -1 -1 -1
#> 8 0 -1 -1
#> 9 1 -1 -1
#> 10 2 -1 -1
....

After initializing the linear model and design grid, use the $optimize() method to calculate an approximate optimal design:

Nonlinear/Linear mixed model optimal design

Sun, 18 Oct 2026 00:00:00 +0000

Nonlinear model optimal design

In vignette("lm_design"), the use of the lm_design R6-class is demonstrated to construct optimal designs for linear models. This vignette introduces the nlm_design and lme_design R6-classes to calculate approximate and exact optimal designs for nonlinear models and random intercept linear mixed models.

The methods available in the nlm_design class are largely equivalent to those in the lm_design class. The principal difference is that the information matrix of a nonlinear model $f$ is not independent of the model parameters $\boldsymbol{\theta}$ . Denote $F(\boldsymbol{x}_i, \boldsymbol{\theta})$ for the gradient vector of the nonlinear model $f$ with respect to the model parameters $\boldsymbol{\theta}$ evaluated at $\boldsymbol{x}_i$ , i.e. the $i$ -th row of the Jacobian matrix. Then, conditional on the error variance $\sigma_e^2$ , the $(p \times p)$ -information matrix is:

Optimal design augmentation

Sun, 18 Oct 2026 00:00:00 +0000

Introduction

In vignette("lm_design"), the implicit assumption is that the optimal design is calculated before observing the data. In practice, experimentation may take place in multiple consecutive steps, in which case it is desirable to construct a new augmented optimal design $\boldsymbol{w}^*$ given an initial (approximate or exact) design $\boldsymbol{w}_0 = (w_1^{(0)}, \ldots, w_{n}^{(0)})$ . Optimal design augmentation is supported by the $augment() method, which is demonstrated through several examples in this vignette.

ethanol

Mon, 01 Jan 0001 00:00:00 +0000

Description

This dataset reports 88 measurements from an experiment in which ethanol was burned in a single cylinder automobile test engine. The data originates from the SemiPar package, which is now archived.

Usage

ethanol

Format

A data.frame containing 88 measurements with columns:

NOx: numeric, concentration of nitric oxide (NO) and nitrogen dioxide (NO2) in engine exhaust, normalized by the work done by the engine
C: numeric, the compression ratio of the engine
E: numeric, the equivalence ratio at which the engine was run – a measure of the richness of the air/ethanol mix

Source

Brinkman, N.D. (1981). Ethanol fuel – a single-cylinder engine study of efficiency and exhaust emissions. SAE transactions Vol. 90, No 810345, 1410–1424.

fullerenes

Mon, 01 Jan 0001 00:00:00 +0000

Description

This dataset reports the production of o-xylenyl adducts of Buckminster fullerenes. Three process conditions (temperature, reaction time and ratio of sultine to C60) are varied to maximize the mole fraction of the desired ’ product.

Usage

fullerenes

Format

A data.frame containing 120 experiments with columns:

reaction_time: numeric, reaction time in flow reactor (min)
sultine_amount: numeric, relative concentration of sultine to C60
temperature: numeric, temperature of the reaction (Celsius)
product_amount: numeric, mole fraction of the desired product

Source

Olympus: Fullerenes dataset

gam_design

Mon, 01 Jan 0001 00:00:00 +0000

Description

An R6 class for managing penalized locally optimal experiment designs for generalized additive models. This class encapsulates the generalized additive model at a set of nominal parameter values, the discrete design grid, optimization of an approximate continuous design, augmentation of an existing design, and optimal rounding to exact integer designs.

Optimal design problem

The following penalized optimization problem is solved by $optimize():

w^* \ = \ \arg \max_{w} \Phi(M(w | X, v, \theta, F, g) + \lambda I) - \alpha \cdot c^T w

where,

glm_design

Mon, 01 Jan 0001 00:00:00 +0000

Description

An R6 class for managing penalized locally optimal experiment designs for generalized linear models. This class encapsulates the generalized linear model at a set of nominal parameter values, the discrete design grid, optimization of an approximate continuous design, augmentation of an existing design, and optimal rounding to exact integer designs.

Optimal design problem

The following penalized optimization problem is solved by $optimize():

w^* \ = \ \arg \max_{w} \Phi(M(w | X, v, \theta, F, g) + \lambda I) - \alpha \cdot c^T w

where,

imidazoles

Mon, 01 Jan 0001 00:00:00 +0000

Description

This dataset reports the yield of the desired product and the reaction cost in a direct arylation reaction of imidazoles. Two continuous process conditions (temperature, initial concentration) and three reaction molecules (base, ligand, solvent) are varied to maximize the yield of the desired product while minimizing the cost of the reaction.

Usage

imidazoles

Format

A data.frame containing 1728 experiments with columns:

yield: numeric, experimental yield of the reaction (%)
cost: numeric, cost of the reaction scaled to the unit interval
concentration: numeric, initial substrate concentration (M)
temperature: numeric, temperature of the reaction (Celsius)
base: character, common name of the reaction base molecule
base_smiles: character, SMILES string of the reaction base molecule
solvent: character, common name of the reaction solvent molecule
solvent_smiles: character, SMILES string of the reaction solvent molecule
ligand: character, common name of the reaction ligand molecule
ligand_smiles: character, SMILES string of the reaction ligand molecule

Source

References

Shields B.J. et al. (2021) “Bayesian reaction optimization as a tool for chemical synthesis”, Nature, Vol. 590 (7844). Torres J.A.G. et al. (2022) “A Multi-Objective Active Learning Platform and Web App for Reaction Optimization”, J. Am. Chem. Soc., Vol. 144 (43).

lm_design

Mon, 01 Jan 0001 00:00:00 +0000

Description

An R6 class for managing penalized optimal experiment designs for linear models. This class encapsulates the linear model formula, the discrete design grid, optimization of an approximate continuous design, augmentation of an existing design, and optimal rounding to exact integer designs.

Optimal design problem

The following penalized optimization problem is solved by $optimize():

w^* \ = \ \arg \max_{w} \Phi(M(w | X, v) + \lambda I) - \alpha \cdot c^T w

where,

lme_design

Mon, 01 Jan 0001 00:00:00 +0000

Description

An R6 class for managing penalized optimal experiment designs for random intercept linear mixed models. This class encapsulates a random intercept linear mixed model, the discrete design grid, optimization of an approximate continuous design, augmentation of an existing design, and optimal rounding to exact integer designs including split-plot designs.

Optimal design problem

The following penalized optimization problem is solved by $optimize():

w^* \ = \ \arg \max_{w} \Phi(M(w | X, Z, \eta, v) + \lambda I) - \alpha \cdot c^T w

where,

nlm_design

Mon, 01 Jan 0001 00:00:00 +0000

Description

An R6 class for managing penalized locally optimal experiment designs for nonlinear models. This class encapsulates the nonlinear model at a set of nominal parameter values, the discrete design grid, optimization of an approximate continuous design, augmentation of an existing design, and optimal rounding to exact integer designs.

Optimal design problem

The following penalized optimization problem is solved by $optimize():

w^* \ = \ \arg \max_{w} \Phi(M(w | X, v, \theta) + \lambda I) - \alpha \cdot c^T w

where,

scs_control_dflt

Mon, 01 Jan 0001 00:00:00 +0000

Description

Allow the user to tune the parameters of the SCS solver procedure.

Usage

scs_control_dflt(
 maxiter = 1e+05,
 eps_rel = 1e-05,
 eps_abs = 1e-05,
 eps_infeas = 1e-07,
 alpha_relax = 1.5,
 rho_x = 1e-06,
 scale = 0.1,
 normalize = TRUE,
 acceleration_lookback = 10,
 acceleration_interval = 5,
 adaptive_scale = TRUE,
 adaptive_diag_scale = TRUE,
 time_limit_secs = 60
)

Arguments

maxiter: positive integer, maximum number of iterations to run.
eps_rel: positive numeric, relative feasibility tolerance.
eps_abs: positive numeric, absolute feasibility tolerance.
eps_infeas: positive numeric, infeasibility tolerance (primal and dual).
alpha_relax: numeric value in (0, 2), douglas-Rachford relaxation parameter.
rho_x: positive numeric, primal scale factor.
scale: positive numeric, initial dual scale factor, updated if adaptive_scale is TRUE.
normalize: logical, whether to perform heuristic data rescaling.
acceleration_lookback: positive integer, relative amount of memory used for Anderson acceleration. More memory requires more time to compute but can give more reliable steps. If acceleration_lookback < 1, Anderson acceleration is disabled.
acceleration_interval: positive integer, run Anderson acceleration every acceleration_interval iteration(s).
adaptive_scale: logical, whether to heuristically adapt dual scale through the solve.
adaptive_diag_scale: logical, whether to refine the metric R per row, using the primal residual profile. Requires adaptive_scale (silently disabled without it).
time_limit_secs: positive numeric, time limit for solve run in seconds (can be fractional). The value 0 is interpreted as no limit. The default is 60.

cvxoptdes

Example gallery

Optimization

Volcano height dataset

Generalized linear/additive model optimal design

Introduction

Linear model optimal design

Getting started

Nonlinear/Linear mixed model optimal design

Nonlinear model optimal design

Optimal design augmentation

Introduction

ethanol

Description

Usage

Format

Source

fullerenes

Description

Usage

Format

Source

gam_design

Description

Optimal design problem

glm_design

Description

Optimal design problem

imidazoles

Description

Usage

Format

Source

References

lm_design

Description

Optimal design problem

lme_design

Description

Optimal design problem

nlm_design

Description

Optimal design problem

scs_control_dflt

Description

Usage

Arguments

Seealso