Uses a surrogate modeled augmented Lagrangian (AL) system to optimize an objective function (blackbox or known and linear) under unknown multiple (blackbox) constraints via expected improvement (EI) and variations; a comparator based on EI with constraints is also provided

```
optim.auglag(fn, B, fhat = FALSE, equal = FALSE, ethresh = 1e-2,
slack = FALSE, cknown = NULL, start = 10, end = 100,
Xstart = NULL, sep = TRUE, ab = c(3/2, 8), lambda = 1, rho = NULL,
urate = 10, ncandf = function(t) { t }, dg.start = c(0.1, 1e-06),
dlim = sqrt(ncol(B)) * c(1/100, 10), Bscale = 1, ey.tol = 1e-2,
N = 1000, plotprog = FALSE, verb = 2, ...)
optim.efi(fn, B, fhat = FALSE, cknown = NULL, start = 10, end = 100,
Xstart = NULL, sep = TRUE, ab = c(3/2,8), urate = 10,
ncandf = function(t) { t }, dg.start = c(0.1, 1e-6),
dlim = sqrt(ncol(B))*c(1/100,10), Bscale = 1, plotprog = FALSE,
verb = 2, ...)
```

fn

function of an input (`x`

), facilitating vectorization on a
`matrix`

`X`

thereof, returning a `list`

with elements `$obj`

containing the (scalar) objective value and `$c`

containing a vector of evaluations of the (multiple) constraint function at `x`

.
The `fn`

function must take a `known.only`

argument which is explained
in the note below; it need not act on that argument

B

2-column `matrix`

describing the bounding box. The number of rows
of the `matrix`

determines the input dimension
(`length(x)`

in `fn(x)`

); the first column gives
lower bounds and the second gives upper bounds

fhat

a scalar logical indicating if the objective function should
be modeled with a GP surrogate. The default of `FALSE`

assumes a known
linear objective scaled by `Bscale`

. Using `TRUE`

is an “alpha”
feature at this time

equal

an optional vector containing zeros and ones, whose length equals the number of
constraints, specifying which should be treated as equality constraints (`0`

) and
which as inequality (`1`

)

ethresh

a threshold used for equality constraints to determine validity for progress measures; ignored if there are no equality constraints

slack

A scalar logical indicating if slack variables, and thus exact EI
calculations should be used. The default of `slack = FALSE`

results in Monte
Carlo EI approximation.
One can optionally specify `slack = 2`

to get the `slack = TRUE`

behavior,
with a second-stage L-BFGS-B optimization of the EI acquisition applied at the end,
starting from the best value found on the random search grid

cknown

A optional positive integer vector specifying which of the constraint
values returned by `fn`

should be treated as “known”, i.e., not modeled
with Gaussian processes

start

positive integer giving the number of random starting locations before
sequential design (for optimization) is performed; `start >= 6`

is
recommended unless `Xstart`

is non-`NULL`

; in the current version
the starting locations come from a space-filling design via `dopt.gp`

end

positive integer giving the total number of evaluations/trials in the
optimization; must have `end > start`

Xstart

optional matrix of starting design locations in lieu of, or in addition to,
`start`

random ones; we recommend `nrow(Xstart) + start >= 6`

; also must
have `ncol(Xstart) = nrow(B)`

sep

The default `sep = TRUE`

uses separable GPs (i.e., via `newGPsep`

, etc.)
to model the constraints and objective; otherwise the isotropic GPs are used

ab

prior parameters; see `darg`

describing the prior used on the
lengthscale parameter during emulation(s) for the constraints

lambda

`m`

-dimensional initial Lagrange multiplier parameter for `m`

-constraints

rho

positive scalar initial quadratic penalty parameter in the augmented Lagrangian; the default setting of `rho = NULL`

causes an automatic starting value to be chosen; see rejoinder to Gramacy, et al. (2016) or supplementary material to Picheny, et al. (2016)

urate

positive integer indicating how many optimization trials should pass before each MLE/MAP update is performed for GP correlation lengthscale parameter(s)

ncandf

function taking a single integer indicating the optimization trial number `t`

, where
`start < t <= end`

, and returning the number of search candidates (e.g., for
expected improvement calculations) at round `t`

; the default setting
allows the number of candidates to grow linearly with `t`

dg.start

2-vector giving starting values for the lengthscale and nugget parameters of the GP surrogate model(s) for constraints

dlim

2-vector giving bounds for the lengthscale parameter(s) under MLE/MAP inference,
thereby augmenting the prior specification in `ab`

Bscale

scalar indicating the relationship between the sum of the inputs, `sum(x)`

,
to `fn`

and the output `fn(x)$obj`

; note that at this time only linear
objectives are fully supported by the code - more details below

ey.tol

a scalar proportion indicating how many of the EIs
at `ncandf(t)`

candidate locations must be non-zero to “trust”
that metric to guide search, reducing to an EY-based search instead
[choosing that proportion to be one forces EY-based search]

N

positive scalar integer indicating the number of Monte Carlo samples to be used for calculating EI and EY

plotprog

`logical`

indicating if progress plots should be made after each inner iteration;
the plots show three panels tracking the best valid objective, the EI or EY surface
over the first two input variables (requires `interp`

,
and the parameters of the lengthscale(s) of the GP(s) respectively. When
`plotprog = TRUE`

the `interp.loess`

function is used to
aid in creating surface plots, however this does not work well with fewer than
fifteen points. You may also provide a function as an argument, having similar
arguments/formals as `interp.loess`

. For example, we use
`interp`

below, which would have been the default if not
for licensing incompatibilities

verb

a non-negative integer indicating the verbosity level; the larger the value the more that is printed to the screen

...

additional arguments passed to `fn`

The output is a `list`

summarizing the progress of the evaluations of the
blackbox under optimization

vector giving the best valid (`c(x) < 0`

) value
of the objective over the trials

vector giving the value of the objective for the input under consideration at each trial

`matrix`

giving the input values at which the blackbox function was
evaluated

`matrix`

giving the value of the constraint function for the input
under consideration at each trial

`matrix`

of lengthscale values obtained at the final update of the
GP emulator for each constraint

if `fhat = TRUE`

then this is a `matrix`

of lengthscale values
for the objective obtained at the final update of the GP emulator

a `matrix`

containing `lambda`

vectors used in each “outer
loop” AL iteration

a vector of `rho`

values used in each “outer loop” AL iteration

These subroutines support a suite of methods used to optimize challenging constrained problems from Gramacy, et al. (2016); and from Picheny, et al., (2016) see references below.

Those schemes hybridize Gaussian process based surrogate modeling and expected
improvement (EI; Jones, et., al, 2008) with the additive penalty method (APM)
implemented by the augmented Lagrangian (AL, e.g., Nocedal & Wright, 2006).
The goal is to minimize a (possibly known) linear objective function `f(x)`

under
multiple, unknown (blackbox) constraint functions satisfying `c(x) <= 0`

,
which is vector-valued. The solution here emulates the components of `c`

with Gaussian process surrogates, and guides optimization by searching the
posterior mean surface, or the EI of, the following composite objective
$$
Y(x) = f(x) + \lambda^\top Y_c(x) + \frac{1}{2\rho} \sum_{i=1}^m
\max(0, Y_{c_i}(x))^2,
$$
where \(\lambda\) and \(\rho\) follow updating equations that
guarantee convergence to a valid solution minimizing the objective. For more
details, see Gramacy, et al. (2016).

A slack variable implementation that allows for exact EI calculations and can accommodate equality constraints, and mixed (equality and inequality) constraints, is also provided. For further details, see Picheny, et al. (2016).

The example below illustrates a variation on the toy problem considered in both papers, which bases sampling on EI. For examples making used of equality constraints, following the Picheny, et al (2016) papers; see the demos listed in the “See Also” section below.

Although it is off by default, these functions allow an unknown objective to
be modeled (`fhat = TRUE`

), rather than assuming a known linear one. For examples see
`demo("ALfhat")`

and `demo("GSBP")`

which illustrate the AL and comparators
in inequality and mixed constraints setups, respectively.

The `optim.efi`

function is provided as a comparator. This method uses
the same underlying GP models to with the hybrid EI and probability of satisfying
the constraints heuristic from Schonlau, et al., (1998). See `demo("GSBP")`

and `demo("LAH")`

for `optim.efi`

examples and comparisons between
the original AL, the slack variable enhancement(s) on mixed constraint
problems with known and blackbox objectives, respectively

Picheny, V., Gramacy, R.B., Wild, S.M., Le Digabel, S. (2016). “Bayesian optimization under mixed constraints with a slack-variable augmented Lagrangian”. Preprint available on arXiv:1605.09466; http://arxiv.org/abs/1605.09466

Gramacy, R.B, Gray, G.A, Lee, H.K.H, Le Digabel, S., Ranjan P., Wells, G., Wild, S.M. (2016)
“Modeling an Augmented Lagrangian for Improved
Blackbox Constrained Optimization”, *Technometrics* (with discussion),
58(1), 1-11. Preprint available on arXiv:1403.4890;
http://arxiv.org/abs/1403.4890

Jones, D., Schonlau, M., and Welch, W. J. (1998).
“Efficient Global Optimization of Expensive Black Box Functions.”
*Journal of Global Optimization*, 13, 455-492.

Schonlau, M., Jones, D.R., and Welch, W. J. (1998). “Global Versus Local Search in
Constrained Optimization of Computer Models.” In *New Developments and Applications
in Experimental Design*, vol. 34, 11-25. Institute of Mathematical Statistics.

Nocedal, J. and Wright, S.J. (2006). *Numerical Optimization.*
2nd ed. Springer.

`vignette("laGP")`

, `demo("ALfhat")`

for blackbox objective,
`demo("GSBP")`

for a mixed constraints problem with blackbox objective,
`demo("LAH")`

for mix constraints with known objective,
`optim.step.tgp`

for unconstrained optimization;
`optim`

with `method="L-BFGS-B"`

for box constraints, or
`optim`

with `method="SANN"`

for simulated annealing

# NOT RUN { ## this example assumes a known linear objective; further examples ## are in the optim.auglag demo ## a test function returning linear objective evaluations and ## non-linear constraints aimprob <- function(X, known.only = FALSE) { if(is.null(nrow(X))) X <- matrix(X, nrow=1) f <- rowSums(X) if(known.only) return(list(obj=f)) c1 <- 1.5-X[,1]-2*X[,2]-0.5*sin(2*pi*(X[,1]^2-2*X[,2])) c2 <- rowSums(X^2)-1.5 return(list(obj=f, c=cbind(c1,c2))) } ## set bounding rectangle for adaptive sampling B <- matrix(c(rep(0,2),rep(1,2)),ncol=2) ## optimization (primarily) by EI, change 25 to 100 for ## 99% chance of finding the global optimum with value 0.6 ## library(akima) ## for plotprog=interp below out <- optim.auglag(aimprob, B, end=25) ##, plotprog=interp) ## using the slack variable implementation which is a little slower ## but more precise; slack=2 augments random search with L-BFGS-B out2 <- optim.auglag(aimprob, B, end=25, slack=TRUE) # } # NOT RUN { out3 <- optim.auglag(aimprob, B, end=25, slack=2) # } # NOT RUN { ## for more slack examples and comparison to optim.efi on problems ## involving equality and mixed (equality and inequality) constraints, ## see demo("ALfhat"), demo("GSBP") and demo("LAH") ## for comparison, here is a version that uses simulated annealing ## with the Additive Penalty Method (APM) for constraints # } # NOT RUN { aimprob.apm <- function(x, B=matrix(c(rep(0,2),rep(1,2)),ncol=2)) { ## check bounding box for(i in 1:length(x)) { if(x[i] < B[i,1] || x[i] > B[i,2]) return(Inf) } ## evaluate objective and constraints f <- sum(x) c1 <- 1.5-x[1]-2*x[2]-0.5*sin(2*pi*(x[1]^2-2*x[2])) c2 <- x[1]^2+x[2]^2-1.5 ## return APM composite return(f + abs(c1) + abs(c2)) } ## use SA; specify control=list(maxit=100), say, to control max ## number of iterations; does not easily facilitate plotting progress out4 <- optim(runif(2), aimprob.apm, method="SANN") ## check the final value, which typically does not satisfy both ## constraints aimprob(out4$par) # } # NOT RUN { ## for a version with a modeled objective see demo("ALfhat") # }