cobraPhaseII: Improve the feasible solution by searching new infill points
In SACOBRA: Self-Adjusting COBRA

Description Usage Arguments Value Author(s) See Also Examples

Improve the feasible solution using the SACOBRA optimizer phase II by searching new infill points with the help of RBF surrogate models. May be even called if no feasible solution is found yet, then phase II will try to find feasible solutions.
The problem to solve iteratively is:

\mbox{Minimize}\quad f(\vec{x}) , \vec{x} \in [\vec{a},\vec{b}] \subset \mathbf{R}^d

\mbox{subject to}\quad g_i(\vec{x}) ≤ 0, i=1,…,m

\mbox{~~~~~~~~~~}\quad\quad h_j(\vec{x}) = 0, j=1,…,r.

In this phase the main optimization steps are repeated in a loop as long as the budget is not exhausted. In every iteration the surrogate models are updated and an optimization on the surrogates is done in order to find a better feasible solution.

1	cobraPhaseII(cobra)

cobra

an object of class COBRA, this is a (long) list containing all settings from cobraInit

cobra, an object of class COBRA from cobraInit, enhanced here by the following elements (among others):

`fn`	function accepting a `d`-dimensional vector \vec{x} and returning an `(1+m+r)`-vector `c(`f,g_1,…,g_m,h_1,…,h_r`)`. This function may be a rescaled and plog-transformed version of the original `fn` passed into `cobraInit`. The original `fn` is stored in `cobra$originalFn`.
`df`	data frame with summary of the optimization run (see below)
`df2`	data frame with additional summary information (see below)
`dftr`	data frame with additional summary information for TR (see below)
`A`	`(feval x d)`-matrix containing all evaluated points in input space. If rescale==TRUE, all points are in rescaled input space.
`Fres`	a vector of the objective values of all evaluated points
`Gres`	a `(feval x m)`-matrix of the constraint values of all evaluated points
`predC`	a `(feval x m)`-matrix with the prediction of `cobra$constraintSurrogates` at all evaluated points
`fbest`	the best feasible objective value found
`xbest`	the point in input space yielding the best feasible objective value
`ibest`	the corresponding iteration number (row of cobra$df, of cobra$A)
`PLOG`	If TRUE, then the objective surrogate model is trained on the `plog`-transformed objective function.

Note that cobra$Fres, cobra$fbest, cobra$fbestArray and similar contain always the objective values of the orignial function cobra$fn[1]. (The surrogate models may be trained on a plog-transformed version of this function.)

feval = cobra$feval is the maximum number of function evaluations.

The data frame cobra$df contains one row per iteration with columns

iter: iteration index
y: true objective value Fres
predY: surrogate objective value. Note: The surrogate may be trained on plog-transformed training data, but predY is transformed back to the original objective range. NA for the initial design points.
predSolu: surrogate objective value at best-known solution cobra$solu, if given. If cobra$solu is NULL, take the current point instead. Note: The surrogate may be trained on plog-transformed training data, but predSolu is transformed back to the original objective range. NA for the initial design points.
feasible: boolean indicating the feasibiltiy of infill point
feasPred: boolean indicating if each infill point is feasible for cobra$constraintSurrogates
nViolations: number of violated constraints
maxViolation: maximum constraint violation.
FEval: number of function evaluations in sequential optimizer. NA if it was a repair step
Best: ever-best feasible objective value fbest. As long as there is no feasible point, take among those with minimum number of violated constraints the one with minimum Fres.
optimizer: e.g. "COBYLA"
optimizationTime: in sec
conv: optimizer convergence code
dist: distance of the current point (row of cobra$A) to the true solution cobra$solu in rescaled space. If there is more than one solution, take the one which has the minimum distance element (since this is the solution to which the current run converges).
distOrig: same as dist, but in original space
XI: the DRC element used in the current iteration
seed: the used seed in every run

The data frame cobra$df2 contains one row per phase-II-iteration with columns

iter: iteration index
predY: surrogate objective value. Note: The surrogate may be trained on plog-transformed training data, but predY is transformed back to the original objective range. NA for the initial design points.
predVal: surrogate objective value + penalty
predSolu: surrogate objective value at true solution (see cobra$df$predSolu)
predSoluPenal: surrogate objective value + penalty at true solution (only diagnostics)
sigmaD: the sigmaD element used in the current iteration (see cobraInit)
penaF: penalty factor used in the current iteration (see cobraInit)
XI: the DRC element used in the current iteration
EPS: the current used margin for constraint function modeling (see epsilonInit in cobraInit )

Wolfgang Konen, Samineh Bagheri, Patrick Koch, Cologne University of Applied Sciences

cobraPhaseI, cobraInit

## Initialize cobra. The problem to solve is the unconstrained sphere function sum(x^2).   
 
## In version 1.1 and higher there is no need for defining a dummy 
## constraint function for the unconstrained problems
d=2
fName="sphere"
cobra <- cobraInit(xStart=rep(5,d), fName=fName,
                   fn=function(x){c(obj=sum(x^2))},  
                   lower=rep(-10,d), upper=rep(10,d), feval=40)
                   
## Run cobra optimizer
cobra <- cobraPhaseII(cobra)

## The true solution is at solu = c(0,0)
## where the true optimum is fn(solu)[1] = optim = 0
## The solution found by SACOBRA:
print(getXbest(cobra))
print(getFbest(cobra))

## Plot the resulting error (best-so-far feasible optimizer result - true optimum)
## on a logarithmic scale:
optim = 0
plot(cobra$df$Best-optim,log="y",type="l",ylab="error",xlab="iteration",main=fName)