Estimation of Distribution Algorithm with MACG Distribution

For a function \(f : Gr(k,p) \rightarrow \mathbf{R}\), find the minimizer and the attained minimum value with estimation of distribution algorithm using MACG distribution.

Usage

grassmann.optmacg(func, p, k, ...)

Arguments

func

a function to be minimized, returning one finite numeric value.

p

dimension parameter as in \(Gr(k,p)\).

k

dimension parameter as in \(Gr(k,p)\).

...

extra parameters including

n.start

number of runs; algorithm is executed n.start times (default: 10).

maxiter

maximum number of iterations for each run (default: 100).

popsize

the number of samples generated at each step for stochastic search (default: 100).

ratio

ratio in \((0,1)\) where top ratio*popsize samples are chosen for parameter update (default: 0.25).

print.progress

a logical; if TRUE, it prints each iteration (default: FALSE).

Value

a named list containing:

cost

smallest function value attained during the iterative search runs; a global minimum is not guaranteed.

solution

a \((p\times k)\) matrix that attains the cost.

Validation status

This retained legacy interface is experimental. Its full numerical and statistical contract has not been independently verified across supported inputs. See riem-method-contracts and the installed contract table for method-specific assumptions, restrictions, and evidence scope.

Examples

#-------------------------------------------------------------------
#               Optimization for Eigen-Decomposition
#
# Given (5x5) covariance matrix S, eigendecomposition is can be 
# considered as an optimization on Grassmann manifold. Here, 
# we are trying to find top 3 eigenvalues and compare.
#-------------------------------------------------------------------
# \donttest{
## PREPARE
A = cov(matrix(rnorm(100*5), ncol=5)) # define covariance
myfunc <- function(p){                # cost function to minimize
  return(sum(-diag(t(p)%*%A%*%p)))
} 

## SOLVE THE OPTIMIZATION PROBLEM
Aout = grassmann.optmacg(myfunc, p=5, k=3, popsize=100, n.start=30)
#> Warning: The angular shape iteration did not converge; the last positive-definite estimate is returned.
#> Error: Shape update must be a finite symmetric positive-definite matrix.

## COMPUTE EIGENVALUES
#  1. USE SOLUTIONS TO THE ABOVE OPTIMIZATION 
abase   = Aout$solution
#> Error: object 'Aout' not found
eig3sol = sort(diag(t(abase)%*%A%*%abase), decreasing=TRUE)
#> Error: object 'abase' not found

#  2. USE BASIC 'EIGEN' FUNCTION
eig3dec = sort(eigen(A)$values, decreasing=TRUE)[1:3]

## VISUALIZE
opar <- par(no.readonly=TRUE)
yran = c(min(min(eig3sol),min(eig3dec))*0.95,
         max(max(eig3sol),max(eig3dec))*1.05)
#> Error: object 'eig3sol' not found
plot(1:3, eig3sol, type="b", col="red",  pch=19, ylim=yran,
     xlab="index", ylab="eigenvalue", main="compare top 3 eigenvalues")
#> Error: object 'eig3sol' not found
lines(1:3, eig3dec, type="b", col="blue", pch=19)
#> Error in plot.xy(xy.coords(x, y), type = type, ...): plot.new has not been called yet
legend(1.55, max(yran), legend=c("optimization","decomposition"), col=c("red","blue"),
       lty=rep(1,2), pch=19)
#> Error: object 'yran' not found
par(opar)
# }