ortho_optim {orthoDr}

Orthogonality constrained optimization

Description

A general purpose optimization solver with orthogonality constraint. The orthogonality constrained optimization method is a nearly direct translation from Wen and Yin (2010)'s MATLAB code.

Usage

ortho_optim(
  B,
  fn,
  grad = NULL,
  ...,
  maximize = FALSE,
  control = list(),
  maxitr = 500,
  verbose = FALSE
)

Arguments

B

Initial B values. Must be a matrix, and the columns are subject to the orthogonality constrains. Will be processed by Gram-Schmidt if not orthogonal

fn

A function that calculate the objective function value. The first argument should be B. Returns a single value.

grad

A function that calculate the gradient. The first argument should be B. Returns a matrix with the same dimension as B. If not specified, then numerical approximation is used.

...

Arguments passed to fn and grad

maximize

By default, the solver will try to minimize the objective function unless maximize = TRUE

control

A list of tuning variables for optimization. epsilon is the size for numerically approximating the gradient. For others, see Wen and Yin (2013).

maxitr

Maximum number of iterations

verbose

Should information be displayed

Value

A orthoDr object that consists of a list with named entries of:

B

The optimal B value

fn

The final functional value

itr

The number of iterations

converge

convergence code

References

Wen, Z., & Yin, W. (2013). A feasible method for optimization with orthogonality constraints. Mathematical Programming, 142(1), 397-434. DOI: doi:10.1007/s10107-012-0584-1

Examples

# an eigen value problem
library(pracma)
set.seed(1)
n <- 100
k <- 6
A <- matrix(rnorm(n * n), n, n)
A <- t(A) %*% A
B <- gramSchmidt(matrix(rnorm(n * k), n, k))$Q

fx <- function(B, A) -0.5 * sum(diag(t(B) %*% A %*% B))
gx <- function(B, A) -A %*% B
fit <- ortho_optim(B, fx, gx, A = A)
fx(fit$B, A)

# compare with the solution from the eigen function
sol <- eigen(A)$vectors[, 1:k]
fx(sol, A)