Maximization of quadratic forms in R with respect to a vector of weights

Question

I have a distance (similarity) matrix D, for example,

D <- matrix(c(0.00, 1.00, 1.00, 0.10, 0.05, 1.00, 0.00, 1.00, 1.00, 1.00, 1.00, 1.00, 0.00, 0.90, 0.95, 0.10, 1.00, 0.90, 0.00, 0.15, 0.05, 1.00, 0.95, 0.15, 0.00),5,5)

and a vector of weights w = (w1, ..., wn) such that sum(w) == 1. The values in the vector w are real and between 0 and 1, inclusively. I need to find a vector w such that the sum w*D*t(w) is maximized. Where t(w) is the transpose of w and the symbol "*" denotes matrix multiplication.

Amazingly enough, I can't find a solver that can do this in R.

Thank you

Answer 1

Maybe you can try fmincon from package pracma, e.g.,

library(pracma)
D <- matrix(c(0.00, 1.00, 1.00, 0.10, 0.05, 1.00, 0.00, 1.00, 1.00, 1.00, 1.00, 1.00, 0.00, 0.90, 0.95, 0.10, 1.00, 0.90, 0.00, 0.15, 0.05, 1.00, 0.95, 0.15, 0.00),5,5)
n <- dim(D)[1]
res <- fmincon(rep(1,n),
               fn = function(w) -t(w)%*%D%*%w, 
               A = t(rep(1,n)), 
               b = 1,
               lb = rep(0,n),
               ub = rep(1,n))
w <- res$par

and you will get

> w
[1] 3.333331e-01 3.333338e-01 3.333331e-01 7.008297e-22 0.000000e+00