Perform a SQP algorithm in R

Question

I would like to run following optimization in R:

u.c.: 0 <= x <= 1 and Sum(x)=1

The equation is based on: Efficient Algorithms for Computing Risk Parity Portfolio Weights (equation 10)

The original authors say to use SQP. I would like to follow that but how?

Answer 1

the code could look like this:

fn <- function(w){return( 
      ((w[1] * w %*% Mat[1,]) - (w[1] * w %*% Mat[1,]))^2 +
      ((w[1] * w %*% Mat[1,]) - (w[2] * w %*% Mat[2,]))^2 + 
      ((w[1] * w %*% Mat[1,]) - (w[3] * w %*% Mat[3,]))^2 +

      ((w[2] * w %*% Mat[2,]) - (w[1] * w %*% Mat[1,]))^2 +
      ((w[2] * w %*% Mat[2,]) - (w[2] * w %*% Mat[2,]))^2 + 
      ((w[2] * w %*% Mat[2,]) - (w[3] * w %*% Mat[3,]))^2 +

      ((w[3] * w %*% Mat[3,]) - (w[1] * w %*% Mat[1,]))^2 +
      ((w[3] * w %*% Mat[3,]) - (w[2] * w %*% Mat[2,]))^2 + 
      ((w[3] * w %*% Mat[3,]) - (w[3] * w %*% Mat[3,]))^2
  )
  }

  library(Rsolnp)

  #start values
  w0 <- c(0.3, 0.6, 0.1)

  #constrain function
  eqcon <- function(w){(w[1]+w[2]+w[3])}
  ebcon <- 1

  #optimizer
  sqp <- solnp(pars = w0,
               fun = fn2,
               eqfun = eqcon,
               eqB = ebcon, 
               LB = c(0,0,0),
               UB = c(1,1,1))

  sqp$pars