Portfolio optimization in R with factor tilting while constraining volatility

Question

what optimizer I can use in R to solve the following portfolio optimization problem:

$min(f^Tx)$
st:
1. $ -a \le \sum_{i=1} ^{n} x(i) \le b$
2. $ -c \le x(i) \le d$
3. $ e \le \sum _{i=1} ^n |x(i)| \le f$
4. $\sqrt{x^t\Sigma x} \le g$

a,b,c,d,e,f,g - positive. f is a vector. x - weights. $\Sigma$ var-covar matrix estimated from historical data. The problem without constrain 4 can be solved with linear solver ( with some tricks for condition 3). Condition 4 makes constrain non-linear, but quadratic. Any suggestion would be helpful. Thanks.

Answer 1

You can try the solnp() function from the package Rsolnp, which is a solver for nonlinear optimization. First, you define your function that is to be minimized:

fun1 <- function(x) ( return(as.numeric(fvec %*% x)))

fvec is your fixed vector $f$.

Then you define the inequality function for the inequalities 1,3 and 4 - inequality 2 is covered by the parameter boundaries in the solver.

ineqfun1 <- function(x){
  ineq1 <- sum(x);
  ineq2 <- sum(abs(x));
  ineq3 <- sqrt(x %*% vcovmat %*% x);
  return(c(ineq1,ineq2,ineq3))
}

vcovmat is your var-cov matrix $\Sigma$.

After additionally defining the boundaries a to g and a starting estimate for x, which we call xstart, you can do the optimization:

sol <- solnp(pars=xstart, fun=fun1, ineqfun=ineqfun1,
             ineqLB=c(-a,e,-Inf),ineqUB=c(b,f,g),
             LB=-c, UB=d)

If the solver converges, you can get the resulting weight estimates with sol$pars.
I have no data to try it, but the code should work. You will have to choose a sensible starting value and appropriate boundaries a-g, though.