Limiting norms by inequalities
I have time series data for N stocks.
sample.data<-replicate(10,rnorm(1000))
where each column shows the return on different stocks over time.
I am trying to plot a portfolio weight vector to minimize the variance of returns.
objective function:
min w^{T}\sum w
s.t. e_{n}^{T}w=1
\left \| w \right \|\leq C
where w is a vector of weights, \sum
is a covariance matrix, e_{n}^{T}
is a vector of units, C
is a constant. Where the second constraint ( \left \| w \right \|
) is the inequality constraint.
I used the following code to solve this problem:
library(Rsolnp)
gamma<-1
fn<-function(x) {cov.Rt<-cov(sample.data); return(t(x)%*%cov.Rt%*%x)} #OBJECTIVE FUNCTION TO MINIMIZE
eqn<-function(x){one.vec<-matrix(1,ncol=10,nrow=1); return(one.vec%*%x)} #EQUALITY CONSTRAINT
constraints<-1 #EQUALITY CONSTRAINT
ineq<-function(x){one.vec<-matrix(1,ncol=10,nrow=1); #INEQUALITY CONSTRAINT
z1<-one.vec%*%abs(x)
return(z1)
}
uh<-gamma #UPPER BOUND
lb<-0 #LOWER BOUND
x0<-matrix(0,10,1) #STARTING PARAMETER VECTOR (NOT SURE WHAT STARTING VALUES TO PUT HERE)
sol1<-solnp(x0,fun=fn,eqfun=eqn,eqB=constraints, ineqfun=ineq,ineqUB=gamma,ineqLB=lb)
When you run this code, you receive the following error message:
Error in solve.default(cz,tol = 1e-25) : system is computationally singular: reciprocal condition number = 0 In addition: There were 50 warnings (use warnings() to see the first 50)
warnings()
1: In cbind(temp, funv): number of rows of result is not a multiple of vector length
Any ideas what I might be doing wrong? Is there a problem with the initial parameter vector x0
?
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I changed the code a bit:
library(Rsolnp)
set.seed(1)
sample.data <- matrix(rnorm(10*1000),ncol=10)
gamma <- 1
#OBJECTIVE FUNCTION TO MINIMIZE
fn <- function(x){
cov.Rt <- cov(sample.data);
as.numeric(t(x) %*% cov.Rt %*% x)
}
#EQUALITY CONSTRAINT
eqn <- function(x){
one.vec <- matrix(1, ncol=10, nrow=1)
as.numeric(one.vec %*% x)
}
constraints <- 1
#INEQUALITY CONSTRAINT
ineq <- function(x){
one.vec <- matrix(1, ncol=10, nrow=1);
z1<-one.vec %*% abs(x)
as.numeric(z1)
}
uh <- gamma #UPPER BOUND
lb <- 0 #LOWER BOUND
x0 <- matrix(1, 10, 1) #STARTING PARAMETER VECTOR (NOT SURE WHAT STARTING VALUES TO PUT HERE)
sol1 <- solnp(x0, fun=fn, eqfun=eqn, eqB=constraints)
When we run the above code, we get the solution:
Iter: 1 fn: 0.09624 Pars: 0.08355 0.08307 0.10154 0.09108 0.11745 0.12076 0.09020 0.09435 0.10884 0.10918
Iter: 2 fn: 0.09624 Pars: 0.08354 0.08308 0.10153 0.09107 0.11746 0.12078 0.09021 0.09434 0.10883 0.10918
solnp--> Completed in 2 iterations
But when we add an inequality constraint to an optimization problem, we run into problems:
> sol2 <- solnp(x0, fun=fn, eqfun=eqn, eqB=constraints,
ineqfun=ineq, ineqUB=gamma, ineqLB=lb)
Iter: 1 fn: 0.09624 Pars: 0.08356 0.08305 0.10153 0.09111 0.11748 0.12078 0.09021 0.09431 0.10881 0.10916
solnp-->The linearized problem has no feasible
solnp-->solution. The problem may not be feasible.
Iter: 2 fn: 272.5459 Pars: 4.44541 4.42066 5.40272 4.84595 6.25082 6.42718 4.80020 5.02004 5.79138 5.81029
solnp-->The linearized problem has no feasible
solnp-->solution. The problem may not be feasible.
Iter: 3 fn: 272.5459 Pars: 4.44547 4.42070 5.40274 4.84596 6.25078 6.42712 4.80023 5.02006 5.79138 5.81023
Iter: 4 fn: 0.09624 Pars: 0.08357 0.08304 0.10157 0.09106 0.11744 0.12074 0.09021 0.09432 0.10886 0.10918
Iter: 5 fn: 0.09625 Pars: 0.08354 0.08308 0.10153 0.09107 0.11747 0.12078 0.09021 0.09434 0.10883 0.10919
Iter: 6 fn: 0.09717 Pars: 0.08394 0.08347 0.10201 0.09150 0.11803 0.12135 0.09064 0.09479 0.10935 0.10971
Iter: 7 fn: 0.09624 Pars: 0.08353 0.08307 0.10153 0.09106 0.11747 0.12078 0.09020 0.09433 0.10883 0.10919
Iter: 8 fn: 0.09624 Pars: 0.08353 0.08307 0.10153 0.09106 0.11747 0.12078 0.09020 0.09433 0.10883 0.10919
solnp--> Solution not reliable....Problem Inverting Hessian.
Warning message:
In p0 * vscale[(neq + 2):(nc + np + 1)] :
longer object length is not a multiple of shorter object length
Try changing the gamma slightly:
gamma <- 1.01
> sol2 <- solnp(x0, fun=fn, eqfun=eqn, eqB=constraints,
ineqfun=ineq, ineqUB=gamma, ineqLB=lb)
Iter: 1 fn: 0.09624 Pars: 0.08355 0.08307 0.10153 0.09108 0.11745 0.12076 0.09020 0.09435 0.10884 0.10918
Iter: 2 fn: 0.09624 Pars: 0.08354 0.08308 0.10153 0.09107 0.11746 0.12078 0.09021 0.09434 0.10883 0.10918
solnp--> Completed in 2 iterations
So your inequality constraint appears to be related to an equality constraint. Also looking at the two constraints together seems a little odd to me. My guess is that you probably mis-spelled your inequality in inequality and just want something like a short-circuit constraint for example. weight from 0 to 1. This can be archived according to your method:
ineq <- function(x){ return(x) }
uh <- rep(gamma, 10) #UPPER BOUND
lb <- rep(0, 10) #LOWER BOUND
sol3 <- solnp(x0, fun=fn, eqfun=eqn, eqB=constraints, ineqfun=ineq, ineqUB=uh, ineqLB=lb)
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