Question about optimize() function in Mata

[Nhan Nguyen]

Dear Stata users,

I am playing around with data and I just wonder why I get inconsistent results for one optimization problem when I try different approaches and how to use the KKT condition in optimize(). Let say I have the optimization problem as follow. I have the Var-Covariance matrix as

m Var_Covar = (2,1.3,0.35,0\1.3,1.5,0.8,0\0.35,0.8,0.6,0\0,0,0,1.2)

and I would want to find the combination of stocks (by adjusting how many of them that I should hold in a portfolio) that minimizes the global variance. Therefore, the optimization problem can be expressed as min{wVw'} Subject to rowsum(w) = 1 and w_i >= 0 where w is the (1x4) vector of weights. This is the convex optimization problem because the objective function is convex and continuous. Here is how I use optimize() to tackle the problem.

1. If I relax the inequality constraints and proceed with optimize() then I have two ways of coding which should return the same (or at least similar to some extent) results.
- The first way is using the optimize_init_constraints() to specify the equality constraint (the sum of the proportions of each stock in the portfolio is one).

mata
w1 = (0.2,0.2,0.2,0.2)'
void globalminvar2(minvar, params, sig, y, g, H)
    { 
        y = -params*sig*params'
        if(minvar==1){
            g = (-2:*params*sig)
        }
    }
T = optimize_init()
optimize_init_evaluator(T,&globalminvar2())
optimize_init_evaluatortype(T,"d1")
optimize_init_params(T,(w1'))
optimize_init_argument(T,1,Var_Covar)
optimize_init_constraints(T,(1,1,1,1,1))
p = optimize(T)
p
end

- The second way is adding the Lagrange multiplier to the objective function which means I no longer have to declare my constraint optimize_init_constraints(T,(1,1,1,1,1)). Below is my coding for the problem.

mata
void globalminvar3(minvar, params, sig, y, g, H)
{
    w = params[1..rows(sig)]
    lambda = params[cols(params)]
    y = -w*sig*w'-(rowsum(w)-1)*lambda
    if(minvar==1){
         g[1] = (-2*w*sig[.,1]:-lambda)
         g[2] = (-2*w*sig[.,2]:-lambda)
         g[3] = (-2*w*sig[.,3]:-lambda)
         g[4] = (-2*w*sig[.,4]:-lambda)
         g[5] = -rowsum(w)+1
    }
}
U = optimize_init()
optimize_init_evaluator(U,&globalminvar3())
optimize_init_evaluatortype(U,"d1")
optimize_init_params(U,(w1',0.2))
optimize_init_argument(U,1,Var_Covar)
p = optimize(U)
p
end

The result from using optimize_init_constraints(T,(1,1,1,1,1))is correct because rowsum(p) is equal to 1. However, the result I got from the second way is not binding. You can see by checking rowsum(p[1..4]). At this point, I would want to know which part that I understand wrong. My conjecture is that the problem that results in inconsistent outcome is the initial guess on the parameters because every time I feed in a different parameter for lambda, the result changes. This optimization problem is quite basic and I can do it in other statistical softwares but I mostly use Stata so I truly want to know how to do it.

2. Now I want the weight of each stock to be non-negative (no-short selling), which means now I have to use the KKT conditions. Let theta_i be the Lagrange multiplier for each of the inequality condition, we have the complementary slackness conditions as theta_i > 0, theta_i = 0 if w_i > 0. Below is the implementation of the above idea.

mata
Return = (2.5,2,1,1.5)'
Var_Covar = (2,1.3,0.35,0\1.3,1.5,0.8,0\0.35,0.8,0.6,0\0,0,0,1.2)
Corr_mat = Var_Covar:/(sqrt(diagonal(Var_Covar))#sqrt(diagonal(Var_Covar)'))
w1 = (0.2,0.2,0.2,0.2)'
void globalminvar1(minvar, params, sig, y, g, H)
{
    w = params[1..rows(sig)]
    lambda = params[rows(sig)+1]
    theta = params[rows(sig)+2..cols(params)]
    y = -w*sig*w'+(rowsum(w)-1)*lambda+w*theta'
    if(minvar==1){
        theta[selectindex(w:>0)]=J(rows(selectindex(w:>0)),cols(selectindex(w:>0)),0)
        g = (-2:*w*sig:+lambda:+theta,rowsum(w)-1,theta)
    }
}
S = optimize_init()
optimize_init_evaluator(S,&globalminvar1())
optimize_init_evaluatortype(S,"d1")
optimize_init_params(S,(w1',1,0,0,0,0))
optimize_init_argument(S,1,Var_Covar)
p = optimize(S)
p
end

However, mata could not find the optimum value of this solution. Can anyone help me with this. Thank you in advance.

[Choonjoo Lee]

Quite some time has passed, so I suspect the issue may already have been resolved. I encountered a similar problem in another project and came to the following conclusion. If you found a different approach or interpretation, I would be very interested to hear it.

The multiplier formulation is not an equivalent unconstrained maximization. The Lagrangian has a saddle point and, being linear in lambda, is generally unbounded away from the constraint. optimize() maximizes over every element of params; simply including multipliers as parameters does not make it solve the KKT conditions.

In the inequality formulation, the derivative with respect to theta should also be w, not theta, and neither primal nor dual nonnegativity is imposed. I would therefore use the explicit equality constraint for the first problem and a quadratic-programming or active-set method for the no-short-sales problem.

For the posted covariance matrix, the verified no-short solution is
w = (.0893522, 0, .5897245, .3209233),
with variance .3851080.