Error with FindMinimum. The Message is FindMinimum::cnpcons

Question

I want to run a rather demanding check of feasibility.

I have 3565 non-negativity constraints for my variables, and one that makes them sum to a certain budget they shall not surpass. I also have one inequality constraint that calculates whether or not, for a given set of values of the variables, a given constant is smaller or larger than the Gini coefficient for the current data.

The expression looks like this:

GWSUB[y_, ES_, t_, w_, k_] := Module[{yt, ytw, Gini},
    yt = (y + t) / ES;
    ytw = (w*(y + t)) / ES;
    Gini = -k*(2*(Plus @@ w)*(Plus @@ ytw)) + 
             Sum[ (w[[j]]*(Plus @@ Abs[ytw - yt[[j]]*w]), {j, 1, Length[y]}
  ]

Here are the meanings of the symbols I used:

• The vector of variables is $t$.
• $y$ is a vector of incomes.
• $w$ is a vector of weights.
• ES is a separate vector of weights.
• $k$ defines the candidate value for the Gini coefficient and is a scalar.

The vectors are all of length 3565.

I then tried to run the FindMinimum command. Here T is the vector of variables t[i]. TC are the non-negativity constraints on the t[i]. B is the budget (a scalar).

FindMinimum[
  {
   0*(Plus @@ T),
   GWSUB[Y, ES, T, W, 0.3294] <= 0, 
   Sequence[TC], (Plus @@ T) == B
  },
  T, 
  Method -> "InteriorPoint", WorkingPrecision -> 7, MaxIterations -> 500
]

I get an error of this form:

FindMinimum::cnpcons: Could not process the constraints ...

and then it refers to the GWSUB expression.

Does this mean that I cannot run the computation with the current machine? Is there a way to simplify the expression, so that Mathematica will process it?

My Value Configuration

To reproduce my results you will need the following data set:

https://www.dropbox.com/s/qni6zoecmdm9p1x/data.xls?dl=0

Then import the data set and use the following set up:

Y = Table[dta[[j]][[1]], {j, 1, 100}]; 
W = Table[dta[[j]][[3]], {j, 1, Length[Y]}];
ES = Table[dta[[j]][[2]], {j, 1, Length[Y]}]; 
T = Table[t[i], {i, 1, Length[Y]}]; 
TC = Table[t[i] >= 0., {i, 1, Length[Y]}]; 
TS =  Table[{t[i], 0.}, {i, 1, Length[Y]}];
B = 2791967732

Minor Update

I have tried to let FindMinimum only evaluate numerically by compiling the aforementioned GWSUB function with "RuntimeOptions" -> {"EvaluateSymbolically" -> False}. While the calculations of the GWSUB function are faster, the call to FindMinimum still resulted in the same error message.

Answer 1

Ok, I have not figured out what the specific error code is but I have reformulated the objective function so that MMA can actually process the constraint.

The objective function now is:

 ObJ[y_, ES_, t_, w_, k_] := Module[{yt, ytw, Gini},
 yt = (y + t)/ES;
 ytw = (w*(y + t))/ES;
 If[-(k*(2*(Plus @@ w)*(Plus @@ ytw))) +Sum[ (w[[j]]*(Plus  @@ Abs[ytw - yt[[j]]*w]),{j,1,Lentgth[y]}] <= 0
, 0, \[Infinity] ]
 ]

Now the constraint is part of the objective function and FindMinimum seems to process this just fine.

However, the processing speed is still very slow. I have run the same program in Matlab and it was worked out in a couple of minutes. MMA is still running.