variable in index expression

[Joseph Cullen]

I have  a variable that can take on a discrete set of values, but AMPL will not let me use it as an index. It comes from the set that V is defined on, but it doesn't work.  Any ideas? 

set D:=1..6;

set X:=1..6;

var V{X,D};

transd{D,D};

var xp{X,D} in X;

var EV {i in X, j in D} = sum{ d in D } transd[j,d] * V[ xp[i,j] , d];

The Error:

variable in index expression

context:  var EV {i in X,j in D} = sum{ d in D } transd[j,d] * V[ xp[i,j] ,  >>> d] <<< ;

[victor.z...@gmail.com]

AMPL and most solvers don't support variables in subscripts, so you either have to reformulate it with binary variables or use a IBM/ILGOG CPLEX CP Optimizer (ilogcp) and the element constraint:

  include cp.ampl;

  var x{i in 0..2} >= i integer;

  var y in 0..2 integer;

  minimize o: element({i in 0..2} x[i], y);

  option solver ilogcp;

  solve;

where element({i in 0..2} x[i], y) is equivalent to x[y] and is translated into an IloElement constraint.

HTH,

Victor

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[Joseph Cullen]

Thanks Victor. This is very helpful. I am also interested in the binary variable method. How does using binary variables solve the problem?  It might help if I give a little background on the problem.

I am trying to set up a relatively simple dynamic programming problem in AMPL. 

I have two controls, q and z, plus a stochastic element y. z is discrete. 

The state space is (x,y) where

x = z_t-1

The Bellman for this problem is:

V(x,y) = max_q,z (payoff(x,y,z,q) + beta* E[V(z,y')])

Where the expectation is taken over y'. 

At the optimal q=q* and z=z*, it should be the case that:

V(x,y) =  (payoff(x,y,z*,q*) + beta* E[V(z*,y')])

I setting this up by letting the solver choose q and z to minimize the difference between the left and right hand side of the equation subject to some constraints on q and z. 

The payoff function has an analytical form in q and z. The problem comes with integrating over the expected future value. Since V(z,y) is an unknown function, I have to index it based on the value of chosen for z. I don't see another way around it. How might I use binary variables to allow for this type of indexing?

[victor.z...@gmail.com]

The main idea is to replace each variable in the range 1..N that you use as an index (xp) with N binary variables. The i-th binary variable is one when element i is selected, so you'll also need a constraint that allows exactly one binary variable to be equal to 1. Then subscript V[xp , d] can be replaced with sum{i in 1..N} V[i, d] * xp[i] or something like that.

There is an example of assignment model formulated with binary variables (https://github.com/ampl/ampl.github.io/blob/master/models/logic/assign0.mod) and variable subscripts (https://github.com/ampl/ampl.github.io/blob/master/models/logic/assign2.mod).

HTH,

Victor