Variables with indexes and sums with indexes in mosek

Question

I have to find solutions to an integer programming problem:

I am using Mosek's Fusion API (Python). Now the constrains are easy to put in, I am more worried about the actual objective. The problem for me is: How can I tell mosek that I want to sum by all is, js or ks and define what they are, what are their boundaries, etc.?

This is a simplified version of a self-caching problem in the context of servers. So i here means a server, j means an object to cache, but in this version there's one object, so this I guess is not important. k means server too, so e.g. d(ik) means the distance from the server i to the server k.

But whatever I want to achieve, I don't know how to write this objective. For now I have something like this:

from mosek.fusion import Domain, Model, Expr, ObjectiveSense

alpha = 4 # alpha is the same for all i and j
demand = 1 # w is the same for all i and k
n = 6 # number of servers
distances_matrix = [[...], [...], ...]


with Model("lo1") as M:

    x = M.variable("x", n, Domain.integral(Domain.inRange(0, 1)))
    y = M.variable("y", n, Domain.integral(Domain.inRange(0, 1)))

    alpha_times_x = Expr.mul(alpha, x)
    demand_times_dist_times_y = Expr.mul(demand, distances_matrix, y)

    M.objective("obj", ObjectiveSense.Minimize, )

    M.solve()

    print(x.level())
    print(y.level())

Now of course the demand_times_dist_times_y is wrong, because I want to get the distance from i to k from the matrix. And the x above is fine since xs are: {x0, x1, x2, x3, x4, x5, x6}, but the ys would have to be {y11, y12, y13, y14, y15, y16, y21, y22, ..., y66}, so I guess I defined them wrong.

So e.g. how can I define that i,k are in {1,2,3,4,5,6} and create an Expr.sum by e.g. k? And how would I define those two sums at the beginning of the objective?

Why would you have to define boundaries if you are summing over all i,j,k (at least that it how it looks like from the notation)? Expr.sum() should do it. If you really want to sum over a subset use pick() or range(). It is possible that you want y to be 2-dimensional. Let me know if you need a more expanded answer. — Michal Adamaszek, Jan 06 '20 at 07:53
Yes, but how and where would I define what those i,j and ks are? — dabljues, Jan 07 '20 at 04:31

score 1 · Accepted Answer · answered Jan 07 '20 at 07:39

I don't know if that answers the question, but if you have, say

x = M.variable("x", n, Domain.integral(Domain.inRange(0, 1)))

then sum_i x_i is obtained with

Expr.sum(x)

Similarly, if now alpha is a numerical array of length n then sum_i (alpha_i*x_i) is obtained with

Expr.sum( Expr.mulElm(alpha,x) )

or even

Expr.dot( alpha, x )

and so on. You never explicitly specify the summation index, you are summing all entries of whatever appears inside the Expr.sum and similar methods.

Variables with indexes and sums with indexes in mosek

1 Answers1