Formalizing an allocation problem

[Jack1]

I have a set "S" of 20 elements --->  x_i where i in {1, ..., 20} 

I need to allocate each element to 5 locations  --->  j in {1, ..., 5} 

Define the cost of allocating element "x_i" to location "j" by: 

f_j (x_i)

then, I would like to minimize the total cost:

\sum_j \sum_i f_j (x_i)

subject to the constraint that each element x_i can be allocated to only one location j as well as other constraints on the cost function f_j.   

I would be appreciative of a way to formalize the constraints above and tractably solve the problem via yalmip given that the cost function is differentiable and cvx.  Thank you. 

[Johan Löfberg]

convex is impossible as it is a combinatorial and thus requires binary variables

If d_i encodes that x_i takes value S_i, then the sum simplifies to sum d_i*f(S_i)

See logic programming/implies/modeling if-else etc in the wiki

[Jack1]

thank you for getting back. I reviewed your post here https://yalmip.github.io/modellingif

if I did not have a functional representation for the costs f(S_i), but, these were empirical values, then, how do I separate my space into 5*20 cases? Do I have to find a functional representation or is there another way? 

Also, not sure how to impose the allocation of one element to one location mentioned above - would be much appreciative if you can kindly help with a simple case for me to get started. 

[Johan Löfberg]

empirical values is a function

if x = 3 then f = 5 else if x = 9 then f = -67 etc

is 

binary d1 d2

f = d1*5 + (-67*)*d2

model = implies(d1, x == 3) + implies(d2,x==9)

etc