Devising objective function for integer linear programming

Question

I am working to devise a objective function for a integer linear programming model. The goal is to determine the copy number of two genes as well as if a gene conversion event has happened (where one copy is overwritten by the other, which looks like one was deleted but the net copy number has not changed).

The problem involves two data vectors, P_A and P_B . The vectors contain continuous values larger than zero that correspond to a measure of copy number made at each position. P_{A,i} is not necessarily the same spot across the gene as P_{B,i} is, because the positions are unique to each copy (and can be mapped to an absolute position in the genome).

Given this, my plan was to try and minimize the difference between my decision variables and the measured data across different genome windows, giving me different slices of the two data vectors that correspond to the same region.

Decision variables:

A_w = copy number of A in window in {0,1,2,3,4}
B_w = copy number of B in window in {0,1,2,3,4}
C_w = gene conversion in {-2,-1,0,1,2}

The goal then would be to minimize the difference between the left and right sides of the below equations:

A_w - C_w ~= mean(P_{A,W})
B_w + C_w ~= mean(P_{B,W})

Subject to a handful of constraints such as 2 <- A_w + B_w <= 4

But I am unsure how to formulate this into a function to minimize. I have two equations that are not really a function, and the decision variables have no coefficients.

I am also unsure of how to handle the negative values of C_w .

I also am unsure of how to bring the results back together; after I solve the LP in each window, I still need to merge it into one gene-wide call (and ideally identify which window(s) had non-zero values of C_w .

Answer 1

Create the LpProblem instance:

problem = LpProblem("Another LpProblem", LpMinimize)

Objective (per what you've vaguely described above):

problem += (mean(P_{A,W}) - (A_w - C_w)) + (mean(P_{B,W}) - (B_w + C_w))

This is all I could tell from your really rather vague question. You'll need to be much more specific with what you mean by terms like "bring the results back together", or "handle the negative values in C_w". Add in your current code snippets and the errors you're getting for more details.