Constrained integer optimization in Python

Question

I'd like to minimize a function, which takes a 3x8 matrix of non-negative integers as input. Each row specifies a variable, whereas each column specifies a certain time point in the system. See the input in CSV format below.

   ,Time0,Time1,Time2
U_i,0,0,0
U_o,0,0,0
C_i,0,0,0
C_o,0,0,0
T_i,0,0,0
T_o,0,0,0
D_i,0,0,0
D_o,0,0,0

The constraints for each column is:

C_i + T_i >= U_i
C_o + T_o >= U_o
D_i <= 15
D_o <= 15
D_i = 0 if C_i == 0
D_o = 0 if C_o == 0

And the overall constraint across rows is C_i + C_o + T_i + T_o = 5 . I've looked at scipy.optimize, but cannot find a proper method that handles integers. Could someone give me a hint or a MWE on how to do this?

Answer 1

For smaller problems the AMPL website has a free tool: http://www.ampl.com/TRYAMPL/startup.html . The model file (*.mod) to upload would would be:

set T;

var U_i {T} >= 0 integer;
var U_o {T} >= 0 integer;
var C_i {T} >= 0 integer;
var C_o {T} >= 0 integer;
var T_i {T} >= 0 integer;
var T_o {T} >= 0 integer;
var D_i {T} >= 0 integer;
var D_o {T} >= 0 integer;

minimize obj: sum {t in T} (U_i[t] + U_o[t] + C_i[t] + C_o[t] + T_i[t] + T_o[t] + D_i[t] + D_o[t]);
subject to C1 {t in T}: C_i[t] + T_i[t] >= U_i[t];
subject to C2 {t in T}: C_o[t] + T_o[t] >= U_o[t];
subject to C3 {t in T}: D_i[t] <= 15;
subject to C4 {t in T}: D_o[t] <= 15;
subject to C5 {t in T}: D_i[t] <= C_i[t] / 1000;
subject to C6 {t in T}: D_o[t] <= C_o[t] / 1000;
subject to C7 {t in T}: C_i[t] + C_o[t] + T_i[t] + T_o[t] = 5;

The data file (*.dat):

set T := 1 2 3;

Since you didn't specify an objective function I put in a linear dummy. After uploading both files you can select a solver; I think none of them is an integer solver unfortunately, but some are non-linear. This way you could probably find a lower bound, feasible integer solutions you can get from there by some rounding heuristic (not necessary integer optimal).

Alternatively: Excel solver?