Solving minimization problem over discrete matrices with constraints
I'm trying to solve an order minimization problem with python. Therefore I distribute M orders over N workers. Every worker has a basic energy-level X_i which is gathered in the vector X. Also, every order has a specific energy consumption E_j which is gathered in E. With that being said I'm trying to solve the following problem
where Y is some optimal energy level, with the norm beeing the 2-norm. Under the constraints, that any column adds up to exactly one, since an order should be done and could only be done by one worker. I looked at scipy.optimize but it doesn't seem to support this sort of optimization as far as I can tell.
Does one know any tools in Python for this sort of discrete optimization problem?
The answer depends on the norm. If you want the 2-norm, this is a MIQP (Mixed Integer Quadratic Programming) problem. It is convex, so there are quite a number of solvers around (e.g. Cplex, Gurobi, Xpress -- these are commercial solvers). It can also be handled by an MINLP solver such as BonMin (open source). Some modeling tools that can help are Pyomo and CVXPY.
If you want the 1-norm, this can be formulated as a linear MIP (Mixed Integer Programming) model. There are quite a few MIP solvers such as Cplex, Gurobi, Xpress (commercial) and CBC, GLPK (open source). Some modeling tools are Pyomo, CVXPY, and PuLP.