Q:

How to check, without an object function, if the constraints are feasible?

My professor gave me a binary linear programming problem, but this problem is slightly different from optimization problems I used to solve(i.e. this is probably not maximizing or minimizing the object function.)
The problem is as follows,
Given a matrix M, for entries m_ij != 0, there are corresponding x_ijk variables.
Entries m_ij = 0 can be ignored.
x_ijk is either 0 or 1, and I want to try 5 x_ijk variables for each m_ij (that is, x_ij1, x_ij2, x_ij3, x_ij4, and x_ij5. One of them is 1 and the others are 0) are enough to satisfy some conditions(a set of inequalities).
More simply, this is to check if the set of constraints involving 5 x_ijk variables for each m_ij is a valid(or feasible) constraints.
I have solved some optimization problems, but I have never solved a problem without an objective function.
What should I set as my objective function here?
0? nothing?
I might be using lp_solve or CPLEX.
Thank you in advance for your advice!

A:

That is correct, you can set an arbitrary constant value as an objective function.
Most of the solvers I have tried allow an empty objective function. Simply leave it out from your model.
Depending on the solver and the API you are using, it can happen that you have to set the coefficients of all variables in the objective to zero.
Don't worry, it has to work.

In response to your comment: Yes, constraint programming tools can provide better performance on feasibility problems than LP solvers (such as CPLEX). I have played with the IBM ILOG CPLEX CP Optimizer a few months ago, it is free for Academic users. Both the LP solver and the CP solver failed on my problems. Don't expect a miracle from constraint programming.
Keep in mind the that time needed to solve a constraint program grows exponentially with the size of the problem in the worse case. Sooner or later, your problems will most likely become unsolvable with either tool.
Just for your information: in the end, the constraint programming solver will call the LP solver (for example CPLEX).
My advice is: try the tool you already have / use the problem formulation that is more natural to you. Check whether the tool can solve your problem. Switch tool only if the tool fails and you cannot improve your model.

