Add support to provide an initial primal solution

[hhkramer]

Is your feature request related to a problem? Please describe.
It would be a nice addition to Coluna if an initial primal solution could be provided in terms of the original formulation of the problem using set_start_value

Describe the feature or solution you'd like
Suppose one has a procedure that is able to find a primal solution to a given problem. It could be passed to Coluna after coding the original formulation using set_start_value. Then Coluna would translate such solution in terms of the original variables to columns in the master problem. The advantage is that the algorithm would start with both a primal bound and a feasible solution which could speed up the process to find an optimal solution. Yet, using set_start_value is both simple and familiar to those already using JuMP.

Do you feel able to code this feature or solution?
I'm not sure 😅

[guimarqu]

The transformation of the solution expressed on original variables into columns of the master does not seem obvious to me when multiplicities of the subproblems are different from one. I think it's the main problem to address before implementing the feature.

How do you think we can address this problem ? Do you think we can enumerate all the feasible solutions to a subproblem from the aggregated solution ?

[hhkramer]

Indeed @guimarqu

What do you mean with all feasible solutions? Do you think that enumerating upper_multiplicity columns would be sufficient to have a complete integer feasible solution?

[guimarqu]

I mean that we have to disaggregregate the solution to a subproblem that has multiplcity greater than one. I think we may have to enumerate all the the solutions to the subproblem that when combined can lead to the aggregated solution.

Another solution is to let the user provide the disaggregregated solution.

(what's your opinion @rrsadykov @artalvpes ?)

[rrsadykov]

I agree that it is not trivial to obtain a solution expressed with columns from a solution expressed with original variables (if the subproblems have multiplicity greater than one). What is more important is that Coluna cannot exploit for the moment an initial primal solution. Giving just the value of this solution (initial upper bound) is enough. Therefore, I do not find this feature useful (at least for now).