If one uses Sage's MixedIntegerLinearProgram class for solving an LP, it is useful to access dictionary (tableau) data corresponding to the current (often optimal) basis. For example, one needs this to compute tableau cutting planes such as Gomory's fractional cut.

The Sage interface should, of course, be designed to work consistently across all solvers. There are some subtleties here. Best to be compatible with a library that has already sorted most of it out: In the COIN open solver interface, there are the functions getBInvARow, getBInvRow, getBInvACol, getBInvCol.
Unfortunately these functions do not seem to be implemented in Osi's GLPK backend; so one needs to use one of Osi more complete backends such as CPLEX or CLP to find the precise definition.

#18732 adds the necessary backend functions for the GLPK backend.
#18763 adds the necessary backend functions for the COIN (CBC/CLP) backend.

This kind of information is available in every solver.

See also #18688, and also #18804 (LPBackendDictionary).

Depends on #18732

CC: @nathanncohen @yuan-zhou

Component: numerical

Issue created by migration from https://trac.sagemath.org/ticket/18733