add_col and add_variable both add a variable to the problem; but add_col only allows to add a column with no name; whereas add_variable only allows to add a column with no coefficients.
There should be one function (add_variable, probably - see #20296 for a possible interface) that can do both; should then deprecate add_col.
Note that add_col is not used by MixedIntegerLinearProgram; it is only used in doctests of the backends.
Also, GurobiBackend does not implement add_col (this is fixed in #20424).
(Also compare with add_linear_constraint, which takes a zipped index/coefficient list, whereas add_col takes two parallel lists.)

While making this change, should also clarify the defaulting behavior of the binary, continuous, integer arguments of add_variable. The argument list is:

cpdef int add_variable(self, lower_bound=0.0, upper_bound=None, binary=False, continuous=False, integer=False, obj=0.0, name=None)

Whereas the documentation says:

• binary - True if the variable is binary (default: False).
• continuous - True if the variable is binary [SIC!] (default: True).
• integer - True if the variable is binary [SIC!] (default: False).
  Probably the defaults in the arglist should be None, and defaulting to creating continuous variables should be explained separately.
  (See MIP backends: Make variable type consistent between add_variable, set_variable_type (with deprecation) #20362 for a discussion of a possible better interface and the mismatch with set_variable_type.)

Also #23600 should be fixed: the signatures of the cpdef functions should be the same in all implementations to fix Cython vtable warnings.

CC: @dimpase @nathanncohen @videlec

Component: numerical

Issue created by migration from https://trac.sagemath.org/ticket/20324