Trac #20296: MixedIntegerLinearProgram: New backend using Interactive…

…LPProblem

If one has to solve a small LP with irrational (say, `AA`) data (and
needs access to the exact solution), the only available tool is the
didactical implementation of the simplex method in
`InteractiveLPProblem` (but see #18735). This ticket implements a
`MixedIntegerLinearProgram` backend using `InteractiveLPProblem`.

Example:
{{{
            sage: poly = polytopes.dodecahedron(base_ring=AA)
            sage: lp = poly.to_linear_program(solver='InteractiveLP')
            sage: b = lp.get_backend()
            sage: b.set_objective([1, 1, 1])
            sage: lp.solve()
            2.291796067500631?
}}}
(This example uses backend functions because of #20301; and the
`base_ring=AA` is there because of #13041.)

URL: http://trac.sagemath.org/20296
Reported by: mkoeppe
Ticket author(s): Matthias Koeppe
Reviewer(s): Andrey Novoseltsev, Dima Pasechnik

src/module_list.py

@@ -1147,6 +1147,9 @@ def uname_specific(name, value, alternative):
     Extension("sage.numerical.backends.glpk_graph_backend",
               ["sage/numerical/backends/glpk_graph_backend.pyx"]),
 
+    Extension("sage.numerical.backends.interactivelp_backend",
+              ["sage/numerical/backends/interactivelp_backend.pyx"]),
+
     OptionalExtension("sage.numerical.backends.gurobi_backend",
               ["sage/numerical/backends/gurobi_backend.pyx"],
               libraries = ["stdc++", "gurobi"],

src/sage/numerical/backends/coin_backend.pyx

@@ -683,7 +683,7 @@ cdef class CoinBackend(GenericBackend):
 
         INPUT:
 
-        - ``indices`` (list of integers) -- this list constains the
+        - ``indices`` (list of integers) -- this list contains the
           indices of the constraints in which the variable's
           coefficient is nonzero
 

src/sage/numerical/backends/generic_backend.pxd

@@ -51,4 +51,4 @@ cdef class GenericBackend:
 
     cdef object obj_constant_term
 
-cpdef GenericBackend get_solver(constraint_generation = ?, solver = ?)
+cpdef GenericBackend get_solver(constraint_generation = ?, solver = ?, base_ring = ?)

src/sage/numerical/backends/interactivelp_backend.pxd

@@ -0,0 +1,23 @@
+##############################################################################
+#       Copyright (C) 2010 Nathann Cohen <nathann.cohen@gmail.com>
+#       Copyright (C) 2016 Matthias Koeppe <mkoeppe@math.ucdavis.edu>
+#  Distributed under the terms of the GNU General Public License (GPL)
+#  The full text of the GPL is available at:
+#                  http://www.gnu.org/licenses/
+##############################################################################
+
+from sage.numerical.backends.generic_backend cimport GenericBackend
+
+cdef class InteractiveLPBackend(GenericBackend):
+
+    cdef object lp
+    cdef object row_names
+    cdef object prob_name
+
+    cdef object lp_std_form
+    cdef object final_dictionary
+    cdef int verbosity
+
+    cpdef dictionary(self)
+
+    cpdef interactive_lp_problem(self)