Trac #29493: Failing tests when using system glpk

At the moment there are two failing doctests, when using the glpk from
the system, e.g. on ubuntu eoan
[https://github.com/mkoeppe/sage/runs/542655821]

{{{
sage -t src/sage/numerical/backends/glpk_backend.pyx
**********************************************************************
File "src/sage/numerical/backends/glpk_backend.pyx", line 2287, in
sage.numerical.backends.glpk_backend.GLPKBackend.print_ranges
Failed example:
    p.print_ranges()
Expected:
    glp_print_ranges: optimal basic solution required
    1
Got:
    1
}}}

This doctest was mentioned before in #29317 with a suggestion for a fix.

{{{
 sage -t src/sage/libs/glpk/error.pyx
**********************************************************************
File "src/sage/libs/glpk/error.pyx", line 100, in
sage.libs.glpk.error.setup_glpk_error_handler
Failed example:
    res = p.solve()
Expected:
          0: obj = ...
Got:
    <BLANKLINE>
}}}

The problem is that we have doctests that rely on error-recovery
behavior added by a custom patch
[https://trac.sagemath.org/ticket/20710#comment:18],
which wasn't accepted by upstream. (The doctest for the patch was added
in #20832.)

The present ticket fixes the failures by
 * adding explicit input validation in several functions in the Cython
wrappers so that the GLPK error handler is not reached;
 * changing the default optimization mode from `glp_intopt` to
`glp_simplex_then_intopt`, which is more robust;
 * disabling the test for the error-recovery behavior, which provokes a
crash with unpatched GLPK.

URL: https://trac.sagemath.org/29493
Reported by: gh-kliem
Ticket author(s): Michael Orlitzky, Matthias Koeppe
Reviewer(s): Matthias Koeppe, Michael Orlitzky

src/sage/libs/glpk/error.pyx

@@ -21,7 +21,10 @@ from sage.numerical.mip import MIPSolverException
 
 class GLPKError(MIPSolverException):
     """
-    An error raised by the GLPK library.
+    A low-level error that is raised by ``sage_glpk_term_hook``.
+
+    The GLPK API considers these errors non-recoverable.  User code should not try
+    to catch this exception.
 
     EXAMPLES::
 
@@ -68,9 +71,17 @@ def setup_glpk_error_handler():
     Setup the GLPK error handler. Called automatically when this module
     is imported at Sage startup.
 
+    We install this error handler so that an error does not lead to
+    an immediate error exit of the process.  Instead, we raise a
+    ``GLPKError`` for the convenience of developers.
+
+    The GLPK API considers errors non-recoverable.
+    Therefore, user code should not try to catch this exception.
+
     TESTS::
 
-        sage: cython('''
+        sage: cython(             # optional - glpk_error_recovery_patch
+        ....: '''
         ....: # distutils: libraries = glpk z gmp
         ....: from cysignals.signals cimport sig_on, sig_off
         ....: from sage.libs.glpk.env cimport glp_term_out
@@ -97,8 +108,8 @@ def setup_glpk_error_handler():
         sage: p.add_constraint(3*x + 2*y <= 6)
         sage: p.add_constraint(x >= 0)
         sage: p.set_objective(x + y)
-        sage: res = p.solve()
-              0: obj = ...
+        sage: print('output', flush=True); res = p.solve()
+        output ... 0: obj = ...
         sage: res  # rel tol 1e-15
         2.4
     """

src/sage/libs/glpk/lp.pxd

@@ -90,3 +90,5 @@ cdef extern from "glpk.h":
     double glp_ios_mip_gap(glp_tree *T)
     int glp_ios_best_node(glp_tree *tree)
     double glp_ios_node_bound(glp_tree *T, int p)
+
+    int glp_bf_exists(glp_prob *lp)

src/sage/numerical/backends/glpk_backend.pxd

@@ -29,7 +29,7 @@ cdef class GLPKBackend(GenericBackend):
     cpdef __copy__(self)
     cpdef int print_ranges(self, filename = *) except -1
     cpdef double get_row_dual(self, int variable)
-    cpdef double get_col_dual(self, int variable)
+    cpdef double get_col_dual(self, int variable) except? -1
     cpdef int get_row_stat(self, int variable) except? -1
     cpdef int get_col_stat(self, int variable) except? -1
     cpdef eval_tab_row(self, int k)