correcting max-flow problem in MILP tutorial

[anshu129]

tdf #35497 incorrect max-flow problem formulation in MILP tutorial

I have made changes in two files , just completed the objective function as described in the comments.
p.set_objective(p.sum(f[s,u] for u in g.neighbors_out(s)) - p.sum(f[v,s] for v in g.neighbors_in(s)))

#35497 (comment)

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

[dcoudert]

Please rebase this branch on last beta. I will then review it.

[anshu129]

@dcoudert I have tried rebasing it . if there is any problem. plz let me know.

[dcoudert]

LGTM.

[vbraun]

I'm getting

sage -t --long --warn-long 55.4 --random-seed=123 src/sage/tests/books/computational-mathematics-with-sagemath/lp_doctest.py
**********************************************************************
File "src/sage/tests/books/computational-mathematics-with-sagemath/lp_doctest.py", line 194, in sage.tests.books.computational-mathematics-with-sagemath.lp_doctest
Failed example:
    p.solve()
Exception raised:
    Traceback (most recent call last):
      File "/home/release/Sage/src/sage/doctest/forker.py", line 714, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/home/release/Sage/src/sage/doctest/forker.py", line 1144, in compile_and_execute
        exec(compiled, globs)
      File "<doctest sage.tests.books.computational-mathematics-with-sagemath.lp_doctest[58]>", line 1, in <module>
        p.solve()
      File "sage/numerical/mip.pyx", line 2638, in sage.numerical.mip.MixedIntegerLinearProgram.solve
        self._backend.solve()
      File "sage/numerical/backends/glpk_backend.pyx", line 1142, in sage.numerical.backends.glpk_backend.GLPKBackend.solve
        raise MIPSolverException("GLPK: "+solve_status_msg.get(solve_status, "unknown error during call to GLPK : "+str(solve_status)))
    sage.numerical.mip.MIPSolverException: GLPK: The LP (relaxation) problem has no dual feasible solution
**********************************************************************

[anshu129]

@vbraun @dcoudert @mkoeppe any suggestions, how I can correct the error and close the PR.

[dcoudert]

try to change

 Sage example in ./lp.tex, line 681::
 
-  sage: for e in g.edges(sort=True, labels = False): p.add_constraint( f[e] <= 1 )
+  sage: for e in g.edges(labels=False): p.add_constraint(f[e], min=0, max=1)

[github-actions]

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