DOC: add a tutorial of scipy.optimize.linprog

[AtsushiSakai]

Reference issue

To fix a part of #10358

What does this implement/fix?

I added a tutorial of scipy.optimize.linprog.
I just added a standard linear programing problem as a tutorial in this PR.

Additional information

If this PR is accepted, I would like to add some other tutorials with scipy.optimize.linprog like:

• max-flow program solution with scipy.optimize.linprog
• minimum-cost flow problem solution with scipy.optimize.linprog
  etc.

[mdhaber]

Thanks for working on this. One thing is that bound constraints on the decision variables should use bounds, not A_ub and b_ub. (Actually, this problem doesn't even need explicit bounds, since the default bounds enforce non-negativity.) Also, the tutorial should be more educational than the example in the reference documentation. How do you turn a real problem into equations? How do you turn the equations into matrices? What if it were a maximization problem, or if one of your constraints were >=? It would also be nice to show what some of the options do. Thanks!

[mdhaber]

Also, @Kai-Striega was working on this, so we should see how his was going

[AtsushiSakai]

Thank you for your comment!. I got it. I will try to improve the tutorial.

[Kai-Striega]

@AtsushiSakai I had started writing one, but never finished it. You're welcome to write your own!

[AtsushiSakai]

@mdhaber Hi. I revised the example. PTAL.

[mdhaber]

Much better. The most important comment is to add other kinds of bounds. Besides that, please also show that revised simplex gets a more accurate solution (at the expense of speed) for large problems. Perhaps conclude by showing how the fun, con, and slack reported correspond with the original problem. It would be nice to show that interior-point accepts sparse matrices as input, but perhaps you could leave that for another problem that is actually sparse.

[mdhaber]

Oops I meant "Request changes"

[mdhaber]

I think a feasible problem would be preferable. Better yet, you could keep the existing problem, explain that the infeasible means that there is no solution vector that satisfies all the constraints, then pretend that you realized one of your constraints was wrong. Adjust it, and get a solution, then show that c@x gives the objective function values and x satisfies all the constraints.

[AtsushiSakai]

I agreed. The problem should be feasible. I didn't notice that. I'm sorry. I will try fix the code again based on your comments. I will request a review again when my code is ready.

[AtsushiSakai]

@mdhaber I think the code is ready to review. PTAL

[mdhaber]

I like the navigation. Thanks.
The new math and code for the bounds didn't render properly, though,

and the output doesn't match the new code.

[AtsushiSakai]

@mdhaber I’m sorry for my mistakes. I fixed it, PTAL.

[mdhaber]

If on GitHub you just "commit suggestion" for each of these (and don't change anything else), I can merge without further review.

[AtsushiSakai]

I merged all your suggentions. PTAL.

[mdhaber]

Thanks @AtsushiSakai!

[AtsushiSakai]

@mdhaber Thank you so much too!!. I'm sorry for taking your time many times. I'll try to do better next time.