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Many thanks @Zhengyizhe0209-arch! It's a nice PR.

I think the wording should be a bit more specific because the lecture calls linprog without setting method, so SciPy uses the default method='highs'.

In SciPy, highs is an interface to the HiGHS optimization solver, and it automatically chooses between the HiGHS dual simplex solver and the HiGHS interior-point solver.

Because of this, I would avoid saying that SciPy necessarily converts the problem to standard form by adding slack variables before solving, since that description is mainly tied to the legacy linprog methods.

I think this is also the point that is being raised in #599.

A simpler replacement could be:

SciPy accepts inequality constraints in the form $A_{ub} x \leq b_{ub}$, equality constraints in the form $A_{eq} x = b_{eq}$, and variable bounds.

In this lecture, linprog uses SciPy’s default highs method, which calls the HiGHS optimization solver.

The slack value returned by linprog is a one-dimensional NumPy array whose entries measure the difference $b_{ub} - A_{ub}x$ for each inequality constraint.

Another small comment: you noted that this PR addresses #599, but it only tackles the first part of the issue.

It would be nice to address the entire issue in one PR or explicitly state that it addresses the first or second part of the issue to avoid closing it without fully resolving it.

Many thanks for the detailed explanation and suggestions!

That makes sense. I agree that my current wording is too broad, especially since linprog uses SciPy’s default method='highs' here and therefore calls the HiGHS solver rather than necessarily following the legacy standard-form/slack-variable transformation description.

I will revise the paragraph to avoid implying that SciPy always converts the problem to standard form by adding slack variables before solving. I will use the simpler wording you suggested, focusing on the accepted constraint forms, the default highs method, and the interpretation of the returned slack values.

I will also update the PR description to say that it addresses the first part of #599 rather than the whole issue.

Thanks again for the helpful review！

Thanks for the PR @Zhengyizhe0209-arch — nice work on the first part.

I pushed a follow-up commit that addresses the second part of #599 (the OR-Tools prose had x3/x5 values swapped) and fixes several other issues I noticed while reviewing the lecture:

• OR-Tools prose: corporate bond is $50,000 (not $4,648.825) and year 3 borrows $20,000 (not "bank balance $50,000")
• Removed incorrect note claiming OR-Tools and SciPy give different solutions — the solution is unique (objective is strictly increasing in x1 and x5, so both hit their feasible limits)
• Math error: slack variable transformation had RHS = 0 instead of b_i
• Wrong comment: A_ex2/b_ex2 are equality constraints, not inequality
• Prose: iso-revenue lines show product combinations, not materials/labor; clarified the firm's objective sentence; fixed "creates two variables" for the 5-variable example
• Grammar: "keep invest" → "keep investing", "it's total" → "its total", "where in" → "where"
• Missing constraint: added x, y ≥ 0 to Exercise 2 formulation

This should fully close #599.