[docs] refactor and re-organize the documentation #574
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| global optimality of the resulting solution (if the solver succeeds. For some | ||
| models, the solver may experience numerical difficulty). |
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this is definitely better, but note both solutions said they were "OPTIMAL". I'm not really sure it's just a numeric issue as much as an algorithmic issue; e.g. https://optimization-online.org/wp-content/uploads/2018/07/6695.pdf shows SDPA-GMP failing on some SDP's with a duality gap. Although I've seen SDPA-GMP fail easily on many problems without sufficient presolve so maybe that is a factor there too.
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Thanks for citing my blog! I second this comment: I had in fact a similar issue with MOSEK: the problem was not strictly feasible, and as expected MOSEK failed to solve it, but then falsely claimed that it was "OPTIMAL". I submitted a bug report to MOSEK but only managed to get insulted by the CEO.
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As for the proof, there's a simpler argument: a matrix A is positive semidefinite iff A = B'*B for some matrix B. Now if A has a diagonal element equal to zero, then v'*A*v = 0 for some basis vector v. But v'*A*v = v'*B'*B*v = norm(B*v)^2, so B*v = 0, which implies B'*B*v = A*v = 0, so the entire column must be zero.
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ah, that is a nice argument. It may be worth adding that to your blog post, or at least noting that the entire column is zero (and therefore X13 = 0) rather than jumping to X13=0, since providing the intermediate step can help readers figure it out (or figure out what to look up).
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Fair enough, I added it to the blog post (as a footnote).
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I don't want to derail this PR with a discussion, so I've added a TODO to #344. Once this PR is merged I'm happy to review other changes.
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You could also argue that "if the solver succeeds" is true, and that solvers reporting OPTIMAL are a bug. That's their problem, not ours.
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