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Either you have a neighboring vertex that is strictly better than your current solution, or you have found an optimal solution. I don't think you need to store vertices with equal value, because you never backtrack.

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Ok, I think I'm getting it. This line from the OP confused me:

'But there’s an obvious risk of winding up in a “cycle” where you get back to a vertex that you already hit earlier–in which case the program will never terminate.'

You only get a cycle if multiple vertices have the same objective function value. There's a similar problem of degeneracy, where you pivot back and forth between binding constraints on the same vertex and never go anywhere.

And a way to avoid cycles in simplex is known: https://en.wikipedia.org/wiki/Bland%27s_rule

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