Just chiming in here... I don't know that I would consider this counter-intuitive. Most introductory MIP textbooks make it amply clear that that IP and LP relax

Just chiming in here... I don't know that I would consider this counter-intuitive. Most introductory MIP textbooks make it amply clear that that IP and LP relaxations wouldn't be the same in general (except in special cases like when the constraint matrix is totally unimodular). Even so, sometimes LP relaxations with rounding can produce "good enough" answers if an IP solution is intractable in a reasonable time. Consider the Wolsey example; the optimal integer objective is 5.0, and the LP relaxation (376/193,950/193) objective is an optimistic 5.10, whereas the closest rounded LP relaxation (2,4) objective is 4.56. Even though the x may be in a completely different area than the integer optimal solution, it still manages to achieve roughly the same objective. So depending on the application, the rounded LP relaxation may just be good enough, especially in iterative processes (like control) where you just need to take baby steps toward a larger objective. It's not optimal, but it's not terrible.