Let us say that I have a nonlinear integer program $(P1)$ that is transformed to an integer linear one $(P2)$. Is there a gap between $(P1)$ and $(P2)$? At first, I thought that the gap should be zero (both $(P1)$ and $(P2)$ are equivalent) but I doubt that maybe it is not always zero. Is there any reference that can help me to answer this question?
2026-03-25 20:41:40.1774471300
Is there a gap between a nonlinear program and a linear one?
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For linear programming problems the optimal value of a target function is at the corners while for the non-linear programing the optimal value could occur at any point.
To better understand the difference, consider a piece of paper and cut it into a polygon.
If you keep this paper in various positions and observe the highest and the lowest point, you see that they happen at some corner or the other .
That is not necessarily the case if you fold the paper or cut it arbitrarily or deform it so it looses it flatness.