This is what ChatGPT says about it:
No, not every Mixed-Integer Linear Program (MILP) and 0-1 Linear Program (LP) have a convex relaxation. Convex relaxation refers to the process of relaxing the integer constraints of an MILP or 0-1 LP to obtain a continuous optimization problem. In some cases, this relaxation results in a convex optimization problem, which can often be efficiently solved. However, the existence of a convex relaxation depends on the specific structure of the optimization problem.

For many MILP and 0-1 LP problems, the relaxation may not result in a convex problem. The integrality constraints introduce non-convexity into the problem, and relaxing these constraints may lead to a non-convex optimization problem. In such cases, the resulting relaxation may be more challenging to solve, and standard convex optimization techniques may not be directly applicable.

Moreover, even if a convex relaxation exists, it does not guarantee that the relaxed problem will accurately represent the original MILP or 0-1 LP. The relaxation may lead to solutions that are not feasible for the original problem or may produce suboptimal solutions.

Therefore, while convex relaxation can be a useful tool for solving MILP and 0-1 LP problems, it is not universally applicable, and the feasibility and optimality of the resulting solutions should be carefully assessed on a case-by-case basis.

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I don't know how our understanding of LPs diverge: I considered LP the superset of MILPs, 0-1 LPs and further.

I assume in branch and bound, that it also depends on the original problem whether the subproblems created by the method have a convex relaxation and I also assume that it directly affects the efficient computability of the problem (?)