> Is there any difficulty for lattice based method to be extended to solve MILP?
I don't think that continuous variables are an issue. Even when all the explicit variables are integer, we have implicit continuous variables as soon as we have an inequality: the slack of that inequality. There is probably some linear algebra trick one can use to transform any problem into a form that is convenient for lattice-based algorithms.
> How likely do you think this lattice type algorithm will overcome the difficulties you mentioned and eventually replace B&B, totally or partly (like barrier vs simplex methods)?
Very unlikely in the next 5 years. Beyond that, they could be the next small revolution, maybe. "Cutting planes" were another tool that had some good theory but were thought to be impractical. Then 25 years ago, people found a way to make them work, and they were a huge boost to solvers. We may be due for another big jump.
Lattice-based method are already effective in some niches. Branch-and-bound solvers are horrible at cryptography and number theory problems (those problems are bad fits for floating-point arithmetic in general), and lattice-based methods shine there. There are also some rare dense optimization problems that benefit from lattice-based methods (typically, one would use lattices in a pre-processing step, then pass the reformulated problem to a regular branch-and-bound solver [1]).
I don't think that continuous variables are an issue. Even when all the explicit variables are integer, we have implicit continuous variables as soon as we have an inequality: the slack of that inequality. There is probably some linear algebra trick one can use to transform any problem into a form that is convenient for lattice-based algorithms.
> How likely do you think this lattice type algorithm will overcome the difficulties you mentioned and eventually replace B&B, totally or partly (like barrier vs simplex methods)?
Very unlikely in the next 5 years. Beyond that, they could be the next small revolution, maybe. "Cutting planes" were another tool that had some good theory but were thought to be impractical. Then 25 years ago, people found a way to make them work, and they were a huge boost to solvers. We may be due for another big jump.
Lattice-based method are already effective in some niches. Branch-and-bound solvers are horrible at cryptography and number theory problems (those problems are bad fits for floating-point arithmetic in general), and lattice-based methods shine there. There are also some rare dense optimization problems that benefit from lattice-based methods (typically, one would use lattices in a pre-processing step, then pass the reformulated problem to a regular branch-and-bound solver [1]).
[1] https://link.springer.com/chapter/10.1007/3-540-48777-8_1