Would you tell me some main distinctions between the convex discrete minimization and maximization optimization problems?
In the case of the feasible are bounded then we only need to transform one to another (by multiple -1) and solve only one of them. However, in other cases, we can not do that. Which problem is usually more difficult and why?
For continuous problems, minimizing a convex function on a convex domain is considered an easy problem, because there is only ever one local minimum, and a local minimum is the global minimum. Finding that minimum (when it exists) can be done by local search methods.
On the other hand, maximizing a convex function on a convex domain is a hard problem, because there may be very many local maxima, and local properties give essentially no clue about whether you are close to the global maximum.
But adding discreteness (say, requiring variables to be integers) can make an easy continuous problem into a very hard problem.
If the continuous version of a problem is easy to solve, that can be helpful in e.g. a branch-and-bound algorithm, where you can use the optimal value of the continuous version (with some variables fixed) as a bound on the optimal value of the discrete problem, as well as using a rounding procedure on the optimal solution of the continuous version to get good solutions of the discrete problem. Of course that isn't helpful if the continuous version is hard to solve.
