Modern solvers (like CPLEX, Gurobi, or open-source COIN-OR) use a hybrid approach. They combine the tree search of Branch and Bound with the tightening power of Cutting Planes at every node of the tree.
Several solution methods have been developed to solve integer optimization problems, including: optimization over integers pdf
In applied settings, moderate‑size MIPs (hundreds to thousands of variables) can often be solved to optimality using modern solvers (e.g., Gurobi, CPLEX, SCIP). Large‑scale integer problems typically rely on heuristics or decomposition methods. Modern solvers (like CPLEX, Gurobi, or open-source COIN-OR)
Minimize (or Maximize) f(x) Subject to: g(x) ≤ 0 h(x) = 0 x ∈ ℤ^n Modern solvers (like CPLEX