Optimization Over Integers Pdf May 2026

We consider optimization problems in which the decision variables are restricted to integer values. Unlike continuous optimization, the feasible set is discrete, non‑convex, and often finite. Standard Form A general integer optimization problem can be expressed as:

[ \begin{aligned} \min_{\mathbf{x}} \quad & f(\mathbf{x}) \ \text{s.t.} \quad & g_i(\mathbf{x}) \leq 0, \quad i = 1,\dots,m \ & h_j(\mathbf{x}) = 0, \quad j = 1,\dots,p \ & \mathbf{x} \in \mathbb{Z}^n \end{aligned} ] optimization over integers pdf

Even this linear case is NP‑hard in general. 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. We consider optimization problems in which the decision