An analytical lower bound for a class of minimizing quadratic integer optimization problems

Lower bounds for minimization problems are essential for convergence of both branching-based and iterative solution methods for optimization problems. They also serve an important role in evaluating the quality of feasible solutions by providing conservative optimality gaps. We derive a closed-form analytical lower bound for a class of quadratic optimization problems with binary decision variables. Unlike traditional lower bounds obtained by solving relaxed models, our bound is purely analytical and does not require numerically solving any optimization problem. This is particularly valuable for problem instances that are too large to even formulate or load into a solver due to memory limitations. Further, we propose a greedy heuristic for obtaining feasible solutions. Together, the analytical bound and heuristic provide a provable optimality gap without solving any optimization model. Numerical experiments demonstrate that we can solve real-world large-scale instances, that were previously unsolvable due to memory limitations, in under a minute with provable optimality gaps of under 7%. For smaller instances where the optimal solution is computable, our greedy solutions are about 1% away from the optimal. These results highlight the practical value and scalability of our approach when direct solution methods are computationally prohibitive.