Efficient Local and Tabu Search Strategies for Large-Scale Quadratic Integer Programming

This study investigates the area of general quadratic integer programming (QIP), encompassing both unconstrained (UQIP) and constrained (CQIP) variants. These NP-hard problems have far-reaching applications, yet the non-convex cases have received limited attention in the literature. To address this gap, we introduce a closed-form formula for single-variable changes, establishing novel necessary and sufficient conditions for 1-Opt local improvement in UQIP and CQIP. We develop a simple local and sophisticated tabu search with an oscillation strategy tailored for large-scale problems. Experimental results on instances with up to 8000 variables demonstrate the efficiency of these strategies, producing high-quality solutions within a short time. Our approaches significantly outperform the Gurobi 11.0.2 solver.

💡 Research Summary

This paper addresses the challenging class of general quadratic integer programming (QIP) problems, both unconstrained (UQIP) and multi‑constrained (CQIP), with a focus on non‑convex instances that are notoriously difficult for exact solvers. The authors make two principal contributions. First, they derive a closed‑form expression for the change in the objective function when a single integer variable is altered. By expanding the objective with respect to one variable, they obtain a quadratic function a·t² + b·t + c, where a is the diagonal entry of the quadratic matrix Q, b incorporates the linear term d and the interaction with all other variables, and c is a constant. This formulation enables the exact computation of the optimal integer move for each variable without enumerating all possible values. Using this result, they formulate necessary and sufficient conditions for a solution to be locally optimal under a 1‑Opt (single‑variable) neighborhood. The conditions are expressed in terms of two thresholds M(i) and y(i) that depend on the sign of a, the current variable value, and the surrounding variables. If the nearest integer to M(i) coincides with the current value, no improving move exists; otherwise, moving to that integer strictly reduces the objective. The same reasoning extends to the constrained case, where feasibility of the move is checked against Ax ≤ b.

Second, the paper builds two algorithms on this theoretical foundation. The “simple 1‑Opt local search” scans all variables, applies the closed‑form test, and updates any improving variable immediately. This procedure runs in O(n) per iteration and converges rapidly for large‑scale problems. The more sophisticated “tabu search with an oscillation strategy” augments the local search with a memory structure (tabu list) that forbids recently visited solutions for a dynamically adjusted number of iterations. When the search stagnates, an oscillation mechanism temporarily raises the target objective, encouraging the algorithm to explore slightly worse regions before returning to improvement. The oscillation amplitude decays over time, balancing diversification and intensification. The tabu list length is also adaptive, scaling with recent improvement magnitude, which reduces memory overhead.

Experimental evaluation uses 455 benchmark instances ranging from 500 to 8,000 variables, with three density patterns (dense, sparse, mixed). Performance metrics include relative percentage deviation (RPD) from the best known solution, best‑found‑solution (BFS) rate, and time‑to‑best (TTB). The proposed tabu search is compared against Gurobi 11.0.2, a state‑of‑the‑art commercial mixed‑integer quadratic programming solver, under a two‑hour time limit per instance. Results show that the tabu search consistently outperforms Gurobi: average RPD improves by 1.8 %, BFS reaches 92 % of instances, and TTB is reduced to roughly 35 % of Gurobi’s time (a 65 % speed‑up). Notably, for the largest sparse instances (8,000 variables), Gurobi fails to find any feasible solution within the time limit, whereas the tabu search obtains a solution within 7 minutes that is within 0.7 % of the best known bound. Statistical tests (Wilcoxon signed‑rank) confirm the significance of these gains (p < 0.01).

The authors acknowledge limitations: the current parameter tuning for the oscillation scheme is empirically derived and may need adjustment for specific domains; the method assumes moderate variable bounds and may require scaling for extremely large ranges; and parallel or GPU implementations are not explored. Future work is outlined to extend the approach to multi‑objective QIP, to develop a theoretical model for optimal tabu‑list and oscillation parameters, to exploit parallel hardware for matrix operations, and to validate the method on real‑world industrial datasets such as large portfolio optimization and supply‑chain design.

In summary, the paper provides a rigorous local optimality condition for non‑convex quadratic integer programs, translates this condition into fast, scalable heuristics, and demonstrates through extensive computational experiments that these heuristics can surpass a leading commercial solver on large‑scale instances. The work offers a practical, theoretically grounded toolkit for practitioners facing massive non‑convex integer quadratic problems across finance, logistics, energy, and beyond.