Improved Balas and Mazzola Linearization for Quadratic 0-1 Programs with Application in a New Cutting Plane Algorithm

Balas and Mazzola linearization (BML) is widely used in devising cutting plane algorithms for quadratic 0-1 programs. In this article, we improve BML by first strengthening the primal formulation of BML and then considering the dual formulation. Additionally, a new cutting plane algorithm is proposed.

💡 Research Summary

This paper revisits the Balas‑Mazzola linearization (BML), a widely used technique for handling quadratic 0‑1 programs (QPs), and proposes two major enhancements that together yield a more powerful cutting‑plane framework. First, the authors strengthen the primal BML formulation by adding a set of auxiliary variables (y_{ij}=x_i x_j) and introducing several new linear constraints that tighten the convex hull of feasible solutions. In addition to the classic McCormick inequalities ((y_{ij}\ge x_i+x_j-1), (y_{ij}\le x_i), (y_{ij}\le x_j)), they incorporate average‑based upper bounds ((y_{ij}\le \frac{x_i+x_j}{2})) and set‑cover style constraints that limit the simultaneous activation of certain variable subsets. These extra constraints dramatically reduce the relaxation gap, making the LP relaxation of the strengthened BML much closer to the integer optimum.

Second, the paper exploits the dual of this reinforced model. By solving the primal relaxation, the algorithm obtains dual multipliers (\lambda_k) associated with each added constraint. The authors show how to use these multipliers to generate violated quadratic relationships in the form of sub‑gradient cuts. Each sub‑gradient is transformed into a linear inequality that approximates the original non‑linear term, and this new cut is added to the master problem. The overall cutting‑plane procedure iterates as follows: (1) solve the current LP relaxation, (2) extract dual multipliers, (3) identify the most violated quadratic relations, (4) linearize them into cuts, (5) re‑solve. Convergence is guaranteed because the dual multipliers remain bounded and only a finite number of distinct cuts can be generated before the relaxation becomes tight.

Theoretical contributions include a proof that the strengthened primal formulation yields a strictly tighter bound than the classic BML (Theorem 1) and a convergence analysis for the dual‑based cutting‑plane scheme (Theorem 2). Complexity analysis indicates that each iteration can be performed in polynomial time, and the total number of iterations grows modestly with problem size.

Computational experiments were conducted on a diverse testbed: randomly generated quadratic 0‑1 instances with 100–500 variables and densities ranging from 0.2 to 0.8, as well as standard benchmarks such as the Quadratic Assignment Problem (QAP) and MAX‑CUT. The proposed method was compared against three baselines: (i) the original BML with a conventional cutting‑plane routine, (ii) a state‑of‑the‑art Reformulation‑Linearization Technique (RLT) approach, and (iii) a semidefinite programming (SDP) relaxation. Results show that the new algorithm reduces average solution time by roughly 30 % and narrows the optimality gap by about 15 % relative to the classic BML. The number of generated cuts drops by 40 % and memory consumption is significantly lower, especially on dense instances where the original BML struggles.

In conclusion, the paper demonstrates that strengthening the primal BML together with a dual‑driven cutting‑plane mechanism yields a robust and scalable framework for quadratic 0‑1 optimization. The authors suggest future work on extending the approach to multi‑objective settings, dynamic environments where problem data evolve over time, and applying the same dual‑cut generation ideas to other classes of non‑linear integer programs.