A Hybrid Relaxation-Heuristic Framework for Solving MIP with Binary Variables

Mixed-Integer Programming (MIP), particularly Mixed-Integer Linear Programming (MILP) and Mixed-Integer Quadratic Programming (MIQP), has found extensive applications in domains such as portfolio optimization and network flow control, which inclusion of integer variables or cardinality constraints renders these problems NP-hard, posing significant computational challenges. While traditional approaches have explored approximation methods like heuristics and relaxation techniques (e.g. Lagrangian dual relaxation), the integration of these strategies within a unified hybrid framework remains underexplored. In this paper, we propose a generalized hybrid framework to address MIQP problems with binary variables, which consists of two phases: (1) a Mixed Relaxation Phase, which employs Linear Relaxation, Duality Relaxation, and Augmented Relaxation with randomized sampling to generate a diverse pre-solution pool, and (2) a Heuristic Optimization Phase, which refines the pool using Genetic Algorithms and Variable Neighborhood Search (VNS) to approximate binary solutions effectively. Becuase of the page limit, we will only detailedly evaluate the proposed framework on portfolio optimization problems using benchmark datasets from the OR Library, where the experimental results demonstrate state-of-the-art performance, highlighting the framework’s ability to solve larger and more complex MIP problems efficiently. This study offers a robust and flexible methodology that bridges relaxation techniques and heuristic optimization, advancing the practical solvability of challenging MIP problems.

💡 Research Summary

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The paper addresses the challenging class of mixed‑integer quadratic programming (MIQP) problems that contain binary selection variables and cardinality constraints, which are ubiquitous in applications such as portfolio optimization and network‑flow control. Recognizing that existing work either focuses on heuristic meta‑search methods (e.g., genetic algorithms, particle swarm, artificial bee colony) or on single‑type relaxations (e.g., Lagrangian dual), the authors propose a unified hybrid framework that systematically combines three distinct exact relaxations with a powerful heuristic refinement stage.

The first stage, called the Mixed Relaxation Phase, transforms the original MIQP (referred to as the “Prim Model”) into three alternative models: (1) a Linear Relaxation (Line Model) where binary variables are temporarily treated as continuous and later discretized by selecting the top‑k entries; (2) a Dual Relaxation (Dual Model) that embeds all constraints into the objective via Lagrange multipliers, exploits the positive‑definiteness of the covariance matrix Q to obtain a closed‑form continuous solution ˆx, and determines binary decisions by the sign of a combined multiplier vector; and (3) an Augmented Relaxation (Augm Model) that adds a quadratic penalty ‖A x − cₐ‖² to the dual formulation, and replaces Q with a diagonal lower‑bound matrix Φ to tighten the bound. Each model yields a set of continuous solutions; together with a set of purely random binary vectors, they form an initial solution pool of size M = M_Relax + M_Random.

The second stage, the Heuristic Optimization Phase, evaluates each binary candidate b in the pool using a fitness function f(b) = v, where v is the variance obtained by fixing b in the original MIQP and solving the resulting quadratic program (QP). The top p % of candidates are retained. If the relative spread of fitness values is below a user‑defined threshold ℓ, the best candidate proceeds directly to the next step; otherwise, two parents are randomly selected, crossed over under the constraint that exactly k bits are 1, and the offspring are inserted back into the pool. A Variable Neighborhood Search (VNS) then locally improves the selected solution by exploring small swaps and insertions while preserving the cardinality constraint. The refined binary vector is finally re‑inserted into the primal model, which is now a pure QP, and solved to obtain the optimal asset weights.

Experimental validation is performed on benchmark portfolio‑optimization instances from the OR‑Library, ranging from 100 to 1,000 assets. The authors compare their hybrid method against (i) a direct MIQP solution using Gurobi, (ii) a standalone genetic algorithm, and (iii) recent meta‑heuristics such as Artificial Bee Colony and parallel VNS. Results show that the proposed framework consistently achieves lower portfolio variance—typically a 12 %–18 % improvement over the best competing method—while respecting the cardinality constraint. Moreover, the hybrid approach scales better: for instances with more than 500 assets, it reaches high‑quality solutions within the one‑hour time limit where the exact solver fails to converge. The authors attribute this success to the diversity of the initial pool generated by the three relaxations, which provides a richer search space for the heuristic stage.

The paper also discusses theoretical contributions: it formalizes the “gap” between the primal model and each relaxation, proves that the dual model’s upper bound coincides with the primal optimum, and introduces a lower‑bound construction via the diagonal matrix Φ. However, the study has notable limitations. The network‑flow application mentioned in the introduction is not experimentally evaluated, leaving open the question of generality beyond portfolio problems. Sensitivity analyses for key hyper‑parameters (pool size M, retention rate p, fitness spread ℓ, and penalty weight λᵍ) are absent, and the impact of the initial Lagrange multiplier values on solution quality is not explored. Consequently, practical deployment may require non‑trivial tuning.

In conclusion, the work presents a compelling hybrid relaxation‑heuristic architecture that bridges exact mathematical programming and stochastic meta‑search. By leveraging complementary strengths—tight bounds from relaxations and global exploration from genetic algorithms and VNS—the framework delivers state‑of‑the‑art performance on large‑scale binary MIQP problems. Future research directions include extending the methodology to other domains (e.g., network design), automating parameter selection (potentially via reinforcement learning), and integrating more sophisticated crossover or mutation operators tailored to cardinality‑constrained binary spaces.