Lagrangian Relaxation approach solves quadratic integer programming. The example is from reference [1]. The detail of algorithm can be seen https://zhuanlan.zhihu.com/p/145944142/
[1] Bragin M A, Luh P B, Yan J H, et al. Convergence of the Surrogate Lagrangian Relaxation Method[J]. Journal of Optimization Theory and Applications, 2015, 164(1): 173-201.