SIAM Journal on Optimization, Volume 30, Issue 1, Page 781-797, January 2020.
Mixed integer quadratic programming (MIQP) is the problem of minimizing a quadratic function over mixed integer points in a rational polyhedron. This paper focuses on the augmented Lagrangian dual (ALD) for MIQP. ALD augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function. Finally, we prove a polynomial bound on the weight on the penalty term to obtain a zero gap.

中文翻译：

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混合整数二次规划的精确增强拉格朗日对偶

SIAM优化杂志，第30卷，第1期，第781-797页，2020年1月。
混合整数二次规划（MIQP）是使有理多面体中混合整数点上的二次函数最小化的问题。本文重点关注MIQP的增强拉格朗日对偶（ALD）。ALD通过对二重约束施加加权非线性罚分来增强通常的Lagrangian对偶。我们首先证明，在惩罚函数的某些温和条件下，随着惩罚上的权重达到无穷大，ALD将渐近达到零对偶间隙。接下来我们表明，当我们使用任何范数作为惩罚函数时，有限的惩罚权重足以实现零差距。最后，我们证明了惩罚项权重的多项式边界以获得零间隙。