AI 中文总结
本文针对混合整数二阶锥规划，通过几何分析提出平衡分离深度与角度新颖性的割生成策略，并引入渐进整数性外近似框架，结合二者可显著降低外近似的计算工作量。
AI 中文摘要
混合整数二阶锥规划（MISOCP）通常通过多面体外近似（OA）求解，该方法通过割平面迭代强化锥可行域的线性松弛。本文研究如何更高效地构建此类近似：首先，分析新生成割相对于松弛中已有割的边际贡献，基于违反度和体积度量表明，当新支撑方向趋近于已有方向时，该贡献至少呈线性下降；受此几何分析启发，提出平衡分离深度与角度新颖性的割生成策略，该策略可实现闭式构造。其次，引入渐进整数性OA框架，其从线性规划（LP）松弛出发，经部分整数松弛后到达完整混合整数线性规划（MILP），利用低成本早期迭代在后续混合整数求解前强化近似。对CBLIB实例及大规模机组组合模型的计算实验表明，所提割策略与渐进整数性具有互补效益，可大幅降低外近似的计算工作量。
英文摘要
Mixed-integer second-order cone programs are commonly solved by polyhedral outer approximation (OA), which iteratively strengthens a linear relaxation of the conic feasible region through cutting planes. We study how such approximations can be constructed more efficiently. First, we analyze the marginal contribution of a newly generated cut relative to cuts already present in the relaxation. Using violation- and volume-based measures, we show that this contribution decreases at least as fast as linearly as the new supporting direction approaches an existing one. Motivated by this geometric analysis, we develop a cut-generation strategy that balances separation depth with angular novelty and admits closed-form constructions. Second, we introduce a progressive-integrality OA framework that proceeds from an LP relaxation through partially integral relaxations before reaching the full MILP, thereby using inexpensive early iterations to strengthen the approximation before later mixed-integer solves. Computational experiments on CBLIB instances and large-scale AC unit-commitment models demonstrate complementary benefits from the proposed cut strategy and progressive integrality, substantially reducing the computational effort of outer approximation.
