基于决策图的混合整数二次优化的凸化
Convexification of mixed-integer quadratic optimization via decision diagrams
AI总结：
该研究针对带指示变量的混合整数二次优化问题，提出基于决策图的统一框架，可构造理想凸包扩展形式，在特定条件下具多项式规模，实验验证其有效性。
AI中文摘要：
我们研究带有指示变量的混合整数二次优化（MIQO）问题。我们提出了一个基于决策图的统一框架，该框架既用于求解相关优化问题，又用于构造基础混合整数集凸包闭包的理想二次锥扩展形式。该构造适用于任意二次函数以及任何可接受易处理动态规划表示的组合约束。当二次函数为低秩，或者其海森矩阵或其逆的支撑图为树时，所得的图及后续的凸包描述具有多项式规模，可恢复并推广文献中的若干结果。对于结构化稀疏和逆稀疏二次函数，我们证明近似决策图的规模与维度呈线性关系，同时能产生具有任意低最优性间隙的解。计算实验证明了所提方法的有效性。
英文摘要：
We study mixed-integer quadratic optimization (MIQO) problems with indicator variables. We propose a unified framework, based on decision diagrams, that serves both to solve the associated optimization problems and to construct ideal conic quadratic extended formulations of the closure of the convex hull of the underlying mixed-integer set. The construction applies to arbitrary quadratics and to any combinatorial constraints admitting a tractable dynamic programming representation. The resulting diagrams and the ensuing convex hull descriptions are of polynomial size when the quadratic is low-rank, or when the support graph of the Hessian or of its inverse is a tree, recovering and generalizing several results from the literature. For structured sparse and inverse-sparse quadratics, we show that approximate decision diagrams have size linear in the dimension while yielding solutions with arbitrarily low optimality gap. Computational experiments demonstrate the effectiveness of the proposed approach.
