In the rank-constrained optimization problem (RCOP), it minimizes a linear objective function over a prespecified closed rank-constrained domain set and $m$ generic two-sided linear matrix inequalities. Motivated by the Dantzig-Wolfe (DW) decomposition, a popular approach of solving many nonconvex optimization problems, we investigate the strength of DW relaxation (DWR) of the RCOP, which admits the same formulation as RCOP except replacing the domain set by its closed convex hull. Notably, our goal is to characterize conditions under which the DWR matches RCOP for any m two-sided linear matrix inequalities. From the primal perspective, we develop the first-known simultaneously necessary and sufficient conditions that achieve: (i) extreme point exactness -- all the extreme points of the DWR feasible set belong to that of the RCOP; (ii) convex hull exactness -- the DWR feasible set is identical to the closed convex hull of RCOP feasible set; and (iii) objective exactness -- the optimal values of the DWR and RCOP coincide. The proposed conditions unify, refine, and extend the existing exactness results in the quadratically constrained quadratic program (QCQP) and fair unsupervised learning. These conditions can be very useful to identify new results, including the extreme point exactness for a QCQP problem that admits an inhomogeneous objective function with two homogeneous two-sided quadratic constraints and the convex hull exactness for fair SVD.

翻译：在秩约束优化问题（RCOP）中，目标是在预设的闭秩约束域集和m个通用双侧线性矩阵不等式上最小化线性目标函数。受丹齐格-沃尔夫（DW）分解（一种求解许多非凸优化问题的流行方法）启发，我们研究了RCOP的DW松弛（DWR）的强度，其与RCOP具有相同表述，仅将域集替换为其闭凸包。值得注意的是，我们的目标是刻画对于任意m个双侧线性矩阵不等式，DWR与RCOP相匹配的条件。从原始视角出发，我们首次提出了同时实现以下目标的充要条件：（i）极值点精确性——DWR可行集的所有极值点均属于RCOP的可行集；（ii）凸包精确性——DWR可行集与RCOP可行集的闭凸包相同；（iii）目标精确性——DWR与RCOP的最优值一致。所提出的条件统一、精炼并扩展了二次约束二次规划（QCQP）和公平无监督学习中的现有精确性结果。这些条件对于发现新结果非常有用，包括具有非齐次目标函数和两个齐次双侧二次约束的QCQP问题的极值点精确性，以及公平奇异值分解（SVD）的凸包精确性。