A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective subject to multiple two-sided linear matrix inequalities intersected with a low-rank and spectral constrained domain set. Although solving LSOP is, in general, NP-hard, its partial convexification (i.e., replacing the domain set by its convex hull) termed "LSOP-R," is often tractable and yields a high-quality solution. This motivates us to study the strength of LSOP-R. Specifically, we derive rank bounds for any extreme point of the feasible set of LSOP-R and prove their tightness for the domain sets with different matrix spaces. The proposed rank bounds recover two well-known results in the literature from a fresh angle and also allow us to derive sufficient conditions under which the relaxation LSOP-R is equivalent to the original LSOP. To effectively solve LSOP-R, we develop a column generation algorithm with a vector-based convex pricing oracle, coupled with a rank-reduction algorithm, which ensures the output solution satisfies the theoretical rank bound. Finally, we numerically verify the strength of the LSOP-R and the efficacy of the proposed algorithms.

翻译：低秩谱优化问题（LSOP）旨在最小化一个线性目标函数，同时满足多个双侧线性矩阵不等式，并与低秩及谱约束域集相交。尽管求解LSOP通常是NP难的，但其部分凸化（即用其凸包替代域集）称为"LSOP-R"，往往易于处理并能获得高质量解。这促使我们研究LSOP-R的强度。具体而言，我们推导了LSOP-R可行集任意极点的秩界，并证明了对于不同矩阵空间下的域集，这些秩界是紧的。所提出的秩界从全新角度恢复了文献中两个著名结果，同时使我们能够推导出松弛问题LSOP-R与原问题LSOP等价的充分条件。为有效求解LSOP-R，我们开发了一种基于向量凸定价原语的列生成算法，并结合秩约简算法，确保输出解满足理论秩界。最后，我们通过数值实验验证了LSOP-R的强度及所提算法的有效性。