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）旨在最小化线性目标函数，该问题受多个双侧线性矩阵不等式以及低秩与谱约束域集的共同约束。尽管求解LSOP通常是NP难的，但其部分凸化（即用域集的凸包替换原域集）——记为"LSOP-R"——往往是可解的，并能得到高质量的解。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，并证明了这些界在不同矩阵空间的域集上的紧性。所提出的秩界从一个新角度重现了文献中的两个著名结果，并使我们能够推导出松弛问题LSOP-R与原问题LSOP等价性的充分条件。为有效求解LSOP-R，我们开发了一种列生成算法，该算法结合基于向量的凸定价预言机与秩约简算法，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。