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A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objectivesubject to multiple two-sided linear matrix inequalities intersected with alow-rank and spectral constrained domain set. Although solving LSOP is, ingeneral, NP-hard, its partial convexification (i.e., replacing the domain setby its convex hull) termed "LSOP-R," is often tractable and yields ahigh-quality solution. This motivates us to study the strength of LSOP-R.Specifically, we derive rank bounds for any extreme point of the feasible setof LSOP-R and prove their tightness for the domain sets with different matrixspaces. The proposed rank bounds recover two well-known results in theliterature from a fresh angle and also allow us to derive sufficient conditionsunder which the relaxation LSOP-R is equivalent to the original LSOP. Toeffectively solve LSOP-R, we develop a column generation algorithm with avector-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 ofthe proposed algorithms.

On the Partial Convexification for Low-Rank Spectral Optimization: Rank Bounds and Algorithms