As a network engineer working on optimization problems, I've observed that the literature suggests convex separable programming problems are not significantly more difficult to solve than linear optimization problems, as noted by Hochbaum and Shanthikumar [1].
This observation has led me to consider the complexity relationship in the non-convex domain. Specifically, I would like to inquire about two related aspects:
From a computational complexity perspective, are non-convex separable programming problems demonstrably harder or easier to solve than non-separable non-convex programming problems?
Can the separable property be effectively exploited to develop more efficient algorithms for non-convex separable programming, similar to how separability is leveraged in convex optimization?
I would appreciate insights from those with expertise in mathematical optimization, particularly regarding theoretical complexity results or practical algorithmic approaches relevant to these questions. Note that in my field, engineers care more about recovering feasible solutions in the nick of time rather than focusing on exotic relaxation techniques.
From what I found in So and Zhou's paper [2], separable QCQP has a known bound for the rank of its SDP relaxation as:
$$\sum\limits_{i = 1}^k {\frac{{{\mathop{\rm rank}\nolimits} ({\bf{X}}_i^ \star )({\mathop{\rm rank}\nolimits} ({\bf{X}}_i^ \star ) + 1)}}{2}} \le m.$$
Where the SDP is characterized by an $n \times n$ matrix variable and $m$ linear constraints. This suggests that separability might offer structural advantages even in non-convex settings, but I'm looking for more comprehensive insights.
Thank you for your consideration.
[1]: Hochbaum, D.S., Shanthikumar, J.G. (1990). Convex separable optimization is not much harder than linear optimization. Journal of the ACM, 37(4), 843-862. https://dl.acm.org/doi/abs/10.1145/96559.96597
[2]: So, A.M.C., Zhou, Z. (2012). Semidefinite relaxation approaches for approximating quadratic optimization problems. IEEE Signal Processing Magazine. http://www.se.cuhk.edu.hk/~manchoso/papers/sdrapp-SPM.pdf
