Exact time complexity of semidefinite programming

Determine the exact time complexity of semidefinite programming, including semidefinite feasibility, under an explicitly specified computational model.

Background

The paper discusses the computational complexity of semidefinite programming in both the Turing model and the real-number model of Blum, Shub, and Smale. Although various membership relations and polynomial-time approximate algorithms under assumptions are known, the paper explicitly states that the exact time complexity of an SDP is unresolved. This open problem concerns a foundational complexity classification for semidefinite optimization.

References

To the best of our knowledge, the exact time complexity of an SDP is unknown.

— Further analysis and extension of the higher-order Newton method of Ahmadi, Chaudhry, and Zhang  (2609.01001 - Voert et al., 1 Sep 2026) in Section 2, subsection “SOS-convex polynomial”