Title: Response to "Counterexample to global convergence of DSOS and SDSOS hierarchies",
Abstract: In a recent note [8], the author provides a counterexample to the global
convergence of what his work refers to as "the DSOS and SDSOS hierarchies" for
polynomial optimization problems (POPs) and purports that this refutes claims
in our extended abstract [4] and slides in [3]. The goal of this paper is to
clarify that neither [4], nor [3], and certainly not our full paper [5], ever
defined DSOS or SDSOS hierarchies as it is done in [8]. It goes without saying
that no claims about convergence properties of the hierarchies in [8] were ever
made as a consequence. What was stated in [4,3] was completely different: we
stated that there exist hierarchies based on DSOS and SDSOS optimization that
converge. This is indeed true as we discuss in this response. We also emphasize
that we were well aware that some (S)DSOS hierarchies do not converge even if
their natural SOS counterparts do. This is readily implied by an example in our
prior work [5], which makes the counterexample in [8] superfluous. Finally, we
provide concrete counterarguments to claims made in [8] that aim to challenge
the scalability improvements obtained by DSOS and SDSOS optimization as
compared to sum of squares (SOS) optimization.
[3] A. A. Ahmadi and A. Majumdar. DSOS and SDSOS: More tractable alternatives
to SOS. Slides at the meeting on Geometry and Algebra of Linear Matrix
Inequalities, CIRM, Marseille, 2013. [4] A. A. Ahmadi and A. Majumdar. DSOS and
SDSOS optimization: LP and SOCP-based alternatives to sum of squares
optimization. In proceedings of the 48th annual IEEE Conference on Information
Sciences and Systems, 2014. [5] A. A. Ahmadi and A. Majumdar. DSOS and SDSOS
optimization: more tractable alternatives to sum of squares and semidefinite
optimization. arXiv:1706.02586, 2017. [8] C. Josz. Counterexample to global
convergence of DSOS and SDSOS hierarchies. arXiv:1707.02964, 2017.