A priori determination of which proportional constraint is redundant in the sub-SDPR of the p=3 homogeneous block

Determine, prior to solving the full semidefinite programming relaxation (SDPR) of equation (18), which of the fourth or fifth inequality constraints in the subproblem SDPR (equation (15)) for the p=3 separable homogeneous sub-quadratically constrained quadratic program is redundant, given that the coefficient matrices satisfy C_5^q = α C_4^q (α > 0) and that the right-hand sides δ_4^3 and δ_5^3 depend on the optimal solution of the full SDPR.

Background

In Example labeled example:main0 (Section 5), the authors construct a separable QCQP by horizontally connecting three heterogeneous sub-QCQPs: a convex QCQP (p=1), a sign-pattern QCQP (p=2), and a separable homogeneous QCQP (p=3). The overall problem has an SDPR given by equation (18), while the p=3 component admits a sub-SDPR of the form (15).

For the p=3 subproblem, the 4th and 5th constraints have proportional coefficient matrices (C_5^q = α C_4^q with α > 0), implying that one of these two constraints is redundant depending on whether δ_4^3 ≤ δ_5^3/α holds. However, δ_4^3 and δ_5^3 depend on the optimal solution of the full SDPR (18), so the analysis cannot a priori identify which of the two constraints can be removed in the sub-SDPR (15). The authors explicitly note this unresolved point.