Identify a constructive rule to choose the DCA subgradient when the sufficient separation condition fails

Develop an effective procedure to identify a vector c in the Euclidean ball B(0, sqrt(d_min)) (equivalently a subgradient w^k in ∂||Q^{1/2}V·||(0) with w^k = c) such that E(l^k) c ∉ V^T ∂T(0) whenever the inequality sqrt(d_min)·E(l^k) ≤ min{T(v_{i_m}), T_1(v_{\bar{i}_m})} holds in the PS-DCA algorithm for the generalized graph Fourier mode fractional program min_{x ≠ 0} T(Vx)/||Q^{1/2}Vx||. The goal is to deterministically select such a c (when it exists) so that the one-step DCA started from 0 improves the candidate point l^k, instead of relying on random selection.

Background

Within PS-DCA, after computing a candidate point l^k for the generalized graph Fourier mode problem min_{x ≠ 0} T(Vx)/||Q^{1/2}Vx||, a one-step DCA is applied with subgradient w^k ∈ ∂||Q^{1/2}V·||(0) to try to reduce the ratio further. The authors derive a simple sufficient condition (involving sqrt(d_min)·E(l^k) and the quantities T(v_{i_m}) and T_1(v_{\bar{i}_m})) under which one can construct a specific c ∈ B(0, sqrt(d_min)) giving E(l^k) c ∉ V^T ∂T(0), which guarantees improvement by the DCA step.

However, when this sufficient condition does not hold, the authors note that a suitable c may still exist but they do not have a method to find it, and therefore resort to random selection from a finite set. A constructive strategy to identify such a c (whenever it exists) would strengthen PS-DCA by reliably enabling DCA-based improvement even in this case.