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Systematic selection of the regularization parameter ρ_k in the one-step DCA of PS-DCA

Develop systematic, theoretically grounded rules or adaptive strategies for choosing the regularization parameter ρ_k > 0 in the strongly convex d.c. decomposition g̃^k(x) = T(x) + (ρ_k/2)||x||^2 and h̃^k(x) = E(l^k) B(x) + (ρ_k/2)||x||^2 used by the one-step DCA inside PS-DCA, for a given difference-of-convex decomposition of T(x) − E(l^k) B(x).

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Background

PS-DCA employs a one-step DCA on a regularized difference-of-convex decomposition with parameter ρ_k to ensure strong convexity of both components, which influences convergence behavior and practical performance. Despite its importance, the paper notes the absence of systematic guidelines for tuning ρ_k across d.c. decompositions.

A principled method to select ρ_k (e.g., based on problem structure, local Lipschitz/curvature surrogates, or adaptive criteria) would improve robustness and performance of PS-DCA beyond heuristic tuning.

References

The choice of regularization parameter \rho_k in param-rho depends on the d.c. decomposition employed by the d.c. function. Unfortunately, there lacks systematic research on how to tune this parameter for a given d.c. decomposition.

param-rho:

g~k(x)=T(x)+ρk2∥x∥2,  h~k(x)=E(lk)B(x)+ρk2∥x∥2,\tilde{g}^k(x)=T(x)+\frac{\rho_k}{2}\|x\|^2,~~\tilde{h}^k(x) =E(l^k)B(x)+\frac{\rho_k}{2}\|x\|^2,

— A proximal algorithm incorporating difference of convex functions optimization for solving a class of single-ratio fractional programming (2510.19408 - Qi et al., 22 Oct 2025) in Section 7, Numerical Experiments (parameter setting discussion)