Asymptotic exactness of DCA rates for regimes $p_5$ and $p_6$

Let $L_1,L_2$ be the smoothness parameters and $\mu_1,\mu_2$ the strong-convexity or hypoconvexity parameters of the two DCA terms, and assume

$$L_1 > \mu_2,\qquad L_2 > \mu_1.$$

Let $p_5$ and $p_6$ be the regimes whose domains are defined in the paper's one-step regime table, let $g_1^k\in\partial f_1(x^k)$ and $g_2^k\in\partial f_2(x^k)$, and let $N$ be sufficiently large. Define

$$p_5^{\infty}:=\frac{(L_2+\mu_1)(\mu_1+\mu_2)}{(L_2+\mu_2)\mu_1^2},\qquad p_6^{\infty}:=\frac{(L_1+\mu_2)(\mu_1+\mu_2)}{(L_1+\mu_1)\mu_2^2}.$$

Asymptotic-rate conjecture. The exact sublinear rates for regimes $p_5$ and $p_6$ correspond to

$$\frac{1}{2}\min_{0\leq k\leq N}\|g_1^k-g_2^k\|^2 \leq \frac{F(x^0)-F(x^N)}{a(L_1,L_2,\mu_1,\mu_2)N+b},$$

where $a\in\{p_5^{\infty},p_6^{\infty}\}$ and $b$ is independent of $N$.

The conjecture addresses the asymptotic sharpness of the sublinear DCA rates in the two regimes where numerical investigations indicate that the finite-iteration bounds are not tight beyond one iteration. The leading constants are proposed explicitly, but the additive term $b$ remains unidentified.