{{\bf{p}}_i^{(k)}-{\bf{q}}_i^{(k+1)}} \right\|^2 = \left\| {{\bf{p}}_i^{(k)}-{\bf{q}}_i^{(k)}} \right\|^2 - \left\| {{\bf{q}}_i^{(k + 1)}-{\bf{q}}_i^{(k)}} \right\|^2.
\label{eq_8}$$

In the ${\bf{p}}^{(k+1)}$ updating stage, the updated $\bf{R}$ and $\mu$ make the objective function smaller. Hence,

$$\sum\limits_{i = 1}^N {{{\left\| {{\bf{p}}_i^{(k + 1)} - {\bf{q}}_i^{(k + 1)}} \right\|}^2}}  \le \sum\limits_{i = 1}^N {{{\left\| {{\bf{p}}_i^{(k)} - {\bf{q}}_i^{(k + 1)}} \right\|}^2}} .
\label{eq_9}$$

According to , , and the objective function ,

$$\begin{array}{l}
\begin{array}{l}
f{({\bf{p}}_i^{(k + 1)},{\bf{q}}_i^{(k + 1)})} \le f{({\bf{p}}_i^{(k)},{\bf{q}}_i^{(k)})} - \sum\limits_{i = 1}^N {{{\left\| {{\bf{q}}_i^{(k + 1)} - {\bf{q}}_i^{(k)}} \right\|}^2}}
\end{array}.
\end{array}
\label{eq_10}$$

Hence, the objective function will strictly decrease until ${\bf{q}}_i^{(k + 1)} = {\bf{q}}_i^{(k)}$ when not using $1/\lambda_i$ as weights. However, when $1/\lambda_i$ is applied in the objective funtion, the above convergence proof is not rigorous in mathematics because $\lambda_i^{(k+1)} \neq \lambda_i^{(k)}$. As the iteration process, the changes of $\lambda_i$ become small, which makes the formula hold. In addition, our experimental results in this paper also support the assumption that our algorithm is convergent.

Local Minima Avoidance
----------------------

We have concluded that the iterative process of R1PP$n$P is convergent. However, we still need to address situations that R1PP$n$P may get stuck in local minima. To demonstrate the iterative process more intuitively, we introduce a 1D 