Please refer attached 6-page short paper for details.
Let $M(y)=y+2r\nabla g_r(y)$.
Given $\ g_r(M(y^k))>g_r(y^k)+r|\nabla g_r(y^k)|^2, \forall k. \ $ (equation 36 in the paper)
and $ \ \lim_{k \rightarrow \infty}[g_r(y^{k+1})-g_r(M(y^k))]=0. $ (equation 33 in the paper)
Can anyone tell me how we can conclude $\lim_{k \rightarrow \infty}|\nabla g_r(y^{k})|=0 \ $?
Since there's a gap of index shift between these two equations and the result, I don't understand his logic. The author claimed to also use "$g_r$ bounded above" to reach this conclusion, but I don't see how.
Source: Page 561 of The multiplier method of Hestenes and Powell applied to convex programming, by R T Rockafellar
http://sites.math.washington.edu/~rtr/papers/rtr051-MultMethod.pdf