I have a sequence of vector $x^k = \frac{1}{N} \sum_{i=1}^N x_i^k$, and a set of functions $\{f_i\}_{i=1}^N$ convex functions. From the convexity of $f_i$, I know that $\langle \nabla f_i(x_i^{k+1}) - \nabla f_i(x_i^k), x_i^{k+1} -x_i^{k} \rangle \geq 0$.
Is there a way to show that $\langle \nabla f_i(x^{k+1}) - \nabla f_i(x^k), x_i^{k+1} -x_i^{k} \rangle \geq 0$?