  Nonlinear convex problems arise in various areas of applied mathematics and
engineering. Classical techniques such as the relaxed proximal point algorithm
(PPA) and the prediction correction (PC) method were proposed for linearly
constrained convex problems. However, these methods have not been investigated
for nonlinear constraints. In this paper, we customize the varying proximal
matrix to develop the relaxed PPA for nonlinear convex problems. We also extend
the PC method to nonlinear convex problems. As both methods are an extension of
the PPA-based contraction method, their sequence convergence can be directly
established. Moreover, we theoretically demonstrate that both methods can
achieve a convergence rate of $O(1/t)$. Numerical results once again support
the theoretical analysis.
