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

Nonlinear Convex Optimization: From Relaxed Proximal Point Algorithm to Prediction Correction Method