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

TL;DR

This paper extends classical convex optimization methods, the relaxed proximal point algorithm and prediction correction, to nonlinear convex problems, providing convergence guarantees and numerical validation.

Contribution

It customizes the proximal matrix for nonlinear problems and extends the PC method, establishing their convergence and rate of $O(1/t)$.

Findings

Both methods achieve $O(1/t)$ convergence rate.

Numerical results support theoretical convergence.

Extensions to nonlinear constraints are effective.

Abstract

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.