Research on the descent direction of prediction correction algorithms for pseudo-convex/convex optimization problems

TL;DR

This paper studies the descent directions in prediction-correction algorithms for pseudo-convex and convex optimization, revealing optimal adjustment coefficients and providing convergence proofs.

Contribution

It introduces a new perspective on adjustment coefficients, establishing their ranges for pseudo-convex and convex problems, and offers rigorous convergence analysis.

Findings

Adjustment coefficient range for pseudo-convex problems is (1/2,1].

Adjustment coefficient range for convex problems is [0,1].

Algorithms perform best near differential equation-inspired coefficients.

Abstract

Prediction-correction algorithms are a highly effective class of methods for solving pseudo-convex optimization problems. The descent direction of these algorithms can be viewed as an adjustment to the gradient direction based on the prediction step. This paper investigates the adjustment coefficients of these descent directions and offers explanations from the perspective of differential equations. Unlike existing algorithms where the adjustment coefficient is always set to 1, we establish that the range of the adjustment coefficient lies within (1/2,1] for pseudo-convex optimization problems, and [0,1] for convex optimization problems. We also provide rigorous convergence proofs for these proposed algorithms. Numerical experiment results show that the algorithms perform best when the value of the adjustment coefficient makes the algorithm approach or equal to those in differential equations with higher-order global discrete error.