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

TL;DR

The paper analyzes prediction-correction algorithms for pseudo-convex and convex optimization by introducing a tunable adjustment coefficient β that modifies the descent direction. It connects the choice of β to discrete differential equation discretizations, revealing that β≈1/2 (the trapezoidal case) often yields superior performance, while providing rigorous convergence results under adaptive and constant steps. The authors propose IPC-C and IPC-A for pseudo-convex problems and convex-IPC for convex problems, proving Fejér monotonicity and convergence with explicit β-ranges and line-search mechanisms. Numerical experiments on fractional programming and a smooth convex task confirm that choosing β near 0.5 can significantly improve convergence speed compared to standard methods such as gradient descent or extra-gradient. Overall, the work clarifies how adjusting the descent direction via β influences both theory and practice in PC algorithms.

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.

Figures (3)

• Figure 1: Comparison of iterating one step with different adjustment coefficient in Algorithm (\ref{['eqn1-13']})
• Figure 2: Numerical performance of Algorithm 2 with different $\beta$ for Experiment 1
• Figure 3: Numerical performance of Algorithm 3 with different $\beta$ for Experiment 2