# American Institute of Mathematical Sciences

January  2022, 27(1): 555-568. doi: 10.3934/dcdsb.2021054

## A learning-enhanced projection method for solving convex feasibility problems

 School of Mathematics, Monash University, 9 Rainforest Walk, Victoria 3800 Australia

* Corresponding author: Janosch Rieger

Received  June 2020 Revised  November 2020 Published  January 2022 Early access  February 2021

We propose a generalization of the method of cyclic projections, which uses the lengths of projection steps carried out in the past to learn about the geometry of the problem and decides on this basis which projections to carry out in the future. We prove the convergence of this algorithm and illustrate its behavior in a first numerical study.

Citation: Janosch Rieger. A learning-enhanced projection method for solving convex feasibility problems. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 555-568. doi: 10.3934/dcdsb.2021054
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Methods MCP, MRP and PAM applied to toy problem. Top row: Iterates red, subspaces blue. Bottom row: Frequencies (yellow = high, blue = low) of transitions from set $C_m$ to set $C_n$
Trajectories and frequencies of PAM as in Example 4(ⅰ)
Trajectories and frequencies of PAM as in Example 4(ⅱ)
Trajectories and frequencies of PAM as in Example 5
Error plots of methods applied to toy model with varying parameters. Solid black line MCP, dashed black line MRP, solid red line PAM $\omega = N/4$, dashed red line PAM $\omega = N/2$, dash-dotted red line PAM $\omega = N$. More details given in Example 6
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