On modified methods for the generalized Fermat-Torricelli problem and the split variational inequality problem

Abstract

Fermat-Torricelli problem (FTP) is widely regarded as one of the fundamental problems in statistics, location science, optimization, and wireless communications. In an excellent work, Reich and Tuyen (Journal of Optimization Theory and Applications 196(1), 78-97 (2023)) generalized the FTP to real Hilbert spaces and termed it the "Generalized Fermat-Torricelli Problem" (in short, GFTP). They proposed an iterative method based on the subdifferential of the involved operator to solve the GFTP.
They also proved the existence of a weakly converging subsequence of the generated sequence to a solution of the considered problem. However, the strong convergence of the generated sequence itself remains unexplored. To address this gap, we introduce an iterative method to solve the GFTP and a split variational inequality problem (SVIP). Initially, we prove strong convergence of the generated sequence under suitable conditions on the involved parameters. Moreover, we employ a variant of the proposed method to solve a split feasibility problem with multiple output sets (SFPMOS) and the SVIP, a special case of our primary problem. To conclude the study, we conduct some numerical experiments to validate the stated results.