Successive linear programing approach for solving the nonlinear split feasibility problem

The Split Feasibility Problem (SFP), which was introduced by Censor and Elfving, consists of finding a point in a set C in one space such that its image under a linear transformation belongs to another set Q in the other space. This problem was well studied both theoretically and practically as it was also used in practice in the area of Intensity-Modulated Radiation Therapy (IMRT) treatment planning. Recently Li et. al. extended the SFP to the non-linear framework. Their algorithm tries to follow the algorithm for the linear case. But, unlike the linear case, the involved proximity function is not necessarily convex. Therefore in order to use Baillon-Haddad and Dolidze Theorems, the authors assume convexity in order to prove convergence of the projected gradient method. Since convexity of the proximity function is too restrictive, we consider here a Successive Linear Programing (SLP) approach in order to obtain local optima for the non-convex case. We also aim to intro duce a non-linear version of the Split Variational Inequality Problem (SVIP).