The Split Feasibility Problem and Beyond

Vu Thi Huong

Vietnam Academy of Science and Technology, Institute of Mathematics

Friday, September 22, 2023, 11:00 - 11:59

Room 02-012, Georges-Köhler Allee 102, Freiburg 79110, Germany

Given nonempty, closed and convex sets C ⊂ H, Q ⊂ K in Hilbert spaces H, K and a bounded linear operator A : H → K, the split feasibility problem (SFP) is to find x ∈ C such that Ax ∈ Q. The problem was introduced by Censor and Elfving [Numer. Algorithms 1994] to model phase retrieval and other image restoration problems in signal processing. During the last three decades, many efforts have been made to design solution algorithms for SFP. Interestingly, this feasibility problem can be reformulated as a fixed point problem or a convex minimization one; hence, advanced tools from operator theory, convex analysis, and optimization machinery can be fully exploited. In the first part of the talk, we will review some basic solution algorithms for SFP resulting from this approach, and then discuss further the gradient projection method with Polyak’s stepsize.The second part of the talk is about solution stability of SFP with respect to small changes of input data, where SFP is reformulated as a parametric generalized equation to which set-valued and variational analysis techniques are applied.