MULTIPLE-SETS SPLIT QUASI-CONVEX FEASIBILITY PROBLEMS: ADAPTIVE SUBGRADIENT METHODS WITH CONVERGENCE GUARANTEE.

In this paper, we consider a multiple-sets split quasi-convex feasibility problem (MSSQFP), which is to find a point such that itself and its image under a linear transformation fall within two families of sublevel sets of quasi-convex functions in the space and the image space, respectively. A unified framework of the adaptive subgradient methods with general control schemes is proposed to solve the MSSQFP. This paper is contributed to establish the quantitative convergence theory of adaptive subgradient methods with several general control schemes. An interesting finding is disclosed by the iteration complexity results that the stochastic control enjoys both advantages of low computational cost requirement and low iteration complexity. In addition, a notion of the H¨older-type bounded error bound property for the MSSQFP is introduced, and the linear/sublinear convergence rates for the adaptive subgradient methods to a feasible solution of the MSSQFP is established. [ABSTRACT FROM AUTHOR]