Adaptive subgradient method for the split quasi-convex feasibility problems

In this paper, we consider a type of the celebrated convex feasibility problem, named as split quasi-convex feasibility problem (SQFP). The SQFP is to find a point in a sublevel set of a quasi-convex function in one space and its image under a bounded linear operator is contained in a sublevel set of another quasi-convex function in the image space. We propose a new adaptive subgradient algorithm for solving SQFP problem. We also discuss the convergence analyses for two cases: the first case where the functions are upper semicontinuous in the setting of finite dimensional, and the second case where the functions are weakly continuous in the infinite-dimensional settings. Finally some numerical examples in order to support the convergence results are given.

[1]  C. Zălinescu,et al.  Elements of quasiconvex subdifferential calculus. , 2000 .

[2]  Xiaoqi Yang,et al.  Inexact subgradient methods for quasi-convex optimization problems , 2015, Eur. J. Oper. Res..

[3]  P. L. Combettes,et al.  The Convex Feasibility Problem in Image Recovery , 1996 .

[4]  Wataru Takahashi,et al.  The split common null point problem and the shrinking projection method in Banach spaces , 2016 .

[5]  Alfred O. Hero,et al.  Energy-based sensor network source localization via projection onto convex sets , 2005, IEEE Transactions on Signal Processing.

[6]  J. B. G. Frenk,et al.  Introduction to Convex and Quasiconvex Analysis , 2001 .

[7]  Y. Censor,et al.  The multiple-sets split feasibility problem and its applications for inverse problems , 2005 .

[8]  Xinmin Yang,et al.  On the Existence of Minimizers of Proximity Functions for Split Feasibility Problems , 2015, J. Optim. Theory Appl..

[9]  Yair Censor,et al.  Algorithms for the quasiconvex feasibility problem , 2006 .

[10]  Hyunjoong Kim,et al.  Functional Analysis I , 2017 .

[11]  Q. Ansari,et al.  Split hierarchical variational inequality problems and related problems , 2014, Fixed Point Theory and Applications.

[12]  Songnian He,et al.  A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem , 2015 .

[13]  Pini Gurfil,et al.  Convex feasibility modeling and projection methods for sparse signal recovery , 2012, J. Comput. Appl. Math..

[14]  Yair Censor,et al.  A multiprojection algorithm using Bregman projections in a product space , 1994, Numerical Algorithms.

[15]  Chih-Sheng Chuang.  Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem , 2016 .

[16]  Rudong Chen,et al.  A Relaxed CQ Algorithm for Solving Split Feasibility Problem , 2011, 2011 International Conference on Control, Automation and Systems Engineering (CASE).

[17]  Y. Censor,et al.  A unified approach for inversion problems in intensity-modulated radiation therapy , 2006, Physics in medicine and biology.

[18]  Igor V. Konnov,et al.  On Convergence Properties of a Subgradient Method , 2003, Optim. Methods Softw..

[19]  Erik G. Ström,et al.  Cooperative Wireless Sensor Network Positioning via Implicit Convex Feasibility , 2013, IEEE Transactions on Signal Processing.

[20]  Qingzhi Yang.  The relaxed CQ algorithm solving the split feasibility problem , 2004 .

[21]  Siegfried Schaible,et al.  Handbook of Generalized Convexity and Generalized Monotonicity , 2005 .

[22]  C. Byrne,et al.  Iterative oblique projection onto convex sets and the split feasibility problem , 2002 .

[23]  A. Iusem,et al.  Set-valued mappings and enlargements of monotone operators , 2008 .

[24]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[25]  J. Penot.  Are Generalized Derivatives Sseful for Generalized Convex Functions , 1998 .

[26]  Hong-Kun Xu.  Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces , 2010 .

[27]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..