In this paper, we adopt an iterative approach to solve the class of optimization problem for the sum of finite functions over split equality optimization problems for the sum of two functions. This type of problem contains many optimization problems, and bilevel problems, as well as split equality problems, and split feasibility problems as special cases. Here, we are able to establish a strong convergence theorem for an iterative method for solving this problem. As consequences of this convergence theorem, we study the following problems: optimization for the sum of finite functions over the common solution set of optimization problems for the sum of two functions; optimization for the sum of finite functions; optimization for the sum of finite functions with split equality inconsistent feasibility constraints; optimization for the sum of finite functions over the solution set for split equality constrained quadratic signal recovery problem; optimization for the sum of finite functions over the solution set of generalized split equality multiple set feasibility problem, and optimization for the sum of finite functions over the solution set of split equality linear equations problem. We use simultaneous iteration to establish strong convergence theorems for these problems. Our results generalize and improve many existing theorems for these types of problems in the literature and will have applications in nonlinear analysis, optimization problems and signal processing problems.
Optimization for the sum of finite functions Split equality optimization problem Split equality inconsistent feasibility problem Split equality quadratic signal recovery problem
Mathematics Subject Classification
47H06 47H09 47H10 47J25 65K15 90C35 90C25
This is a preview of subscription content, log in to check access.
The author wishes to express his gratitude to the referees for their valuable suggestions during the preparation of this paper.
Chuang, C.S., Yu, Z.T., Lin, L.J.: Mathematical programming for the sum of two convex functions with applications to lasso problems, split feasibility problems and image deblurring problem. Fixed Point Theory Appl. 2015, 143 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
Lin, L.J.: Simultaneous iteration for variational inequalities over common solutions for finite families of nonlinear problems. J. Nonlinear Sci. Appl. 11, 394–416 (2018)MathSciNetCrossRefGoogle Scholar
Chuang, C.S., Lin, L.J., Yu, Z.T.: Mathematical programming over the solution set of the minimization problem for the sum of two convex functions. J. Nonlinear Convex Anal. 17, 2105–2118 (2016)MathSciNetzbMATHGoogle Scholar
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)zbMATHGoogle Scholar
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
Bargetz, C., Reich, S., Zalas, R.: Convergence properties of dynamic string averaging projection methods in the presence of perturbations. Numer. Algorithms 77, 185–209 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
Bailon, J.B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Huston J. Math. 4, 1–9 (1978)MathSciNetGoogle Scholar
Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Huston J. Math. 3, 459–470 (1977)MathSciNetzbMATHGoogle Scholar
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North Holland Publishing Company, Amsterdam (1976)zbMATHGoogle Scholar
Takahashi, W.: Nonlinear Functional Analysis, Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000)zbMATHGoogle Scholar
Lee, G.M., Lin, L.J.: Variational inequalities over split equality fixed point sets of strongly quasinonexpansive mappings with applications. J. Nonlinear Convex Anal. 18, 1781–1800 (2017)MathSciNetzbMATHGoogle Scholar
Yu, Z.T., Lin, L.J., Chuang, C.S.: Mathematical programing with multiple sets split monotone variational inclusion constraints. Fixed Point Theory Appl. 2014, 20 (2014)CrossRefzbMATHGoogle Scholar
Chang, S.S., Wang, L., Tang, Y.K., Wang, G.: Moudafi’s open question and simultaneous iterative algorithm for general split equality optimization. Fixed Point Theory Appl. 2014, 215 (2014)CrossRefzbMATHGoogle Scholar
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
Combettes, P.L., Bondon, P.: Hard constrained inconsistent signal feasibility problems. IEEE Trans. Signal Process 47, 2460–2468 (1999)CrossRefzbMATHGoogle Scholar
Commettes, P.L.: A block-iterative surrogate constraint splitting method for quadratic signal recovery. IEEE Trans. Signal Process 51, 1771–1782 (2003)MathSciNetCrossRefGoogle Scholar
Masad, E., Reich, S.: A note on the multiple-set convex set split feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)MathSciNetzbMATHGoogle Scholar
Xu, H.K.: A variable Krasnosel’skiĭ–Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)CrossRefzbMATHGoogle Scholar