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Iterative algorithms with some control conditions for quadratic optimizations. (English) Zbl 1165.47058

Given \(N\) nonexpansive mappings \(T_i\), \(i=1,\dots,N\), defined in a Hilbert space \(H\), and their sets of fixed points \(F_i:=\operatorname{Fix}(T_i)\), \(i=1,\dots,N\), such that \(\bigcap_{i=1}^N F_i\not=\emptyset\), the author considers the problem of minimizing a quadratic function subject to the constraint set \(C:=\bigcap_{i=1}^N F_i\). He introduces an iterative inexact method for solving this problem, and establishes the strong convergence of the method.
The author also considers a more general optimization problem, in which the objective function \(\theta\) is convex and twice differentiable on an open set \(U\supset \bigcup_{i=1}^N T_i(H)\), with constraint set \(C\) as defined above. Under some additional technical assumptions on \(\theta\), the author establishes the strong convergence of an inexact method for this problem.
The results presented here improve those found in [F.Deutsch and I.Yamada, Numer.Funct.Anal.Optimization 19, No.1–2, 33–56 (1998; Zbl 0913.47048); K.–H.Xu, J. Lond.Math.Soc., II.Ser.66, No.1, 240–256 (2002); Zbl 1013.47032; H.–K.Xu, J.Optimization Theory Appl.116, No.3, 659–678 (2003; Zbl 1043.90063); I.Yamada, N.Ogura, Y.Yamashita and K.Sakaniwa, Numer.Funct.Anal.Optimization 19, No.1–2, 165–190 (1998; Zbl 0911.47051)].

MSC:

47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49M05 Numerical methods based on necessary conditions
65K05 Numerical mathematical programming methods
90C20 Quadratic programming
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