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Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space. (English) Zbl 0911.47051

Finding an optimal point in the intersection of the fixed point sets of a family of nonexpansive mappings is a frequent problem in various areas of mathematical science and engineering. Let \(T_i\) \((i= 1,2,\dots, N)\) be nonexpansive mappings on a Hilbert space \({\mathcal H}\), and let \(\Theta:{\mathcal H}\to\mathbb{R}\) be a quadratic function defined by \(\Theta(x):= {1\over 2}\langle Ax,x\rangle- \langle b,x\rangle\) for all \(x\in{\mathcal H}\), where \(A:{\mathcal H}\to{\mathcal H}\) is a strongly positive bounded selfadjoint linear operator. Then, for each sequence of scalar parameters \((\lambda_n)\) satisfying certain conditions, the authors propose an algorithm that generates a sequence converging strongly to a unique minimizer \(u^*\) of \(\Theta\) over the intersection of the fixed point sets of all the \(T_i\)’s.
In particular, the minimization of \(\Theta\) over the intersection \(\bigcap^N_1 C_i\) of closed convex sets \(C_i\) can be handled by taking \(T_i\) to the metric projection \(P_{C_i}\) onto \(C_i\) without introducing any special inner product that depends on \(A\). The authors also propose an algorithm that generates a sequence converging to a unique minimizer of \(\Theta\) over \(K_\phi:= \{u\in K\mid \phi(u)= \inf\phi(K)\}\neq \emptyset\), where \(K\) is a given closed convex set and \(\phi(x):= \sum^N_{i=1} w_id(x, C_i)\) for positive weights \(w_i\) \((i= 1,\dots, N)\). This is applicable to the inconsistent case \(\bigcap^N_1 C_i= \emptyset\) as well.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
90C25 Convex programming
65K05 Numerical mathematical programming methods
65K10 Numerical optimization and variational techniques
90C30 Nonlinear programming
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