A multiprojection algorithm using Bregman projections in a product space.

*(English)*Zbl 0828.65065Generalized distances give rise to generalized projections into convex sets. An important question is whether or not one can use, within the same projection algorithm, different types of such generalized projections. This question has practical consequences in the area of signal detection and image recovery in situations that can be formulated mathematically as a convex feasibility problem.

Using an extension of Pierra’s product space formalism, it is shown that a multiprojection algorithm converges. The algorithm is fully simultaneous, i.e., it uses all sets of the convex feasibility problem in each iterative step. Different multiprojection algorithms can be derived from this algorithmic scheme by a judicious choice of the Bregman functions that govern the process.

As a by-product of the investigation, the authors obtain block-iterative schemes for certain kinds of linearly constrained optimization problems.

Using an extension of Pierra’s product space formalism, it is shown that a multiprojection algorithm converges. The algorithm is fully simultaneous, i.e., it uses all sets of the convex feasibility problem in each iterative step. Different multiprojection algorithms can be derived from this algorithmic scheme by a judicious choice of the Bregman functions that govern the process.

As a by-product of the investigation, the authors obtain block-iterative schemes for certain kinds of linearly constrained optimization problems.

Reviewer: J.Guddat (Berlin)

##### Keywords:

convergence; signal detection; image recovery; convex feasibility problem; Pierra’s product space; multiprojection algorithm; Bregman functions; block-iterative schemes; linearly constrained optimization
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\textit{Y. Censor} and \textit{T. Elfving}, Numer. Algorithms 8, No. 2--4, 221--239 (1994; Zbl 0828.65065)

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