## A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets.(English)Zbl 1164.47399

Summary: The method of projections onto convex sets, to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such a corridor consists in taking a big step at different moments during the iteration, because in that way the monotonous behavior that is responsible for the slow convergence may be interrupted. In this paper, we present a technique that may introduce interruption of the monotony for a sequential algorithm, but at the same time guarantees convergence of the constructed sequence to a point in the intersection of the sets. Concerning convergence speed, we compare experimentally the behavior of the new algorithm with that of an existing monotonous algorithm.

### MSC:

 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 40A99 Convergence and divergence of infinite limiting processes 49M30 Other numerical methods in calculus of variations (MSC2010) 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 90C25 Convex programming
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### References:

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