Algorithms for Computing of Optimization Problems with a Common Minimum-Norm Fixed Point with Applications

This research is aimed to study a two-step iteration
process defined over a finite family of σ-asymptotically
quasi-nonexpansive nonself-mappings. The strong convergence
is guaranteed under the framework of Banach spaces with some
additional structural properties including strict and uniform
convexity, reflexivity, and smoothness assumptions. With similar
projection technique for nonself-mapping in Hilbert spaces, we
hereby use the generalized projection to construct a point within
the corresponding domain. Moreover, we have to introduce the use
of duality mapping and its inverse to overcome the unavailability
of duality representation that is exploit by Hilbert space theorists.
We then apply our results for σ-asymptotically quasi-nonexpansive
nonself-mappings to solve for ideal efficiency of vector optimization
problems composed of finitely many objective functions. We also
showed that the obtained solution from our process is the closest to
the origin. Moreover, we also give an illustrative numerical example
to support our results.