8 research outputs found
Convergence and Perturbation Resilience of Dynamic String-Averaging Projection Methods
We consider the convex feasibility problem (CFP) in Hilbert space and
concentrate on the study of string-averaging projection (SAP) methods for the
CFP, analyzing their convergence and their perturbation resilience. In the
past, SAP methods were formulated with a single predetermined set of strings
and a single predetermined set of weights. Here we extend the scope of the
family of SAP methods to allow iteration-index-dependent variable strings and
weights and term such methods dynamic string-averaging projection (DSAP)
methods. The bounded perturbation resilience of DSAP methods is relevant and
important for their possible use in the framework of the recently developed
superiorization heuristic methodology for constrained minimization problems.Comment: Computational Optimization and Applications, accepted for publicatio
String-Averaging Projected Subgradient Methods for Constrained Minimization
We consider constrained minimization problems and propose to replace the
projection onto the entire feasible region, required in the Projected
Subgradient Method (PSM), by projections onto the individual sets whose
intersection forms the entire feasible region. Specifically, we propose to
perform such projections onto the individual sets in an algorithmic regime of a
feasibility-seeking iterative projection method. For this purpose we use the
recently developed family of Dynamic String-Averaging Projection (DSAP) methods
wherein iteration-index-dependent variable strings and variable weights are
permitted. This gives rise to an algorithmic scheme that generalizes, from the
algorithmic structural point of view, earlier work of Helou Neto and De Pierro,
of Nedi\'c, of Nurminski, and of Ram et al.Comment: Optimization Methods and Software, accepted for publicatio
Weak and Strong Superiorization: Between Feasibility-Seeking and Minimization
We review the superiorization methodology, which can be thought of, in some
cases, as lying between feasibility-seeking and constrained minimization. It is
not quite trying to solve the full fledged constrained minimization problem;
rather, the task is to find a feasible point which is superior (with respect to
an objective function value) to one returned by a feasibility-seeking only
algorithm. We distinguish between two research directions in the
superiorization methodology that nourish from the same general principle: Weak
superiorization and strong superiorization and clarify their nature.Comment: Revised version. Presented at the Tenth Workshop on Mathematical
Modelling of Environmental and Life Sciences Problems, October 16-19, 2014,
Constantza, Romania. http://www.ima.ro/workshop/tenth_workshop