Strict Fejér Monotonicity by Superiorization of Feasibility-Seeking Projection Methods
Abstract: We consider the superiorization methodology, which can be thought of 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 the objective function value) to one returned by a feasibility-seeking only algorithm. Our main result reveals new information about the mathematical behavior of the superiorization methodology. We deal with a constrained minimization problem with a feasible region, which is the intersection of finitely many closed convex constraint sets, and use the dynamic string-averaging projection method, with variable strings and variable weights, as a feasibility-seeking algorithm. We show that any sequence, generated by the superiorized version of a dynamic string-averaging projection algorithm, not only converges to a feasible point but, additionally, either its limit point solves the constrained minimization problem or the sequence is strictly Fejér monotone with respect to a subset of the solution set of the original problem.
Keywords: Bounded perturbation resilience, constrained minimization, convex feasibility problem, dynamic string-averaging projections, strict Fejér monotonicity, subgradients, superiorization methodology, superiorized version of an algorithm.
Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )
Category 2: Optimization Software and Modeling Systems (Parallel Algorithms )
Citation: Journal of Optimization Theory and Applications, accepted for publication.
Entry Submitted: 05/28/2014
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