176 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
A generalized projection-based scheme for solving convex constrained optimization problems
In this paper we present a new algorithmic realization of a projection-based
scheme for general convex constrained optimization problem. The general idea is
to transform the original optimization problem to a sequence of feasibility
problems by iteratively constraining the objective function from above until
the feasibility problem is inconsistent. For each of the feasibility problems
one may apply any of the existing projection methods for solving it. In
particular, the scheme allows the use of subgradient projections and does not
require exact projections onto the constraints sets as in existing similar
methods.
We also apply the newly introduced concept of superiorization to optimization
formulation and compare its performance to our scheme. We provide some
numerical results for convex quadratic test problems as well as for real-life
optimization problems coming from medical treatment planning.Comment: Accepted to publication in Computational Optimization and
Application
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
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