582 research outputs found
Linear Superiorization for Infeasible Linear Programming
Linear superiorization (abbreviated: LinSup) considers linear programming
(LP) problems wherein the constraints as well as the objective function are
linear. It allows to steer the iterates of a feasibility-seeking iterative
process toward feasible points that have lower (not necessarily minimal) values
of the objective function than points that would have been reached by the same
feasiblity-seeking iterative process without superiorization. Using a
feasibility-seeking iterative process that converges even if the linear
feasible set is empty, LinSup generates an iterative sequence that converges to
a point that minimizes a proximity function which measures the linear
constraints violation. In addition, due to LinSup's repeated objective function
reduction steps such a point will most probably have a reduced objective
function value. We present an exploratory experimental result that illustrates
the behavior of LinSup on an infeasible LP problem.Comment: arXiv admin note: substantial text overlap with arXiv:1612.0653
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
Bounded perturbation resilience of projected scaled gradient methods
We investigate projected scaled gradient (PSG) methods for convex
minimization problems. These methods perform a descent step along a diagonally
scaled gradient direction followed by a feasibility regaining step via
orthogonal projection onto the constraint set. This constitutes a generalized
algorithmic structure that encompasses as special cases the gradient projection
method, the projected Newton method, the projected Landweber-type methods and
the generalized Expectation-Maximization (EM)-type methods. We prove the
convergence of the PSG methods in the presence of bounded perturbations. This
resilience to bounded perturbations is relevant to the ability to apply the
recently developed superiorization methodology to PSG methods, in particular to
the EM algorithm.Comment: Computational Optimization and Applications, accepted for publicatio
Superiorization: An optimization heuristic for medical physics
Purpose: To describe and mathematically validate the superiorization
methodology, which is a recently-developed heuristic approach to optimization,
and to discuss its applicability to medical physics problem formulations that
specify the desired solution (of physically given or otherwise obtained
constraints) by an optimization criterion. Methods: The underlying idea is that
many iterative algorithms for finding such a solution are perturbation
resilient in the sense that, even if certain kinds of changes are made at the
end of each iterative step, the algorithm still produces a
constraints-compatible solution. This property is exploited by using permitted
changes to steer the algorithm to a solution that is not only
constraints-compatible, but is also desirable according to a specified
optimization criterion. The approach is very general, it is applicable to many
iterative procedures and optimization criteria used in medical physics.
Results: The main practical contribution is a procedure for automatically
producing from any given iterative algorithm its superiorized version, which
will supply solutions that are superior according to a given optimization
criterion. It is shown that if the original iterative algorithm satisfies
certain mathematical conditions, then the output of its superiorized version is
guaranteed to be as constraints-compatible as the output of the original
algorithm, but it is superior to the latter according to the optimization
criterion. This intuitive description is made precise in the paper and the
stated claims are rigorously proved. Superiorization is illustrated on
simulated computerized tomography data of a head cross-section and, in spite of
its generality, superiorization is shown to be competitive to an optimization
algorithm that is specifically designed to minimize total variation.Comment: Accepted for publication in: Medical Physic
Feasibility-Seeking and Superiorization Algorithms Applied to Inverse Treatment Planning in Radiation Therapy
We apply the recently proposed superiorization methodology (SM) to the
inverse planning problem in radiation therapy. The inverse planning problem is
represented here as a constrained minimization problem of the total variation
(TV) of the intensity vector over a large system of linear two-sided
inequalities. The SM can be viewed conceptually as lying between
feasibility-seeking for the constraints and full-fledged constrained
minimization of the objective function subject to these constraints. It is
based on the discovery that many feasibility-seeking algorithms (of the
projection methods variety) are perturbation-resilient, and can be proactively
steered toward a feasible solution of the constraints with a reduced, thus
superiorized, but not necessarily minimal, objective function value.Comment: Contemporary Mathematics, accepted for publicatio
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