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

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

Improved analysis of algorithms based on supporting halfspaces and quadratic programming for the convex intersection and feasibility problems

This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs (QPs) and approximate projections arising from partially solving the QPs are sufficient for multiple-term superlinear convergence for nonsmooth problems. Second, we identify additional regularity, which we call the second order supporting hyperplane property (SOSH), that gives multiple-term quadratic convergence. Third, we show that these fast convergence results carry over for the convex inequality problem. Fourth, we show that infeasibility can be detected in finitely many operations. Lastly, we explain how we can use the dual active set QP algorithm of Goldfarb and Idnani to get useful iterates by solving the QPs partially, overcoming the problem of solving large QPs in our algorithms.Comment: 27 pages, 2 figure