Zero-convexity, perturbation resilience, and subgradient projections for feasibility-seeking methods

Invited abstract in session MB-54: Projection methods in optimization problems 2, stream Convex Optimization.

Area: Continuous Optimization

Monday, 10:30-12:00
Room: Building PA, Room B

Authors (first author is the speaker)

1. Daniel Reem
Math, The Technion
2. Yair Censor
Department of Mathematics , University of Haifa


The convex feasibility problem (CFP) is at the core of the
modeling of many problems in various areas of science. Subgradient
projection methods are important tools for solving the CFP because
they enable the use of subgradient calculations instead of orthogonal
projections onto the individual sets of the problem. Working in a real
Hilbert space, we show that the sequential subgradient projection method
is perturbation resilient. By this we mean that under appropriate
conditions the sequence generated by the method converges weakly, and
sometimes also strongly, to a point in the intersection of the given
subsets of the feasibility problem, despite certain perturbations which
are allowed in each iterative step. Unlike previous works on solving
the convex feasibility problem, the involved functions, which induce
the feasibility problem's subsets, need not be convex. Instead, we allow
them to belong to a wider and richer class of functions satisfying a
weaker condition, that we call zero-convexity. This class, which is
introduced and discussed here, holds a promise to solve optimization
problems in various areas, especially in non-smooth and non-convex
optimization. The relevance of this study to approximate minimization
and to the recent superiorization methodology for constrained optimization
is explained.

Ref: [Math. Prog. (Ser. A) 152 (2015), 339-380, arXiv:1405.1501].


Status: accepted

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