2016/10/31 by Daniel Reem, Simeon Reich
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Convergence (economics) #Convex optimization #Generalization #Iterative method #Limit (mathematics) #Nonlinear system #Optimization and Variational Analysis #Optimization problem #Resolvent #Stochastic Gradient Optimization Techniques #Variational inequality #acm:47H05 #acm:47J25 #acm:49M37 #acm:65K05 #acm:90C30 #acm:90C31 #math.FA #math.OC #msc:47H05 #msc:47J25 #msc:49M37 #msc:65K05 #msc:90C30 #msc:90C31
paper · pdf · doi:10.1007/s12215-017-0318-6
published as Rend. Circ. Mat. Palermo (2) 67 (2018), 337--371 · To appear in Rendiconti del Circolo Matematico di Palermo Series 2; several remarks which discuss fully Legendre functions were moved from Section 13 to Section 3 and were improved slightly; correction of minor inaccuracies, mainly ones related to some references; added DOI and one additional reference
openalex created_date 2016/10/14 · openalex publication_date 2017/08/18 · arxiv created 2017/08/22 · arxiv updated 2018/10/23 · openalex updated_date 2026/08/05
Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been devised in order to achieve this task. A lot of these algorithms are inexact in the sense that they allow perturbations to appear during the iterative process, and hence they enable one to better deal with noise and computational errors, as well as superiorization. For many years a certain fundamental question has remained open regarding many of these known inexact algorithmic schemes in various finite and infinite dimensional settings, namely whether there exist sequences satisfying these inexact schemes when errors appear. We provide a positive answer to this question. Our results also show that various theorems discussing the convergence of these inexact schemes have a genuine merit beyond the exact case. As a by-product we solve the standard and the strongly implicit inexact resolvent inclusion problems, introduce a promising class of functions (fully Legendre functions), establish continuous dependence (stability) properties of the solution of the inexact resolvent inclusion problem and continuity properties of the protoresolvent, and generalize the notion of strong monotonicity.