Uniqueness results for strongly monotone operators related to Gauss measure

2019 ◽  
Vol 233 (1) ◽  
pp. 297-310
Author(s):  
Maria Francesca Betta ◽  
Filomena Feo ◽  
Maria Rosaria Posteraro
Keyword(s):  
Monotone Operators ◽  
Gauss Measure ◽  
Strongly Monotone ◽  
Uniqueness Results ◽  
Strongly Monotone Operators

scholarly journals Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Caiping Yang ◽  
Songnian He
Keyword(s):  
Variational Inequality ◽  
Unique Solution ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Monotone Operators ◽  
Strongly Monotone ◽  
Lipschitz Constants ◽  
Strongly Monotone Operators ◽  
Original Domain ◽  
Self Adaptive

Consider the variational inequalityVI(C,F)of finding a pointx*∈Csatisfying the property〈Fx*,x-x*〉≥0for allx∈C, whereCis a level set of a convex function defined on a real Hilbert spaceHandF:H→His a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets ofH) and strongly monotone operator. He and Xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see He and Xu, 2009). In this paper, relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. Since our algorithms avoid calculating the projectionPC(calculatingPCby computing a sequence of projections onto half-spaces containing the original domainC) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants ofF(i.e., Lipschitz constants on some bounded subsets ofH)), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.

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