arXiv subjects

Salihah Alwadani

Publications and source records attributed to Salihah Alwadani.

5 recordsLinked to original sources

arXiv2021

The difference vectors for convex sets and a resolution of the geometry conjecture

The geometry conjecture, which was posed nearly a quarter of a century ago, states that the fixed point set of the composition of projectors onto nonempty closed convex sets in Hilbert space is actually equal to the intersection of certain translations of the underlying sets. In this paper, we provide a complete resolution of the geometry conjecture. Our proof relies on monotone operator theory. We revisit previously known results and provide various illustrative examples. Comments on the numerical computation of the quantities involved are also presented.

math.OC↗

arXiv2021

Attouch-Théra Duality, Generalized Cycles and Gap Vectors

Using the Attouch-Théra duality, we study the cycles, gap vectors and fixed point sets of compositions of proximal mappings. Sufficient conditions are given for the existence of cycles and gap vectors. A primal-dual framework provides an exact relationship between the cycles and gap vectors. We also introduce the generalized cycle and gap vectors to tackle the case when the classical ones do not exist. Examples are given to illustrate our results.

math.OC↗

arXiv2021

Resolvents and Yosida approximations of displacement mappings of isometries

Maximally monotone operators are fundamental objects in modern optimization. The main classes of monotone operators are subdifferential operators and matrices with a positive semidefinite symmetric part. In this paper, we study a nice class of monotone operators: displacement mappings of isometries of finite order. We derive explicit formulas for resolvents, Yosida approximations, and (set-valued and MoorePenrose) inverses. We illustrate our results by considering certain rational rotators and circular shift operators.

math.FA↗

arXiv2020

Fixed points of compositions of nonexpansive mappings: finitely many linear reflectors

Nonexpansive mappings play a central role in modern optimization and monotone operator theory because their fixed points can describe solutions to optimization or critical point problems. It is known that when the mappings are sufficiently "nice", then the fixed point set of the composition coincides with the intersection of the individual fixed point sets. In this paper, we explore the situation for compositions of linear reflectors. We provide positive results, upper bounds, and limiting examples. We also discuss classical reflectors in the Euclidean plane.

math.FA↗

arXiv2018

On the asymptotic behaviour of the Aragon Artacho-Campoy algorithm

Aragón Artacho and Campoy recently proposed a new method for computing the projection onto the intersection of two closed convex sets in Hilbert space; moreover, they proposed in 2018 a generalization from normal cone operators to maximally monotone operators. In this paper, we complete this analysis by demonstrating that the underlying curve converges to the nearest zero of the sum of the two operators. We also provide a new interpretation of the underlying operators in terms of the resolvent and the proximal average.

math.OC↗