An iterative row-action method for interval convex programming

Censor, Yair; Lent, Arnold

doi:10.1007/bf00934676

Cited by 421 publications

(186 citation statements)

References 15 publications

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“…While convergence of Bregman projections has been discussed extensively in the literature (see for instance Censor and Lent (1981); Bauschke and Borwein (1996)), the authors are unaware of a result along these lines that would apply directly to the situation considered in the present paper. Below we use an alternative perspective, which relates PEFF solutions to positive eigenvectors of a nonlinear mapping.…”

Section: Introductionmentioning

confidence: 92%

The composite iteration algorithm for finding efficient and financially fair risk-sharing rules

Pazdera

Schumacher

Werker

2017

Journal of Mathematical Economics

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The composite iteration algorithm for finding efficient and financially fair risk-sharing rules Pazdera, J.; Schumacher, J.M.; Werker, B.J.M. Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. AbstractWe consider the problem of finding an efficient and fair ex-ante rule for division of an uncertain monetary outcome among a finite number of von Neumann-Morgenstern agents. Efficiency is understood here, as usual, in the sense of Pareto efficiency subject to the feasibility constraint. Fairness is defined as financial fairness with respect to a predetermined pricing functional. We show that efficient and financially fair allocation rules are in one-to-one correspondence with positive eigenvectors of a nonlinear homogeneous and monotone mapping associated to the risk sharing problem. We establish relevant properties of this mapping. On the basis of this, we obtain a proof of existence and uniqueness of solutions via nonlinear Perron-Frobenius theory, as well as a proof of global convergence of the natural iterative algorithm. We argue that this algorithm is computationally attractive, and discuss its rate of convergence.

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Section: Introductionmentioning

confidence: 92%

The composite iteration algorithm for finding efficient and financially fair risk-sharing rules

Pazdera

Schumacher

Werker

2017

Journal of Mathematical Economics

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“…The information-geometric view that we take also shows that some of the algorithms we study, including AdaBoost, fit into a family of algorithms described in 1967 by Bregman (1967), and elaborated upon by Censor and Lent (1981), for satisfying a set of constraints. 1 Our work is based directly on the general setting of Lafferty, Della Pietra, and Della Pietra (1997) in which one attempts to solve optimization problems based on general Bregman distances.…”

Section: Previous Workmentioning

confidence: 99%

Untitled

2002

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Abstract. We give a unified account of boosting and logistic regression in which each learning problem is cast in terms of optimization of Bregman distances. The striking similarity of the two problems in this framework allows us to design and analyze algorithms for both simultaneously, and to easily adapt algorithms designed for one problem to the other. For both problems, we give new algorithms and explain their potential advantages over existing methods. These algorithms are iterative and can be divided into two types based on whether the parameters are updated sequentially (one at a time) or in parallel (all at once). We also describe a parameterized family of algorithms that includes both a sequential-and a parallel-update algorithm as special cases, thus showing how the sequential and parallel approaches can themselves be unified. For all of the algorithms, we give convergence proofs using a general formalization of the auxiliary-function proof technique. As one of our sequential-update algorithms is equivalent to AdaBoost, this provides the first general proof of convergence for AdaBoost. We show that all of our algorithms generalize easily to the multiclass case, and we contrast the new algorithms with the iterative scaling algorithm. We conclude with a few experimental results with synthetic data that highlight the behavior of the old and newly proposed algorithms in different settings.

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“…[13,20]). The Bregman distance does not satisfy the well-known properties of a metric, but it does have the following important property, which is called the three point identity: for any x ∈ dom f and y, z ∈ int dom f ,…”

Section: Page 619])mentioning

confidence: 99%

Right Bregman nonexpansive operators in Banach spaces

Martín-Márquez¹,

Reich²,

Sabach³

2012

Nonlinear Analysis: Theory, Methods & Applications

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We introduce and study new classes of Bregman nonexpansive operators in reflexive Banach spaces. These classes of operators are associated with the Bregman distance induced by a convex function. In particular, we characterize sunny right quasi-Bregman nonexpansive retractions, and as a consequence, we show that the fixed point set of any right quasi-Bregman nonexpansive operator is a sunny right quasi-Bregman nonexpansive retract of the ambient Banach space.

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An iterative row-action method for interval convex programming

Cited by 421 publications

References 15 publications

The composite iteration algorithm for finding efficient and financially fair risk-sharing rules

The composite iteration algorithm for finding efficient and financially fair risk-sharing rules

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Right Bregman nonexpansive operators in Banach spaces

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