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Set-Valued and Variational Analysis

, Volume 27, Issue 4, pp 863–893 | Cite as

Progressive Decoupling of Linkages in Optimization and Variational Inequalities with Elicitable Convexity or Monotonicity

  • R. Tyrrell RockafellarEmail author
Article

Abstract

Algorithms for problem decomposition and splitting in optimization and the solving of variational inequalities have largely depended on assumptions of convexity or monotonicity. Here, a way of “eliciting” convexity or monotonicity is developed which can get around that limitation. It supports a procedure called the progressive decoupling algorithm, which is derived from the proximal point algorithm through passing to a partial inverse, localizing and rescaling. In the optimization setting, elicitability of convexity corresponds to a new and very general kind of second-order sufficient condition for a local minimum. Applications are thereby opened up to problem decomposition and splitting even in nonconvex optimization, moreover with augmented Lagrangians for subproblems assisting in the implementation.

Keywords

Convex/nonconvex optimization Monotone/nonmonotone variational inequalities Linkage problems Progressive decoupling Progressive hedging Problem decomposition Splitting Elicitable convexity Elicitable monotonicity Variational convexity Variational second-order sufficiency Proximal point algorithm Method of partial inverses Proximal methods of multipliers Augmented Lagrangians 

Mathematics Subject Classification (2010)

65K10 65K15 

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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