Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces

2016/04/09 by Nobusumi Sagara, Sagara, Nobusumi · 1 citation
Computer Science · Economics, Econometrics and Finance · #28B20 #46G10 #91B50 #Economic theories and models #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC) #Optimization and Variational Analysis #Primary: 28B05 #Secondary: 49J30

paper · pdf · doi:10.48550/arxiv.1604.02514

openalex publication_date 2016/04/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We formulate bang-bang, purification, and minimization principles in dual Banach spaces with Gelfand integrals and provide a complete characterization of the saturation property of finite measure spaces. We also present a new application of the relaxation technique to large economies with infinite-dimensional commodity spaces, where the space of agents is modeled as a finite measure space. We propose a "relaxation" of large economies, which is regarded as a reasonable convexification of original economies. Under the saturation hypothesis, the relaxation and purification techniques enable us to prove the existence of Pareto optimal allocations without convexity assumptions.

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