Nonconvex integration using ϵ-subdifferentials

ABSTRACT

We give an integration criterion for nonconvex functions defined in locally convex spaces. We prove that an inclusion-type relationship between the ϵ-subdifferentials, for small amount of of any two functions is sufficient for the equality of the associated closed convex envelopes, up to an additive constant and to a recession term, that is related to the asymptotic behaviour of the functions. This recession term is dropped out when the functions are convex. We use these results to represent both the values of closed convex envelopes and their ϵ-subdifferentials by means of ϵ-subdifferentials of the original function.

Keywords:

• Integration of nonconvex functions
• ϵ-subdifferentials
• lower semicontinuous convex envelopes

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 26B05
• 26J25
• 49H05

Acknowledgments

We are thankful to both referees for their insightful suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Research supported Project Fondo Nacional de Desarrollo Científico y Tecnológico (Fondecyt) no. 1150909 and 1151003, Project Mathamsud 17-MATH-06, Comisión Nacional de Investigación Científica y Tecnológica Conicyt-Redes no. 150040, and CONCYTEC, Consejo Nacional de Ciencia, Tecnología e Innovación Tecnológica (Concytec) EE020–Math Amsud project no. 003–2017.