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Mathematical Programming

, Volume 148, Issue 1–2, pp 49–88 | Cite as

Applications of convex analysis within mathematics

  • Francisco J. Aragón ArtachoEmail author
  • Jonathan M. Borwein
  • Victoria Martín-Márquez
  • Liangjin Yao
Full Length Paper Series B

Abstract

In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator  Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis.

Keywords

Convex function Chebyshev set Fenchel conjugate Monotone operator Fitzpatrick function Autoconjugate representer 

Mathematics Subject Classification (2000)

Primary 47N10 90C25 Secondary 47H05 47A06 47B65 

Notes

Acknowledgments

The authors are grateful to the three anonymous referees for their pertinent and constructive comments. The authors also thank Dr. Hristo S. Sendov for sending them the manuscript [60]. The authors were all partially supported by various Australian Research Council grants.

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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2013

Authors and Affiliations

  • Francisco J. Aragón Artacho
    • 1
    Email author
  • Jonathan M. Borwein
    • 1
    • 2
  • Victoria Martín-Márquez
    • 3
  • Liangjin Yao
    • 1
  1. 1.Centre for Computer Assisted Research Mathematics and Its Applications (CARMA)University of NewcastleCallaghanAustralia
  2. 2.King Abdul-Aziz UniversityJeddahSaudi Arabia
  3. 3.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de SevillaSevillaSpain

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