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Set-Valued and Variational Analysis

, Volume 19, Issue 1, pp 75–96 | Cite as

Characterizing Convexity of a Function by Its Fréchet and Limiting Second-Order Subdifferentials

  • N. H. Chieu
  • T. D. Chuong
  • J.-C. YaoEmail author
  • N. D. Yen
Article

Abstract

The Fréchet and limiting second-order subdifferentials of a proper lower semicontinuous convex function \(\varphi: \mathbb R^n\rightarrow\bar{\mathbb R}\) have a property called the positive semi-definiteness (PSD)—in analogy with the notion of positive semi-definiteness of symmetric real matrices. In general, the PSD is insufficient for ensuring the convexity of an arbitrary lower semicontinuous function φ. However, if φ is a C 1,1 function then the PSD property of one of the second-order subdifferentials is a complete characterization of the convexity of φ. The same assertion is valid for C 1 functions of one variable. The limiting second-order subdifferential can recognize the convexity/nonconvexity of piecewise linear functions and of separable piecewise C 2 functions, while its Fréchet counterpart cannot.

Keywords

Convexity Characterization Positive semi-definite property Fréchet second-order subdifferential Limiting second-order subdifferential 

Mathematics Subject Classifications (2000)

49K40 49J40 49J52 49J53 

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • N. H. Chieu
    • 1
  • T. D. Chuong
    • 2
  • J.-C. Yao
    • 3
    Email author
  • N. D. Yen
    • 4
  1. 1.Department of MathematicsVinh UniversityVinhVietnam
  2. 2.Department of MathematicsDong Thap UniversityCao Lanh CityVietnam
  3. 3.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan
  4. 4.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam

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