ncku-talk2009-04-30

NCKU Math colloquium

DATE 2009-04-30　15:30-16:10

PLACE 國家理論科學中心 204 室

SPEAKER Prof. Thai Doan Chuong（Department of Mathematics, Dong Thap University）

TITLE Characterizing Convexity of A Function by Its Frechet And Limiting Second-order Subdifferentials

ABSTRACT The second-order Fr\'echet and limiting subdifferentials of a proper lower semicontinous convex function $\varphi: \bar{\mathbb R}^n\rightarrow\bar{\mathbb R}$ 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 arbitrarily lower semicontinous function $\varphi$. However, if $\varphi$ 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 $\varphi.$ The same assertion is valid for $C^1$ function of one variable. The second-order limiting subdifferential can recognize the convexity/nonconvexity of separable piecewise $C^2$ functions and of piecewise linear functions, while its Fr\'echet counterpart cannot.

NCKU Math colloquium

DATE	2009-04-30　15:30-16:10

PLACE	國家理論科學中心 204 室

SPEAKER	Prof. Thai Doan Chuong（Department of Mathematics, Dong Thap University）

TITLE	Characterizing Convexity of A Function by Its Frechet And Limiting Second-order Subdifferentials

ABSTRACT	The second-order Fr\'echet and limiting subdifferentials of a proper lower semicontinous convex function $\varphi: \bar{\mathbb R}^n\rightarrow\bar{\mathbb R}$ 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 arbitrarily lower semicontinous function $\varphi$. However, if $\varphi$ 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 $\varphi.$ The same assertion is valid for $C^1$ function of one variable. The second-order limiting subdifferential can recognize the convexity/nonconvexity of separable piecewise $C^2$ functions and of piecewise linear functions, while its Fr\'echet counterpart cannot.