IJPAM: Volume 51, No. 1 (2009)

THE PROPERTIES OF THE $\mathcal{U}$-LAGRANGIAN
AND ITS MINIMUM SET OF THE PROPER
CONVEX FUNCTION

Yuan Lu$^1$, Wei Wang$^2$
$^{1,2}$CORA, Department of Applied Mathematics
Dalian University of Technology
Dalian, 116024, P.R. CHINA
$^1$e-mail: luyuan626@yahoo.com.cn
$^2$e-mail: wangwei60@dl.cn


Abstract.The analysis of the $\mathcal{U}$-Lagrangian function and its minimum set is critical for giving the $\mathcal{VU}$-theory of the proper convex function. Based on that, this paper studies the $\mathcal{U}$-Lagrangian of the proper convex function. We analyze the properties of the $\mathcal{U}$-Lagrangian and give the expression of its subdifferential. Also, we demonstrate the sufficient condition for the nonempty of the minimum set. Further more, we prove the minimum set is outer semicontinuity.

Received: January 22, 2009

AMS Subject Classification: 90C25, 52A41, 26B25, 90C30

Key Words and Phrases: $\mathcal{VU}$-decomposition, $\mathcal{U}$-Lagrangian, conjugate function, recession function, outer semicontinuity

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 51
Issue: 1