The U-Lagrangian of a convex function

Citation
C. Lemarechal et al., The U-Lagrangian of a convex function, T AM MATH S, 352(2), 2000, pp. 711-729
Citations number
32
Categorie Soggetti
Mathematics
Journal title
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
ISSN journal
00029947 → ACNP
Volume
352
Issue
2
Year of publication
2000
Pages
711 - 729
Database
ISI
SICI code
0002-9947(200002)352:2<711:TUOACF>2.0.ZU;2-T
Abstract
At a given point (p) over bar, a convex function f is differentiable in a c ertain subspace U (the subspace along which partial derivative f((p) over b ar) has 0-breadth). This property opens the way to defining a suitably rest ricted second derivative of f at (p) over bar. We do this via an intermedia te function, convex on U. We call this function the U-Lagrangian; it coinci des with the ordinary Lagrangian in composite cases: exact penalty, semidef inite programming. Also, we use this new theory to design a conceptual patt ern for superlinearly convergent minimization algorithms. Finally, we estab lish a connection with the Moreau-Yosida regularization.