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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The $\mathcal{U}$-Lagrangian of a convex function


Authors: Claude Lemaréchal, François Oustry and Claudia Sagastizábal
Journal: Trans. Amer. Math. Soc. 352 (2000), 711-729
MSC (1991): Primary 49J52, 58C20; Secondary 49Q12, 65K10
Published electronically: September 21, 1999
MathSciNet review: 1487623
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Abstract: At a given point ${\overline{p}}$, a convex function $f$ is differentiable in a certain subspace $\mathcal{U}$ (the subspace along which $\partial f({\overline{p}})$ has 0-breadth). This property opens the way to defining a suitably restricted second derivative of $f$ at ${\overline{p}}$. We do this via an intermediate function, convex on $\mathcal{U}$. We call this function the $\mathcal{U}$-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.


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Additional Information

Claude Lemaréchal
Affiliation: INRIA, 655 avenue de l’Europe, 38330 Montbonnot, France
Email: Claude.Lemarechal@inria.fr

François Oustry
Affiliation: INRIA, 655 avenue de l’Europe, 38330 Montbonnot, France
Email: Francois.Oustry@inria.fr

Claudia Sagastizábal
Affiliation: INRIA, BP 105, 78153 Le Chesnay, France
Email: Claudia.Sagastizabal@inria.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02243-6
PII: S 0002-9947(99)02243-6
Keywords: Nonsmooth analysis, generalized derivative, second-order derivative, composite optimization
Received by editor(s): July 18, 1996
Received by editor(s) in revised form: August 1, 1997
Published electronically: September 21, 1999
Article copyright: © Copyright 1999 American Mathematical Society