The 𝒰-Lagrangian of a convex function

Lemaréchal, Claude; Oustry, François; Sagastizábal, Claudia

doi:10.1090/S0002-9947-99-02243-6

The $\mathcal U$-Lagrangian of a convex function
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by Claude Lemaréchal, François Oustry and Claudia Sagastizábal

Trans. Amer. Math. Soc. 352 (2000), 711-729

DOI: https://doi.org/10.1090/S0002-9947-99-02243-6

Published electronically: September 21, 1999

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Abstract:

At a given point $\bar {p}$, a convex function $f$ is differentiable in a certain subspace $\mathcal {U}$ (the subspace along which $\partial f(\bar {p})$ has 0-breadth). This property opens the way to defining a suitably restricted second derivative of $f$ at $\bar {p}$. 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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