Open Access
December 2005 Minimization with the affine normal direction
Hsiao-Bing Cheng, Li-Tien Cheng, Shing-Tung Yau
Commun. Math. Sci. 3(4): 561-574 (December 2005).


In this paper, we consider minimization of a real-valued function $f$ over $\bold R\sp {n+1}$ and study the choice of the affine normal of the level set hypersurfaces of $f$ as a direction for minimization. The affine normal vector arises in affine differential geometry when answering the question of what hypersurfaces are invariant under unimodular affine transformations. It can be computed at points of a hypersurface from local geometry or, in an alternate description, centers of gravity of slices. In the case where $f$ is quadratic, the line passing through any chosen point parallel to its affine normal will pass through the critical point of $f$. We study numerical techniques for calculating affine normal directions of level set surfaces of convex $f$ for minimization algorithms.


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Hsiao-Bing Cheng. Li-Tien Cheng. Shing-Tung Yau. "Minimization with the affine normal direction." Commun. Math. Sci. 3 (4) 561 - 574, December 2005.


Published: December 2005
First available in Project Euclid: 7 April 2006

zbMATH: 1095.53012
MathSciNet: MR2188684

Primary: 58Exx
Secondary: 53Axx

Rights: Copyright © 2005 International Press of Boston

Vol.3 • No. 4 • December 2005
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