Communications in Mathematical Sciences

Volume 3 (2005)

Number 4

Minimization with the affine normal direction

Pages: 561 – 574



Hsiao-Bing Cheng

Li-Tien Cheng

Shing-Tung Yau


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.

2010 Mathematics Subject Classification

Primary 58Exx. Secondary 53Axx.

Published 1 January 2005