## Communications in Mathematical Sciences

### Minimization with the affine normal direction

#### Abstract

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.

#### Article information

Source
Commun. Math. Sci., Volume 3, Number 4 (2005), 561-574.

Dates
First available in Project Euclid: 7 April 2006

https://projecteuclid.org/euclid.cms/1144429332

Mathematical Reviews number (MathSciNet)
MR2188684

Zentralblatt MATH identifier
1095.53012

Subjects
Primary: 58Exx: Variational problems in infinite-dimensional spaces
Secondary: 53Axx: Classical differential geometry

#### Citation

Cheng, Hsiao-Bing; Cheng, Li-Tien; Yau, Shing-Tung. Minimization with the affine normal direction. Commun. Math. Sci. 3 (2005), no. 4, 561--574. https://projecteuclid.org/euclid.cms/1144429332