## Communications in Mathematical Sciences

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

### Minimization with the affine normal direction

Hsiao-Bing Cheng, Li-Tien Cheng, and Shing-Tung Yau

#### 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

**Permanent link to this document**

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