# An Efficient Rescaled Perceptron Algorithm for Conic Systems

Published

Journal Article

The classical perceptron algorithm is an elementary row-action/relaxation algorithm for solving a homogeneous linear inequality system Ax > 0. A natural condition measure associated with this algorithm is the Euclidean width t of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by 1/t 2 [see Rosenblatt, F. 1962. Principles of Neurodynamics. Spartan Books, Washington, DC]. Dunagan and Vempala [Dunagan, J., S. Vempala. 2007. A simple polynomial-time rescaling algorithm for solving linear programs. Math. Programming 114(1) 101-114] have developed a rescaled version of the perceptron algorithm with an improved complexity of O(n ln(1/t)) iterations (with high probability), which is theoretically efficient in t and, in particular, is polynomial time in the bit-length model. We explore extensions of the concepts of these perceptron methods to the general homogeneous conic system Ax ∈ int k, where K is a regular convex cone. We provide a conic extension of the rescaled perceptron algorithm based on the notion of a deep-separation oracle of a cone, which essentially computes a certificate of strong separation. We show that the rescaled perceptron algorithm is theoretically efficient if an efficient deep-separation oracle is available for the feasible region. Furthermore, when K is the cross-product of basic cones that are either half-spaces or second-order cones, then a deep-separation oracle is available and, hence, the rescaled perceptron algorithm is theoretically efficient. When the basic cones of K include semidefinite cones, then a probabilistic deep-separation oracle for K can be constructed that also yields a theoretically efficient version of the rescaled perceptron algorithm. © 2009 INFORMS.

### Full Text

### Duke Authors

### Cited Authors

- Belloni, A; Freund, RM; Vempala, S

### Published Date

- August 1, 2009

### Published In

### Volume / Issue

- 34 / 3

### Start / End Page

- 621 - 641

### Electronic International Standard Serial Number (EISSN)

- 1526-5471

### International Standard Serial Number (ISSN)

- 0364-765X

### Digital Object Identifier (DOI)

- 10.1287/moor.1090.0388

### Citation Source

- Scopus