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