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 of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by , see Rosenblatt 1962 [14]. Dunagan and Vempala [4] have developed a re-scaled version of the perceptron algorithm with an improved complexity of iterations (with high probability), which is theoretically efficient in , 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 *K* where* K* is a regular convex cone. We provide a conic extension of the re-scaled perceptron algorithm based on the notion of a deep-separation oracle of a cone, which essentially computes a certificate of strong separation. We give a general condition under which the re-scaled perceptron algorithm is itself theoretically efficient; this includes the cases when *K* is the cross-product of half-spaces, second-order cones, and the positive semi-definite cone.

By:* Alexandre Belloni; Robert M. Freund; Santosh Vempala*

Published in: RC24073 in 2006

**LIMITED DISTRIBUTION NOTICE:**

**This Research Report is available. This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). I have read and understand this notice and am a member of the scientific community outside or inside of IBM seeking a single copy only.**

Questions about this service can be mailed to reports@us.ibm.com .