- An Efficient Re-scaled Perceptron Algorithm for Conic Systems Alexandre Belloni (bellonimit.edu) Robert M. Freund (rfreundmit.edu) Santosh Vempala (vempalamath.mit.edu) Abstract: 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 1962. Dunagan and Vempala have developed a re-scaled 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 \in int 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. Keywords: Convex Cones ; Perceptron ; Conic System ; Separation Oracle Category 1: Linear, Cone and Semidefinite Programming Citation: Technical Report RC24073, October/2006 IBM T.J. Watson Research Center 1101 Kitchawan Road, Yorktown Heights, NY 10598 Download: [PDF]Entry Submitted: 10/10/2006Entry Accepted: 10/10/2006Entry Last Modified: 10/10/2006Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.