ON THE SECOND-ORDER FEASIBILITY CONE: PRIMAL-DUAL REPRESENTATION AND EFFICIENT PROJECTION

We study the second-order feasibility cone F = {y is an element of R-n : parallel to M-y parallel to <= g(T)y} for given data (M, g). We construct a new representation for this cone and its dual based on the spectral decomposition of the matrix (MM)-M-T - gg(T). This representation is used to efficiently solve the problem of projecting an arbitrary point x is an element of R-n onto F: min(y){parallel to y - x parallel to : parallel to My parallel to <= g(T)y}, which aside from theoretical interest also arises as a necessary subroutine in the rescaled perceptron algorithm. We develop a method for solving the projection problem to an accuracy epsilon, whose computational complexity is bounded by O(mn(2)+ n ln ln(1/epsilon) + n ln ln(1/min{width(F), width(F*)})) operations. Here width(F) and width(F*) denote the width of F and F*, respectively. We also perform computational tests that indicate that the method is extremely efficient in practice.