On the Second-Order Feasibility Cone: Primal-Dual Representation and Efficient Projection

We study the second-order feasibility cone F = {y IRn : My gT 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 MT M - ggT . This representation is used to efficiently solve the problem of projecting an arbitrary point x IRn onto F: miny{ y-x : My gT y}, which aside from theoretical interest also arises as a necessary subroutine in the re-scaled perceptron algorithm. We develop a method for solving the projection problem to an accuracy whose computational complexity is bounded by O(mn2 + n ln ln(1/) + n ln ln(1/ min{width(F), width(F)})) operations. Here the width(F), width(F) denotes the widths of F and F, respectively. We also perform computational tests that indicate that the method is extremely efficient in practice. Key words. second-order cone, convex cone, projection, computational complexity, Newton method AMS subject classifications. 90C60, 90C51, 90C25, 49M15, 49M29

• Type: Journal
• Year: 2008
• Where: SIAMJO
• Authors: Alexandre Belloni, Robert M. Freund
• Keywords: Re-scaled Perceptron Algorithm, Second-order Feasibility Cone, SIAMJO 2008, Spectral Decomposition