Projecting onto intersections of halfspaces and hyperplanes

Improved analysis of algorithms based on supporting halfspaces and quadratic programming for the convex intersection and feasibility problems

This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs (QPs) and approximate projections arising from partially solving the QPs are sufficient for multiple-term superlinear convergence for nonsmooth problems. Second, we identify additional regularity, which we call the second order supporting hyperplane property (SOSH), that gives multiple-term quadratic convergence. Third, we show that these fast convergence res…

Nonconvex set intersection problems: From projection methods to the Newton method for super-regular sets

The problem of finding a point in the intersection of closed sets can be solved by the method of alternating projections and its variants. It was shown in earlier papers that for convex sets, the strategy of using quadratic programming (QP) to project onto the intersection of supporting halfspaces generated earlier by the projection process can lead to an algorithm that converges multiple-term superlinearly. The main contributions of this paper are to show that this strategy can be effective for super-regular sets, which are structured nonconvex sets introduced by Lewis, Luke and Malick. Manif…

Set intersection problems: Integrating projection and quadratic programming algorithms

Abstract. The Set Intersection Problem (SIP) is the problem of finding a point in the intersection of convex sets. This problem is typically solved by the method of alternating projections. To accelerate the convergence, the idea of using Quadratic Programming (QP) to project a point onto the intersection of halfspaces generated by the projection process was discussed in earlier papers. This paper looks at how one can integrate projection algorithms together with an active set QP algorithm. As a byproduct of our analysis, we show how to accelerate an SIP algorithm involving box constraints, an…

The Block-wise Circumcentered-Reflection Method

The elementary Euclidean concept of circumcenter has recently been employed to improve two aspects of the classical Douglas--Rachford method for projecting onto the intersection of affine subspaces. The so-called circumcentered-reflection method is able to both accelerate the average reflection scheme by the Douglas--Rachford method and cope with the intersection of more than two affine subspaces. We now introduce the technique of circumcentering in blocks, which, more than just an option over the basic algorithm of circumcenters, turns out to be an elegant manner of generalizing the method of…

快速投影到带帽单纯形及其在生物信息学稀疏回归中的应用

Fast Projection onto the Capped Simplex with Applications to Sparse Regression in Bioinformatics

本文提出一种基于牛顿法的O(n)复杂度快速投影算法，用于k-带帽单纯形投影，在生物信息学大规模稀疏回归中显著加速PQN方法，较现有方法快3-8倍。