Birkhoff多面体上投影矩阵的广义雅可比矩阵的有效计算
On the efficient computation of a generalized Jacobian of the projector over the Birkhoff polytope
摘要：
我们导出了一个显式公式，以及一个有效的程序，用于构造一个广义方阵的投影到伯克霍夫多面体，即，双随机阵的集合。为了保证我们的程序的高效率，一个半光滑牛顿法求解投影问题的对偶提出并有效地实现。通过将其性能与其他强大的求解器（如商业软件Guidelines和学术代码PPROJ [{ \sc Hager and Zhang}，SIAM Journal on Optimization，26（2016），pp.~ 1773- 1798年]。特别是，我们的算法是能够解决超过十亿个变量和非负约束的投影问题，以一个非常高的精度在小于15分钟在一个普通的台式电脑。更重要的是，基于我们对投影及其广义雅可比矩阵的有效计算，我们可以设计一种高效的增广拉格朗日方法（ALM）来求解一类Birkhoff多面体约束的凸二次规划（QP）问题.所得到的ALM被证明是更有效的解决所产生的二次指派问题的放松QP问题的集合比Guidelines。
英文摘要：
We derive an explicit formula, as well as an efficient procedure, for constructing a generalized Jacobian for the projector of a given square matrix onto the Birkhoff polytope, i.e., the set of doubly stochastic matrices. To guarantee the high efficiency of our procedure, a semismooth Newton method for solving the dual of the projection problem is proposed and efficiently implemented. Extensive numerical experiments are presented to demonstrate the merits and effectiveness of our method by comparing its performance against other powerful solvers such as the commercial software Gurobi and the academic code PPROJ [{\sc Hager and Zhang}, SIAM Journal on Optimization, 26 (2016), pp.~1773--1798]. In particular, our algorithm is able to solve the projection problem with over one billion variables and nonnegative constraints to a very high accuracy in less than 15 minutes on a modest desktop computer. More importantly, based on our efficient computation of the projections and their generalized Jacobians, we can design a highly efficient augmented Lagrangian method (ALM) for solving a class of convex quadratic programming (QP) problems constrained by the Birkhoff polytope. The resulted ALM is demonstrated to be much more efficient than Gurobi in solving a collection of QP problems arising from the relaxation of quadratic assignment problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
