A feasible method for optimization with orthogonality constraints
A feasible method for optimization with orthogonality constraints
复制标题
一种可行的正交约束优化方法
DOI：
10.1007/s10107-012-0584-1
复制
发表时间：
2013-12-01
影响因子：
2.7
通讯作者：
Yin, Wotao
中科院分区：
文献类型：
--
作者：
Wen, Zaiwen;Yin, Wotao
Minimization with orthogonality constraints (e.g., X-inverted perpendicular X = I) and/or spherical constraints (e. g., parallel to x parallel to(2) = 1) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p谐波流，1位压缩感测，矩阵秩最小化等。这些问题很困难因为限制不仅是非凸的，而且在迭代过程中保留的数值昂贵。为了应对这些困难，我们应用了Cayley Transform-a曲柄nicolson样更新方案，以保留约束并基于它，与基于预测和大地测量学的曲面相比，开发具有较低拖球的曲线搜索算法。在各种测试问题上证明了所提出算法的效率。特别是，对于最大问题，它准确地解决了SDP松弛的分解配方。为了进行多项式优化，最接近的相关矩阵估计和极端的特征值问题，所提出的算法运行非常快，返回解决方案不比其最新算法的算法差。对于二次分配问题，可以在典型的笔记本电脑上5分钟内达到QAPLIB中最大问题“ TAI256C”的最著名解决方案的差距为0.842％。
Minimization with orthogonality constraints (e.g.,) and/or spherical constraints (e.g.,) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform—a Crank-Nicolson-like update scheme—to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 % to the best known solution on the largest problem “tai256c” in QAPLIB can be reached in 5 min on a typical laptop.