本项目主要研究大规模优化问题的 Krylov子空间算法。研究内容为：利用 Krylov子空间算法求解大规模二次约束二次规划(QCQP)问题、正交约束问题、变系数二次规划问题和特征值互补问题等优化中的经典问题。我们的主要创新点在于：1)块Krylov 子问题做为投影子空间，把原问题投影为小问题进行求解；2)利用近似积极集的想法。我们用一阶方法得到最优解积极集的近似估计，利用近似积极约束的法向量形成块Krylov子空间，这样计算效率更高；3)用线性正则性质进行算法的误差估计；4)Krylov子空间算法的缺点是正交性容易失去，造成算法精度下降。我们采用嵌套Krylov子空间的方法可以克服这个困难。目前我们已经有了一些前期工作，其中包括块Krylov子空间算法求解广义信赖域问题、CDT子问题。其中广义信赖域问题的块Krylov子空间算法，论文发表在SIOPT。

This project is devoted to Krylov subspace algorithms for large-scale optimization problems. The main research contents of this project is to use the Krylov subspace method to solve the important subjects in optimization such as large scale quadratically constrained quadratic programing problems (QCQP), the optimal problems with orthogonality constraints, varying coefficient matrix quadratic programming and cone eigenvalue complementarity problems, etc. Our main innovation lies in that: 1) Project the original problem to a small-scale problem. The projection subspace is a block Krylov subspace; 2) We use the idea of approximate active set to form the block Krylov subspace. The approximate active set of the optimal solution is computed by the first order method. Then we use the normal vectors of the approximate active constraints to form the block Krylov subspace. The advantage of this method is that we can form a small dimension Krylov subspace and therefore is more effective; 3) We use the linear regularity property to give error bounds of the Krylov subspace algorithm; 4) The drawback of the Krylov subspace algorithm is the lost of orthogonality, which results in the lower accuracy of the solution. We can use the nested Krylov subspace method to overcome this difficulty. Now we have already done some works, which include solving the extended trust region problem and the CDT subproblem by Krylov subspace lagorithms. The work of block Krylov subspace algorithm of solving the extended trust region subproblem is published in SIOPT.