Numerical Methods For Large Scale Sparse Minimax Problems

In the thesis, symmetrically consistent group-wise updating Newton-like methods, inexact Newton methods and inexact symmetrically consistent group-wise updating Newton-like methods for large scale sparse minimax problems are studied. Main results obtained are sketched as follows:1. In Chapter 2, we begin with smooth problems and study some improved versions of symmetrically consistent group-wise updating Newton-like methods and symmetrically consistent group-wise updating Cholesky factor techniques. And an inexact symmetrically consistent group-wise updating Newton-like method is presented. Local q-superlinear convergence and global convergence are proven and r-convergence rates of them are estimated.2. In Chapter 3, a symmetrically consistent group-wise updating method for sparse minimax problems is given. Because the objective concerns m functions with Hessians in different sparse structures, different grouping strategies for the Hessians must be taken, and hence periods of the Hessians updating may be different. Without strict complementarity, q-superlinear local convergence and global convergence are proven, and an r-convergence rate is estimated. In addition, efficient methods based on the modified Cholesky factorization are given to solve the nonconvex minimax problems.3. In chapter 4, inexact Newton methods for the minimax problems are studied. In each iteration, a quadratic minimax problem needs to be approximately solved inexactly by solving a sequence of special linear equation systems. Precision controlling criterions for the both stages are given to preserve the superlinear convergence and to save computation of solving the subproblems. Without strict complementarity, local q-superlinear convergence and global convergence are proven and the q-convergence rate is estimated.4. In chapter 5, inexact symmetrically consistent group-wise updating methods for large scale sparse minimax problems are studied. Precision controlling criterions for the both stages are given to preserve the superlinear convergence and to save computation of solving the subproblems. Without strict complementarity, local q-superlinear convergence and global convergence are proven and the convergence rate is estimated.All algorithms proposed in this paper are implemented in C/C++ or Matlab systems and numerical tests are done. Numerical results shows that these methods are efficient.