结构最优化问题的进一步研究

The project focused on the further studies about the unconstrained.structured optimization problems, especially about the least squares problems with zero residual or non-zero residual and Unary optimization problems. We proposed some promising algorithms and analyses their convergent properties for the different.versions in the studies. First of all, aiming at the special structure of the least squares problems, which form a primary and typical examples of structured optimization, we first develop a nonmonotone trust region algorithms for least squares problems with zero residual.This algorithm allows the sequence of objective function values to be nonmonotone.which accelerates the iterating progress, especially in the case where the objective function is ill-conditioned. Some global and local convergence properties of the proposed algorithms are proved under mild conditions. For the least squares.problems with non-zero residual, we proposed a modified truncated Newton method with secant proconditioners which has been proved to be globally convergent and quadratically or superlinearly locally convergent under mild conditions. A set of numerical results is reported for the proposed algorithms, respectively, which shows.that the presented algorithms are promising and superior to the corresponding compared algorithms according to the numbers of gradient and function evaluations. The software packages (Fortran code) are ready for the further researches. Another typical structured optimization problem, unary optimization problem, has also been considered in the project. Based on our discussion on the two replacement criteria proposed by Goldfarb and Wang and preconditioned conjugate gradient method, we proposed two sets of modified replacement criteria which.overcame the poor local convergence which Goldfarb and Wang’s algorithms.achieved based on their replacement criteria. Based on the two classes of modified replacement criteria, we presented two classes of modified truncated Newton-like algorithms with secant preconditioners for solving unconstrained unary optimization.problems. The algorithms proposed only partially updated an approximation to the Hessian matrix in each iteration by utilizing limited times of rank-one updating of the Choleschy factorization. In contrast with the Goldfarb and Wang’s original algorithms, the algorithms not only converge globally, but also possess a locally quadratic or superlinear convergence rate. Furthermore, our numerical experiments show that the.new algorithms outperform Goldfarb and Wang’s algorithms. The software packages (Fortran code) are ready for the further researches. For the general unconstrained structured optimization problems, we first tried to.generalize the research results about the above two typical examples: least squares problems and unary optimization problems to the general case. Then we introduced the secant preconditioner strategy to propose an modified structured secant Newton-like method to improve the efficient of the structured secant algorithms proposed by H. J. Martinez et al Based on our proposed new secant Newton equations and new secant Newton update formula.

结构最优化问题的研究具有重要的理论价值，同时有广泛的应用背景。很多重要的实际问题都可以归结为结构最优化问题，因此对此方向的研究越来越的到国内专家学者的关注。我们拟对起进行进一步的研究。设计一些更有效的结构，拟牛顿算法及其软件，分析其收敛性。推动一般结构最优化理论的进一步发展。...