仿射内点既约投影Hessian算法解非线性约束优化

本文结合非单调内点回代技术,提供了新的仿射信赖域方法解含有非负变量约束和非线性等式约束的优化问题.为求解大规模问题,采用等式约束的Jacobian矩阵的QR分解和两块校正的双边既约Hessian矩阵投影,将问题分解成零空间和值空间两个信赖域子问题.零空间的子问题为通常二次目标函数只带椭球约束的信赖域子问题,而值空间的子问题使用满足信赖域约束参数的值空间投影向量方向.通过引入Fletcher罚函数作为势函数,将由两个子问题结合信赖域策略构成的合成方向,并使用非单调线搜索技术回代于可接受的非负约束内点步长.在合理的条件下,算法具有整体收敛性且两块校正的双边既约Hessian投影法将保持超线性收敛速率.非单调技术将克服高度非线性情况,加快收敛进展.

PROJECTED REDUCED HESSIAN ALGORITHM WITH AFFINE SCAALING INTERIOR POINT FOR NONLINEAR CONSTRAINED OPTIMIZATION

In this paper, the author proposes a new affine scaling trust region algorithm with nonmonotonic backtracking interior point technique for nonlinear equality constrained optimization problems with nonnegative constraints on the variables. In order to deal with large problems, QR decomposition of the Jacobian for equality constraint and a two-piece update of two-side projected reduced Hessian are performed into a pair of trust region sub-problems in horizontal and vertical subspaces. The horizontal subproblem is a general trust region subproblem defined by minimizing a quadratic function subject only to an ellipsoidal constraint, while the vertical subproblem is used by a parameter size of the vertical direction with a trust region constraint. By adopting the Fletcher's penalty function as the merit function, combining trust region strategy and nonmonotone line search technique will switch to backtracking interior point direction composed by the two subproblems to obtain an acceptable interior point step. The global convergence results of the proposed algorithm are proved while maintaining fast local superlinear convergence rate is established by performing a two-piece update of two-side projected reduced Hessian. A nonmonotonic criterion is used to speed up the convergence progress in some highly nonlinear cases.