【作者】 陈小燕；

【导师】 张圣贵；

【作者基本信息】 福建师范大学 ， 运筹学与控制论， 2012， 硕士

【摘要】 本文探讨线性等式约束规划问题的几何算法,主要由三部分组成.第一部分,利用点到线性流形的距离的几何特征,提出了求解目标函数的Hesse矩阵正定并带有线性等式约束的最优化问题的几何算法.与牛顿法相比,该算法避免了Hesse矩阵求逆与矩阵乘积等运算.第二部分,利用点到线性流形的距离的几何特征,再结合BFGS校正,提出了求解带有线性等式约束的一般规划问题的BFGS-几何算法,并给出收敛性证明和数值算例.第三部分,利用点到线性流形的距离的几何特征,再结合广义拟牛顿校正,提出了求解带有线性等式约束的一般规划问题的广义拟牛顿-几何算法.同样地,给出收敛性证明和数值算例.更多还原

【Abstract】 In this paper we discuss the geometric algorithms for programs with constraints of linear equalities. It mainly consists of three parts.In the first part, by means of the geometric characterization of the distance from a point to a linear manifold, an algorithm for programs with positive definite Hesse matrix of the cost functions and constraints of linear equalities is presented. Compared with the Newton’s algorithm, the algorithm here avoids computation of the inverse of the Hesse matrix of the cost function and multiplication of matrices.In the second part, by means of the geometric characterization of the distance from a point to a linear manifold and BFGS update, an algorithm for programs with constraints of linear equalities is proposed. The convergence theorems are proved, and some numerical examples are given.In the third part, by means of the geometric characterization of the distance from a point to a linear manifold and generalized quasi-Newton update, an algorithm for programs with constraints of linear equalities is proposed. The convergence theorems are proved, too. Some numerical examples arc given to show the feasibility of the algorithm.更多还原