An affine scaling interior trust-region method combining with nonmonotone line search filter technique for linear inequality constrained minimization

This paper proposes an affine scaling interior trust-region method in association with nonmonotone line search filter technique for solving nonlinear optimization problems subject to linear inequality constraints. Based on a Newton step which is derived from the complementarity conditions of linear inequality constrained optimization, a trust-region subproblem subject only to an ellipsoidal constraint is defined by minimizing a quadratic model with an appropriate quadratic function and scaling matrix. The nonmonotone schemes combining with trust-region strategy and line search filter technique can bring about speeding up the convergence progress in the case of high nonlinear. A new backtracking relevance condition is given which assures global convergence without using the switching condition used in the traditional line search filter technique. The fast local convergence rate of the proposed algorithm is achieved which is not depending on any external restoration procedure. The preliminary numerical experiments are reported to show effectiveness of the proposed algorithm.     

Shape structural optimization with an interior point nonlinear programming algorithm

The aim of this paper is to study the implementation of an efficient and reliable technique for shape optimization of solids, based on general nonlinear programming algorithms. We also study the practical behaviour for this kind of applications of a quasi-Newton algorithm, based on the Feasible Direction Interior Point Method for nonlinear constrained optimization. The optimal shape of the solid is obtained iteratively. At each iteration, a new shape is generated by B-spline curves and a new mesh is automatically generated. The control point coordinates are given by the design variables. Several illustrative two-dimensional examples are solved in a very efficient way. We conclude that the present approach is simple to formulate and to code and that our optimization algorithm is appropriate for this problem. Received May 12, 1999     

Numerical solution of KKT systems in PDE-constrained optimization problems via the affine scaling trust-region approach†

A recently proposed trust-region approach for bound-constrained nonlinear equations is applied to the Karush-Kuhn-Tucker (KKT) system arising from the discretization of a class of partial differential equation (PDE)-constrained optimization problems. Two different implementations are developed that take into account the large dimension and the special structure of the problems. The linear algebra phase is analysed considering the possibility of solving the arising linear systems by either direct methods or short-recurrence iterative linear solvers. Viability of the approach is proved through several numerical experiments on large KKT systems arising from the discretization of control problems.     

Solution of Systems of Equalities and Inequalities by the Method of Interior Points

A version of the Newton method is presented. In constructing an auxiliary problem, constraints in the form of inequalities are not considered and the classical extremal problem is solved. Inequalities are taken into account owing to a special choice of weighted coefficients and the step length. Local convergence of the proposed algorithm is studied. The convergence of successive approximations to a relatively interior admissible point of a non-linear system is established.     

An effective trust-region-based approach for symmetric nonlinear systems

This paper presents a new trust-region procedure for solving symmetric nonlinear systems of equations having several variables. The proposed approach takes advantage of the combination of both an effective adaptive trust-region radius and a non-monotone strategy. It is believed that the selection of an appropriate adaptive radius and the application of a suitable non-monotone strategy can improve the efficiency and robustness of the trust-region framework as well as decrease the computational costs of the algorithm by decreasing the required number of subproblems to be solved. The global convergence and the quadratic convergence of the proposed approach are proved without the non-degeneracy assumption of the exact Jacobian. The preliminary numerical results of the proposed algorithm indicating the promising behaviour of the new procedure for solving nonlinear systems are also reported.     

具有未知参数的非线性系统动态优化

针对具有未知参数和不等式路径约束的非线性系统动态优化问题,提出一种新颖有效的数值求解方法.首先,将未知参数视为一个动态优化问题的决策变量;其次,利用多重打靶法将无限维的含未知参数动态优化问题转化为有限维的非线性规划问题,进而在不等式路径约束违反的时间段内,用有限多个内点约束替代原不等式路径约束;然后,用内点法求解转化后的非线性规划问题,在路径约束违反的一定容许度下,经过有限多次步数迭代后得到未知参数值的同时得到控制策略,并在理论上对所提出算法的收敛性进行相应证明;最后,对两个经典的含未知参数非线性系统的动态优化问题进行数值仿真以验证所提出算法的有效性.     

On the convergence of variational iteration method for nonlinear coupled system of partial differential equations

In this paper, the sufficient conditions that guarantee the convergence of the variational iteration method when applied to solve a coupled system of nonlinear partial differential equations are presented. Especial attention is given to the error bound of the nth term of the resultant sequence. Numerical examples to show the efficiency of the method are presented.     

解非线性方程组的多目标优化进化算法

如何有效地求解复杂非线性方程组是进化计算领域一个新的研究问题。将非线性方程组等价地转化成多目标优化问题,同时设计了求解的多目标优化进化算法。为了提高算法的搜索能力及避免算法陷入局部最优,采用了自适应Levy变异进化算子和均匀杂交算子。计算机仿真表明该算法对非线性方程组的求解是有效的。