【作者】 王晓东；

【导师】 乌力吉；

【作者基本信息】 内蒙古工业大学 ， 计算数学， 2007， 硕士

【摘要】 非线性约束优化问题(CNLP)是运筹学的重要分支之一，它广泛应用于自然科学、工程和经济领域中通过求解CNLP的KKT-点来得到CNLP的局部极小点的方法是最近较流行的求解CNLP的数值方法之一。这类方法的一般特点是首先把KKT-系统等价地转化为光滑的无约束(或具有简单约束)的最优化问题，然后利用经典最优化方法来求解KKT-点。本文研究的算法也是基于这种思想，但构造目标函数的方法不同，具体内容如下：1.通过构造仅在第一象限定义的NCP-函数，将非线性约束优化问题的KKT-系统等价地转化为带非负约束的光滑优化问题，这里借鉴了处理非线性互补问题的不可行点算法的思想，引入了辅助变量，简化了约束条件利用Qi等提出的效率较高的信赖域方法提供了具体求解KKT-点的算法，并在适当的条件下证明了算法的全局收敛性和局部超线性(二次)收敛性。2.在研究以往NCP-函数的基础上总结出了一个构造NCP-函数的方法，利用该方法构造出来的NCP-函数具有良好的性质.在第三章中，根据该方法构造了两个新的NCP-函数，第一个NCP-函数具有四个象限收敛速度相近的性质，而第二个NCP-函数具有Lipschistz连续性、强半光滑，较好的强制性等良好性质.利用后一个NCP-函数给出了CNLP的KKT-系统新的非负约束优化等价形式并结合Qi等提出的信赖域方法进行了求解.在较弱的条件下得到了全局收敛性和局部超线性(二次)收敛性。数值试验表明此该算法具有较好适应性和稳定性，实际计算效果也令人满意。更多还原

【Abstract】 The constrained nonlinear programming problems(CNLP) is a very important component part in operations research.It has wide application in natural science, en-gineering and economics. In recent years, the method that a local minimum of CNLP is obtained by solving KKT-systems for CNLP becomes one of the efficient numerical methods for CNLP. The main idea of the method is to reformulate the KKT-systems for CNLP as an unconstrained (or simply constrained) smooth minimization problem equivalently. Then a KKT-point for CNLP is obtained by using classical numerical methods for the problem. In this thesis, enlightened by the idea, author equivalently transformed the KKT-systems for CNLP into some new simply constrained smooth minimization problems different from that in literature. The details arc as follows:1.Firstly.author constructed a new NCP-function that is only defined in first quad-rant, then equivalently reformulated the KKT-systems for CNLP as a smooth mini-mization problem with non-negative constraints. As in the infeasible method for non-linear complementarity problem, an additional variable was introduced in order to getting simple constraints- An efficient trust region algorithm for solving KKT point for CNLP was provided via using the method proposed by Qi etc[15]. The global convergence and local superlinear (or quadratic) convergence for the algorithm were shown under suitable conditions.2. Futhermorc. a new method that to construct NCP-functions was proposed. By using the method, author obtained a new NCP-function with many useful properties, such as Lipschitz continuity. Strong semismoothness. good coercivity etc. A new refor-mulation was constructed by using it, and a similar algorithm as above was provided on the reformulation- The global convergence and local superlinear (or quadratic) con-vergence were obtained under weaker assumptions than above. Numerical results show that the algorithm is robust and efficient.更多还原