非线性规划的无惩罚型方法及其理论

Nonlinear programming with nonlinear constraints is of very important applications in many fields. Many efficient penalty function methods exist for solving nonlinear constrained optimization problems. Specific examples include sequential unconstrained optimization methods on various penalty functions, and sequential quadratic programming (SQP) methods that adopt either line search method or trust region strategies. These methods by dint of some penalty function are usually called as penalty-type methods. The effectiveness of these so-called penalty-type methods hinges on how well the initial penalty parameter is chosen and how "intelligently" it is updated during the course of minimization. Therefore, it is of very important theoretical role and very great applied value to research the novel methods without any penalty function, which are called penalty-free-type methods. The filter methods presented by Fletcher et al (2002) are now a class of very important and representative penalty-free-type ones, which have been well studied by many researchers. In this project, we study a clsss of novel penalty-free-type methods without a penalty function or a filter to solving nonlinear constrained optimization problems and their convergent theory and numerical effect. We study the penalty-free-type methods without a penalty function or a filter to solving nonlinear optimization problems with nonlinear equality constraints, ones with nonlinear inequality constraints and ones with general nonlinear constraints. The line search strategies or the trust region framework are adopted. Some interior point penalty-free-type methods are studied for some special nonlinear constrained optimizaion problems. The global convergence and local convergent rate of those penalty-free-type methods are analyzed and studied. The globally convergent penalty-free-type algorithms for nonlinear constrained optimization have the following general form. At a given iterate x, a trial step d is computed in either the primal or primal-dual space based and/or historical information of the original problems. The trial step is then either accepted or rejected based on the some acceptable criteria which is dependent on reductions attained in the nonlinear objective function f(x) and in the measure of constraint infeasibility. Moreover, the penalty-type methods determine a trial step to be accepted or not according to the reductions attained in the some combination of both measures above. The penalty-free-type methods also need to balance reductions in the objective function with reductions in the constraint infeasibilty. The numerical experiments for these methods will be done by using small and medium-size test problems from a constrained and unconstrained testing enviroment (CUTEr).

约束非线性规划在许多领域都有重要应用，传统的求解方法是借助于某个惩罚函数作为效益函数――这一类方法统称为惩罚型方法，惩罚型方法的计算效果取决于能否选择一个"好"的初始罚参数值，以及在迭代过程中能否有效地自动校正罚参数。因此，研究不使用罚函数的新型方法――无惩罚型方法既有重要的理论意义，也有重大的应用价值。Fletcher 等人提出的滤子方法是目前比较成熟且十分重要的一类无惩罚型方法。本项目研究约束非线性规划问题的无滤子无罚函数的一类新型无惩罚型方法、理论及其数值效果。研究非线性等式约束优化问题、非线性不等式约束优化问题和一般约束优化问题的无惩罚型序列线性规划方法、无惩罚型序列二次规划方法，采用直线搜索技巧和信赖域结构，结合内点法技巧的无惩罚型方法，研究这些无滤子无罚函数方法接受尝试步的准则，分析它们的全局收敛性和局部收敛速度，对约束和无约束优化测试环境 CUTE 中的测试问题进行数值试验。

约束非线性规划问题在经济计划、工程设计、交通运输等许多领域都有十分重要的应用，传统的求解方法都是借助于某个罚函数作为效益函数 ——这类方法我们统称为惩罚型方法，这些惩罚型方法的计算效果取决于能否选择一个“好”的初始罚参数值，以及在迭代过程中能否有效地自动校正罚参数。因此，研究不使用罚函数的新型方法――无惩罚型方法既有重要的理论意义，也有重大的应用价值。Fletcher 等人提出的滤子方法是目前比较重要的一类无惩罚型方法，但过滤法中保留滤子集会增大存储量，同时，当滤子集中含有某个 Pareto 解时，过滤法可能无法继续进行迭代。本项目主要研究成果有：提出一个无惩罚型原始对偶内点算法，通过对尝试点的不可行性控制来确保算法的全局收敛性，在通常假设条件下，分析了算法的全局收敛性；对非线性半定规划问题，采用信赖域结构计算尝试步，通过设定两个目标去判别尝试步是否可以被接受，在一定假设条件下，分析了算法的适定性及全局收敛性；对非线性等式约束优化提出一种柔性惩罚方法，这种新方法采用罚参数与无惩罚型接受准则相结合的思想，既保留了罚函数方法处理不相容性和约束规格不成立的优点，又体现了无惩罚型方法的特点；对一般约束优化问题提出一个无惩罚型方法，在没有约束规格也不假定迭代点列有界的情况下分析了算法的全局收敛性；对非线性半定规划问题提出一个无惩罚型方法，分析了算法的全局收敛性，采用二阶校正技术克服 Maratos 效应，证明了算法的超线性收敛性；对非线性不等式约束半定规划问题提出一种新的逐次线性化方法，新算法或者要求违反约束的度量有足够改善，或者在约束违反度的一个合理范围内要求目标函数值充分下降，在通常假设条件下，分析了算法的适定性及全局收敛性；研究与椭圆型边值问题相关的重排优化问题，得到了与之相关的重排优化问题的最优解的存在性；对税收管理问题和库存价格控制问题给出了一些理论结果。