【作者】 何素艳；

【导师】 李兴斯； 张洪武；

【作者基本信息】 大连理工大学 ， 工程力学， 2003， 博士

【摘要】 互补问题的最显著特征是含有互补性条件，即要求两组非负变量对应分量的乘积为零。30多年以来，互补问题已经发展成为许多领域非常重要的数学工具，在数学规划、经济、工程及其它学科具有非常广泛的应用。在经济领域中的应用主要有Walrasian平衡、空间价格平衡和对策论模型等；在工程中的应用有接触力学问题、断裂力学问题、弹塑性问题、障碍和自由边界问题、流体弹性动态润滑问题、最优控制问题及交通平衡问题等。也就是说，上述这些问题都可以模型化为互补问题，从而最终归结为互补问题的求解。可以看出，在互补问题众多的应用中，经济和力学是互补问题应用的两个最大的领域。 本论文的主要工作是互补问题的算法研究及其在力学中的应用，其主要动机为： 虽然互补问题已经有许多算法，但一些力学实际问题往往满足不了算法的限定条件，因此需对某些算法做一些适应性改进。与此同时，也有必要针对力学问题进行一些新算法的研究和探讨； 近年来，互补问题的力学应用远远落后于算法发展，把一些新的算法介绍到力学问题求解，既有助于丰富力学问题的算法工具，也有助于扩充互补问题的应用背景。 在互补问题的算法研究方面，主要针对近年来发展起来的两类重要方法：方程组方法和内点方法，希望为力学问题求解提供一些可行的算法，特别希望能提供一些可求解大型与非线性互补问题的算法。 第二章的四个算法均为方程组类方法，包括两个光滑牛顿型算法和两个光滑迭代算法。前者借助NCP函数把互补问题转化为等价的非光滑(不可微)方程组，再用带参数的光滑(连续可微)方程组近似这些非光滑方程组，最后用牛顿型方法求解所得到的光滑方程组，希望通过光滑参数趋于零得到原来互补问题的解；后者基于等价不动点格式，构造了一个光滑迭代算法和一个具有有限终止性质的算法，虽然这种迭代算法仅有线性收敛速度，但由于其格式简单、存储量小、保稀疏性、非常易于计算机实现等特点，故较适用于求解大规模稀疏问题。 第三章给出了两个新的内点算法。本论文从两个不同角度对原-对偶内点方法通常使用的摄动方程组进行了变化，并据此建立了两个不同的内点算法。首先通过对中心化方程xs=μe实施代数等价变换，得到了新的不同的摄动方程组。我们发现，通过幂变换得到的摄动方程组，可给出彭积明等人提出的大步内点算法的牛顿方程，但代数等价变换的思想要比彭等人算法的思想容易理解得多。在此基础上，我们建立了一个基于幂大连理工大学博士学位论文变换的内点算法。第二个算法利用极大极小(min一Inax)函数本身所具有的“均化”作用，定义了一个新的邻近度量函数，并以其最优性条件代替中心化方程。这样，在摄动方程本身建立了一种自调节机制，从而使牛顿方向能够根据上次迭代点的信息在各个互补对之间做出自适应的调整。基于改造后的摄动方程组，建立了一个具有自调节功能的内点算法。 第四章为弹性摩擦接触问题的数值求解及其算法应用。利用参变量变分原理和参数二次规划法，离散后的接触问题可化为线性互补问题，之后，将第二、三章提出的互补问题算法应用到接触问题的数值求解中。 在第五章，我们采用正交各向异性摩擦定律模型对三维弹塑性摩擦接触问题进行分析，将正交各向异性的摩擦定律线性化后，将参变量变分原理和参数二次规划法应用到这一复杂的力学问题中来，最终将问题化为线性互补问题模型。经验证，本论文提出的算法对该互补模型仍然适用。另外，在对接触实际例题的分析中，还注意比较了下面两种情况:正交各向异性摩擦与各向同性摩擦的结果比较;正交各向异性弹性摩擦接触与弹塑性摩擦接触的结果比较。更多还原

【Abstract】 The distinguishing feature of a complementarity problem is the set of complementarity conditions, which require that the product of two nonnegative quantities should be zero. Over more than thirty years, this class of problems has become increasingly popular in many fields, such as mathematical programming, economics and engineering. Applications from the field of economics include general Walrasian equilibrium, spatial price equilibria, and game-theoretic models. In engineering, complementarity problems arise in contact mechanics, fracture mechanics, elasto-plastic problems, obstacle and free boundary problems, hydrodynamic lubrication, optimal control problems and traffic equilibrium problems. That is, all the above practical problems can be formulated as complementarity problems and solved by proper algorithms. Economics and mechanics are the two major areas among the numerous applications of complementarity problemsThe present dissertation is mainly devoted to studying algorithms of complementarity problems and their applications in mechanics. The motivation of this thesis is based the following considerations:◇ Practical mechanics problems may not satisfy certain restrictive conditions for some existing algorithms for complementarity problems, though they might be very efficient for some mathematical problems. Therefore, on one hand, it is necessary to make a few modifications to some existing algorithms so that they are suitable for solving mechanics problems. On the other hand, we need to develop algorithms directly oriented to characteristics of mechanics problems.◇ Applications of modern algorithms in mechanics dropped far behind the algorithmic developments of complementarity problems. We wish the introduction of some new algorithms to solving mechanics problems should play the role of both enriching the solution tools of mechanics and extending the ranges of complimentarity problems, whereby arousing the further interest of researchers from the two fields.In the study of complementarity algorithms, we restrict ourselves to two major classes of them: equation system based algorithms and interior-point algorithms. We wish to develop effective algorithms for practical problems in mechanics, in particularkeeping large-scale and nonlinear complementarity problems in mind.Two smoothing Newton-type algorithms and two smoothing iterative algorithms are given in Chapter 2. In the first two algorithms, the complementarity problem is reformulated to non-smooth (non-differentiable) equations using so-called NCR-functions, and then solved by applying Newton-type methods to the smoothened equations. The third algorithm is a smooth iterative method based on an equivalent fixed-point format of the complementarity problem. The forth algorithm is the same as the third one with an addition of finite termination criteria. Although the latter two algorithms have only linear rate of convergence, they are especially suitable for large-scale and sparse problems, with features of simple formula, small storage, sparsrty preservation and easy implementation.Two improved interior point algorithms are proposed in Chapter 3. They are designed based on the modifications to the standard perturbed system of primal-dual interior-point algorithms from two different angles. The first is based on the algebraically equivalent transformation to the standard centering equation xs = e. We discovered that the Newton equations used by Peng Jiming et al. in their long-step interior point algorithms could be derived by a power transformed perturbed system. Our approach of algebraically equivalent transformation is much simpler than the one proposed by Peng et al. in their algorithms. Inspired by this observation, an interior point algorithm based on power transformation is developed. Another interior algorithm is to employ the "homogenizing" effect of min-max function, where the standard centering equation is replaced by the optimality condition of a new proximity measure function. A self-adjusting mechanism is added to the new pert更多还原