【作者】 屈彪；

【导师】 王长钰；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2002， 博士

【摘要】 本文主要是对非线性最优化问题中的若干重要算法的理论分析作了探讨，包括约束最优化问题中的梯度投影方法以及求解变分不等式和互补问题的几类算法，主要是集中在这几种算法的收敛性分析上。 第一章是绪论部分，简单介绍了变分不等式与互补问题，本文所研究的内容，以及本文的主要工作。 第二章研究了通过广义D-间隙函数求解变分不等式问题的一种方法的收敛性和误差界估计。我们知道，通过广义D-间隙函数，可以将变分不等式问题转化为一个无约束极小化问题。近来，Peng和Fukushima提出了一种混合Newton类型的方法来极小化一个特殊的广义D-间隙函数，在本章中，我们将这种方法用来极小化一般形式的广义D-间隙函数g_αβ。我们证明了算法具有更强的收敛性质。在合适的条件下，证明了算法的全局收敛性和局部二次收敛性。更进一步，当广义D-间隙函数g_αβ中的参数β取值于某一区间时，证明了函数g_αβ对于强单调变分不等式而言，具有有界的水平集，同时，给出了算法的一个误差界估计，它部分回答了Yamashita等人提出的一个问题。 第三章对求解互补问题的阻尼Gauss-Newton方法作了研究，这种方法最早是由Subramanian提出的。本章主要研究了阻尼Gauss-Newton方法的全局收敛性，在较弱的条件下，获得了一个更强的全局收敛结果，该结果推广了相应文献中算法的全局收敛性结果。同时，我们给出了一种不需要线性搜索的新的步长选择方法，研究了在此步长规则下的阻尼Gauss-Newton方法，得到了一个更好的全局收敛结果。 第四章对求解互补问题的可微的无约束优化法作了研究。在将互补问题转化为一个无约束优化问题的基础上，给出了一种求解互补问题的混合方法，证明了该算法的全局收敛性。 第五章研究了求解约束最优化问题的梯度投影方法，在步长的选取时采用了一种新的策略，这种策略不需要进行传统的线搜索且包含步长取常数这种特例，在较弱的条件下，证明了梯度投影方法的全局收敛性。进一步给出了目标函数f（x）是凸函数和拟凸函数时的更强的收敛性结果。第六章是结论部分，主要对本论文的内容作了简单的概括和分析。更多还原

【Abstract】 Some important algorithms for nonlinear optimization problems are studied in this dissertation. The aim of this dissertation is to give some theoretical analysis.Chapter 1 is the introduction of this dissertation, which introduces the variational inequality problem and the complementarity problem, the context of this dissertation and the main results obtained in this dissertation.A hybrid Newton-type method for solving the variational inequality problem (VIP) is studied in Chapter 2. It is known that the VIP can be reformulated as an unconstrained minimization problem through the D-gap function. Recently, a hybrid Newton-type method was proposed by Peng and Fukushima (1999) for minimizing the D-gap function. In this chapter, the hybrid Newton-type method is extended to minimize a generalized D-gapfunction gαβ in the general form. It is shown that the algorithm has nice convergenceproperties. Under some reasonable conditions, it is proved that the algorithm is locally and globally convergent. Moreover, when the parameter β is chosen in a certain interval, it isproved that the generalized D-gap function gαβ has bounded level sets for the stronglymonotone VIP. An error bound estimation of the algorithm is obtained, which partially gives an answer to the question raised by Yamashita (1997) et al.The authors study the convergence properties of the damped Gauss-Newton algorithm which was originally proposed by Subramanian for the complementarity problem in Chapter 4. The aim of this chapter is to give a global convergence result under weaker conditions. The results here improve and generalize those in the literature. In addition, a new step-size rule, which need not a line search, is presented for the damped Gauss-Newton algorithm. A nice global convergence result is obtained.In Chapter 4, a new algorithm for the solution of nonlinear complementarity problems is developed. The algorithm is based on a reformulation of the complementarity problem as an unconstrained optimization. It is proved that the algorithm is globally convergent.In Chapter 5, the authors study the convergence properties of the gradient projection method for the constrained optimization problem. In this chapter, a new step-size rule, which avoids fulfiling the classical line search and includes choosing a constant as the step size as a special case, is presented and analyzed. Some convergence results of the gradient projectioninmethod under milder conditions are obtained.Finally we conclude the dissertation with some remarks in Chapter 6.更多还原