半无限规划和半无限互补问题的基本理论研究
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摘要
半无限规划和互补问题是数学规划的重要研究课题,在工程设计、最优控制、信息技术、经济均衡等领域有着广泛的应用.随着研究的深入,半无限规划和互补问题得到了进一步推广,如带非紧致集的半无限规划、广义半无限规划及半无限互补问题.它们既是原有问题的延续又呈现出新的特征,并且与变分分析、集值分析、扰动分析等理论有着紧密联系.本文运用变分分析的工具和方法对上述问题展开研究,包括一阶最优性条件、极大值函数的微分性质、全局鞍点存在性定理、误差界、解集的弱强极小性质等.
第一章简要介绍半无限规划和互补问题的研究历史与现状,概述本文的研究内容及主要成果,回顾相关概念和基本结论.
第二章研究带有非紧致集的凸半无限极大极小规划.首先通过变分分析的工具和方法刻画凸极大值函数的次方向导数和次微分的解析式,进而建立了凸半无限极大极小规划的一阶最优性条件.
由于广义半无限规划的指标集为一集值映射,这极大增加了其在理论分析和算法设计中的难度.因此在第三章我们感兴趣的是如何将广义半无限规划转化为标准半无限规划,为进一步利用已有的标准半无限规划的算法进行求解提供理论依据.通过对下层规划进行扰动分析,我们分别利用推广的基本二次修正Lagrangian函数和一类精确罚函数实现这一转化.相应的,建立广义半无限规划新的一阶最优性条件.
第四章对一类新的问题-半无限互补问题-的基本理论展开研究,包括解集性质、问题的等价转化、误差界等.不但许多标准互补问题的重要概念得以推广,而且提出了两类新的误差界;即ω-误差界及弱误差界.
半无限规划解集的局部弱强极小性质已利用Lower-C~1函数特殊的微分性质给出了等价描述.在第五章,我们对上述结论进行推广.首先提出一类新的不可微函数一下可微函数.它既包含光滑函数、凸函数、Lower-C~1函数、积分函数等,又同样可以利用切锥和次方向导数或者法锥和次微分等变分分析的工具对解集的局部弱强极小性质进行等价刻画.此外,鉴于变分不等式与互补问题有着密切联系,我们进一步考虑变分不等式解集的弱强极小性质,并建立它与误差界、MPS性质及线性正则性之间的等价关系.最后在解集具有弱强极小性质的条件下,给出算法具有有限步终止的充要条件.
Semi-infinite Programming (SIP) and Complementarity Problems (CP), as important parts of mathematical programming, have wide applications in many fields such as engineering design, optimal control, information technology, and economic equilibrium. Recently, new extended forms of SIP and CP appear, including the semi-infinite programming with noncompact index sets, generalized semi-infinite programming (GSIP), and semi-infinite complementarity problems (SICP). Compared with the SIP and CP, they not only take on new features, but also have close relations to varia-tional analysis, set-valued analysis, and perturbation analysis. This thesis is mainly concerned with these new extended forms of SIP and CP. Topics cover first-order optimality conditions, differentiability properties of sup-type functions, global saddle points, error bounds, and weak sharp minima.
The opening chapter sets forth a short historical review of SIP and CP, briefly summarizes the main results given in this thesis, and presents some of the fundamental concepts and conclusions which are the main tools for our theoretical analysis.
In Chapter 2, based on some new characterizations of subderivative and subdiffer-ential of sup-type functions, we develop the first-order necessary and sufficient optimality conditions for convex semi-infinite min-max programming problems with noncompact index sets.
The main difference between GSIP and SIP is that the index sets in the former depend additionally on the decision variables, instead of just a constant set as required in the latter. This makes the theoretical treatment of GSIP complicated significantly. Therefore, Chapter 3 mainly focuses on how to convert GSIP into SIP. Once this is effected, GSIP can be solved by using any available semi-infinite programming algorithms, which is very significant in both theory and practical applications. By addressing the perturbation analysis of the lower-level programming, we successfully complete this transformation via generalized essentially quadratic augmented Lagrangian functions or a class of exact penalty functions.
Chapter 4 is devoted to the semi-infinite complementarity problems. In particular, we investigate the solvability, feasibility, equivalent reformulation, subdifferentiability properties of residual functions, and error bounds. Not only are the important concepts in standard complementarity problems extended in a natural way, but also two new variants of error bounds,ε-error bounds and weak error bounds, are introduced.
The full characterizations of local weak sharp minima for SIP have been given by virtue of the special structure of the lower-C~1 function. These nice results are extended in Chapter 5. First, we identify a new class of nonsmooth functions, called inf-differentiable functions, which is broad enough to include many important functions, such as smooth functions, convex functions, lower-C1 functions, and integral functions. One of the most significant features of inf-differentiability functions is that the local weak sharp minima could be completely characterized in terms of normal cone and subdifferential, tangent cone and subderivative, or their mixture. In addition, due to the close relationship between CP and variational inequality problems (VIP), we study the weak sharp minima of VIP, and establish its equivalence to global error bound, minimum principle sufficiency (MPS) property, and linear regularity. Finally, under rather mild assumptions, we develop the necessary and sufficient conditions for the finite termination of a class of algorithms for solving convex programming problems and variational inequality problems.
第一章简要介绍半无限规划和互补问题的研究历史与现状,概述本文的研究内容及主要成果,回顾相关概念和基本结论.
第二章研究带有非紧致集的凸半无限极大极小规划.首先通过变分分析的工具和方法刻画凸极大值函数的次方向导数和次微分的解析式,进而建立了凸半无限极大极小规划的一阶最优性条件.
由于广义半无限规划的指标集为一集值映射,这极大增加了其在理论分析和算法设计中的难度.因此在第三章我们感兴趣的是如何将广义半无限规划转化为标准半无限规划,为进一步利用已有的标准半无限规划的算法进行求解提供理论依据.通过对下层规划进行扰动分析,我们分别利用推广的基本二次修正Lagrangian函数和一类精确罚函数实现这一转化.相应的,建立广义半无限规划新的一阶最优性条件.
第四章对一类新的问题-半无限互补问题-的基本理论展开研究,包括解集性质、问题的等价转化、误差界等.不但许多标准互补问题的重要概念得以推广,而且提出了两类新的误差界;即ω-误差界及弱误差界.
半无限规划解集的局部弱强极小性质已利用Lower-C~1函数特殊的微分性质给出了等价描述.在第五章,我们对上述结论进行推广.首先提出一类新的不可微函数一下可微函数.它既包含光滑函数、凸函数、Lower-C~1函数、积分函数等,又同样可以利用切锥和次方向导数或者法锥和次微分等变分分析的工具对解集的局部弱强极小性质进行等价刻画.此外,鉴于变分不等式与互补问题有着密切联系,我们进一步考虑变分不等式解集的弱强极小性质,并建立它与误差界、MPS性质及线性正则性之间的等价关系.最后在解集具有弱强极小性质的条件下,给出算法具有有限步终止的充要条件.
Semi-infinite Programming (SIP) and Complementarity Problems (CP), as important parts of mathematical programming, have wide applications in many fields such as engineering design, optimal control, information technology, and economic equilibrium. Recently, new extended forms of SIP and CP appear, including the semi-infinite programming with noncompact index sets, generalized semi-infinite programming (GSIP), and semi-infinite complementarity problems (SICP). Compared with the SIP and CP, they not only take on new features, but also have close relations to varia-tional analysis, set-valued analysis, and perturbation analysis. This thesis is mainly concerned with these new extended forms of SIP and CP. Topics cover first-order optimality conditions, differentiability properties of sup-type functions, global saddle points, error bounds, and weak sharp minima.
The opening chapter sets forth a short historical review of SIP and CP, briefly summarizes the main results given in this thesis, and presents some of the fundamental concepts and conclusions which are the main tools for our theoretical analysis.
In Chapter 2, based on some new characterizations of subderivative and subdiffer-ential of sup-type functions, we develop the first-order necessary and sufficient optimality conditions for convex semi-infinite min-max programming problems with noncompact index sets.
The main difference between GSIP and SIP is that the index sets in the former depend additionally on the decision variables, instead of just a constant set as required in the latter. This makes the theoretical treatment of GSIP complicated significantly. Therefore, Chapter 3 mainly focuses on how to convert GSIP into SIP. Once this is effected, GSIP can be solved by using any available semi-infinite programming algorithms, which is very significant in both theory and practical applications. By addressing the perturbation analysis of the lower-level programming, we successfully complete this transformation via generalized essentially quadratic augmented Lagrangian functions or a class of exact penalty functions.
Chapter 4 is devoted to the semi-infinite complementarity problems. In particular, we investigate the solvability, feasibility, equivalent reformulation, subdifferentiability properties of residual functions, and error bounds. Not only are the important concepts in standard complementarity problems extended in a natural way, but also two new variants of error bounds,ε-error bounds and weak error bounds, are introduced.
The full characterizations of local weak sharp minima for SIP have been given by virtue of the special structure of the lower-C~1 function. These nice results are extended in Chapter 5. First, we identify a new class of nonsmooth functions, called inf-differentiable functions, which is broad enough to include many important functions, such as smooth functions, convex functions, lower-C1 functions, and integral functions. One of the most significant features of inf-differentiability functions is that the local weak sharp minima could be completely characterized in terms of normal cone and subdifferential, tangent cone and subderivative, or their mixture. In addition, due to the close relationship between CP and variational inequality problems (VIP), we study the weak sharp minima of VIP, and establish its equivalence to global error bound, minimum principle sufficiency (MPS) property, and linear regularity. Finally, under rather mild assumptions, we develop the necessary and sufficient conditions for the finite termination of a class of algorithms for solving convex programming problems and variational inequality problems.
引文
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