| 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.
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