The Research Of The Projection Operator Onto The Intersection Of A Closed Half Space And A Variable Box And Its Applications

Properties of the metric operators play an essential role in designing algorithms andcarrying out sensitivity analysis of optimization problems, complementarity problems andvariational inequality problems, for example, properties of the metric operators are involvedin the application and convergence analysis of the augmented Lagrangian method, and thesensitivity analysis of optimization problems. During last few years, the study on theoperators onto closed convex sets has attracted many researchers’ interest and the fruit resultson this study also have been achieved. Since the operator onto the intersection of a closedhalf-space and a variable box has many practical applications, based on the known results onthe operator onto the box constraint and the intersection of a hyperplane and a box set, thisthesis focuses on studying some properties of the projection operator onto the intersection of aclosed half-space and a variable box, including its computational formulae and its directionalderivative. The main contents of this thesis can be summarized as the following:The first chapter introduces the background and significance of this thesis.The second chapter presents some preliminary results on convex analysis neededthroughout this thesis.The third chapter studies the methods for finding the projection onto the intersection of aclosed half-space and a variable box. We present an algorithm for finding the explicitformulas of the metric projection based on a parametric approach. Our algorithm terminateswithin linearly finite step.The fourth chapter mainly focuses on the metric projectors onto five classes of convexpolyhedral sets.The fifth chapter first characterizes the critical cone of the projection operator onto theintersection of a closed half-space and a variable box. And then, by using the results on thecritical cones and the obtained results in the fourth chapter, we derive the directionalderivative of the projection operator.The sixth chapter, we report some computational experiments on randomly generated test problems. The numerical results show the efficiency of the algorithm. Finally, we makeconclusions and discussions.