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Title: Some inverse optimization problems and network improvement problems
Other Titles: Yi xie ni you hua wen ti he wang lu gai jin wen ti
Authors: Guan, Xiucui (關秀翠)
Department: Dept. of Mathematics
Degree: Doctor of Philosophy
Issue Date: 2005
Publisher: City University of Hong Kong
Subjects: Mathematical optimization
Trees (Graph theory)
Notes: 137 leaves : ill. ; 30 cm.
CityU Call Number: QA164.G83 2005
Includes bibliographical references (leaves 124-135) and index.
Thesis (Ph.D.)--City University of Hong Kong, 2005
Type: Thesis
Abstract: Knowledge of inverse optimization problems and network improvement problems has become an important aspect in the field of optimization during the past decades. Both of them are to modify the model parameters and can be formulated as optimization problems. The difference is that the inverse optimization problem is to minimize the cost incurred by modifications of the parameters to make given solutions optimal, whereas the network improvement problem is to enhance the performance of a given network by modifying a part of the network. We first focus on a class of Inverse Constrained Bottleneck Problems (ICBP). The constrained bottleneck problem is to find a solution that minimizes the bottleneck measurement subject to a bound constraint on the weight-sum function. On the other hand, in its inverse problem (ICBP), a candidate solution is given, and we wish to modify the weights under bound restrictions so that the given solution becomes an optimal one to the constrained bottleneck problem and the deviation of the weights from their original values, measured by the weighted l∞ norm or weighted l1 norm, is minimum. When the modifications of the two weight vectors are proportional, we present a general method under weighted l∞ norm by using a bottleneck cut problem as a subroutine. Giving a general method in this case under weighted l1 norm is difficult, but when the modifications of the two weight vectors are independent, we propose a general method for the problem (ICBP) under weighted l1 norm. The inverse minimum spanning tree (MST) problem is one of the most popular topics in inverse optimization problems. Almost all existing inverse MST models are edge based ones. However, in this thesis, we propose a class of inverse minimum spanning tree problems under a node-edge upgrade strategy, in which upgrading a node reduces the weights of the edges incident to the node. Two models are considered depending on whether the upgrade costs of all nodes are equal or not. Both of them can be solved in polynomial time by finding a minimum weight node cover in a forest. We further discuss node based bottleneck improvement problems in which our aim is to upgrade a subset of nodes such that the bottleneck weight of a family of subsets of edges is minimum under the constraint that the overall upgrade cost does not exceed a given value. Most of these problems are NP-hard. Firstly, we consider the problem for a special tree-path system (V,E,F), in which T = (V,E) is a rooted tree and F is a set of paths from the root to the leaves. In this system, we present an O(nlogn) algorithm based on the dynamic programming method, where n = |V|. We also consider some variations of this problem and present complexity results and heuristic algorithms. Secondly, we extend the system to a complicated directed tree T with multiple sources and multiple terminals. It is, in fact, a network improvement problem for multicuts. We first propose a linear time algorithm for the case of only two critical nodes, then extend the algorithm to the general case. The main idea of these two algorithms is to obtain the optimal upgrading set based on the upgrading sets for the subtrees rooted at the nodes in a minimum cardinality node multicut of T. Such an algorithm is polynomial when the numbers of sources and terminals are bounded by a given number.
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