Material Information
 Title:
 Reformulation and CuttingPlane Approaches for Solving TwoStage Optimization and Network Interdiction Problems
 Physical Description:
 1 online resource (211 p.)
 Language:
 english
 Creator:
 Shen,Siqian
 Publisher:
 University of Florida
 Place of Publication:
 Gainesville, Fla.
 Publication Date:
 2011
Thesis/Dissertation Information
 Degree:
 Doctorate ( Ph.D.)
 Degree Grantor:
 University of Florida
 Degree Disciplines:
 Industrial and Systems Engineering
 Committee Chair:
 Smith, Jonathan
 Committee Members:
 Geunes, Joseph P
Richard, JeanPhilippe Khargonekar, Pramod
Subjects
 Subjects / Keywords:
 algorithms  decomposition  domination  graph  heuristic  management  mathematics  modelling  networks  np  optimization  stochastic
Industrial and Systems Engineering  Dissertations, Academic  UF
 Genre:
 Industrial and Systems Engineering thesis, Ph.D.
bibliography ( marcgt ) theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) borndigital ( sobekcm ) Electronic Thesis or Dissertation
Notes
 Abstract:
 This dissertation investigates models and algorithms for solving a class of twostage optimization problems arising in a variety of practical problems, including network interdiction applications. Traditional decomposition methods for solving largescale mixedinteger programs (MIP), such as Benders decomposition, cannot be utilized for solving the problems that we consider due to the presence of discrete variables in the secondstage problems. In the problems we study, there exist a finite set of firststage feasible solutions and general mixedinteger variables in the second stage. We employ a decomposition strategy and apply the ReformulationLinearization Technique (RLT) to obtain the convex hull of the formulation in terms of the secondstage variables. For some of the problems, we then describe modified Benders cuts for attaining optimality. For the others, we derive valid inequalities based on the subproblem reformulations or suboptimal polynomialtime dynamic programming (DP) solutions. Our goal is to develop novel formulation and solution methodologies for solving several twostage (stochastic) MIPs within practical time limits.
We first consider a class of twostage stochastic optimization problems arising in the protection of vital arcs in a critical path network. A project is completed after a series of dependent tasks are all finished. We analyze a problem in which task finishing times are uncertain, but can be insured a priori to mitigate potential delays. A decision maker must trade off costs incurred in insuring arcs with expected penalties associated with late project completion times, where lateness penalties are assumed to be lower semicontinuous nondecreasing functions of completion time. We provide decomposition strategies to solve this problem with respect to either convex or nonconvex penalty functions. In particular, for the nonconvex penalty case, we employ RLT to make the problem amenable to solution via Benders decomposition. We also consider a chanceconstrained version of this problem, in which the probability of completing a project on time is sufficiently large. We demonstrate the computational efficacy of our approach by testing a set of randomly generated instances, using the Sample Average Approximation method to guide our scenario generation.
Then, we examine variants of the Critical Node Problem (CNP) on speciallystructured graphs, which aim to identify a subset of nodes whose removal will maximally disconnect the graph. These problems lie in the intersection of network interdiction and graph theory research. The two different connectivity metrics that we consider regard the number of maximal connected components (which we attempt to maximize) and the largest component size (which we attempt to minimize). We develop polynomialtime DP algorithms for solving these problems with respect to each networkconnectivity metric on tree structures and on seriesparallel graphs. We also extend our discussion by expanding the core model to account for node deletion costs and weighted nodes, and also by solving the problems on generalizations of tree structures.
Furthermore, we analyze the CNP on general undirected graphs, and consider a third connectivity metric: the minimum cost required to reconnect the graph after the nodes are deleted (which we attempt to maximize). We show that these problem variants are NPhard in general, and formulate each problem as a mixedinteger program. Valid inequalities are studied for the first two connectivity objectives by examining intermediate DP solutions to $k$hole subgraphs (a type of generalizations of tree structures). We randomly generate a set of test instances, on which we demonstrate the computational efficacy of our approaches.
Finally, we consider an optimization problem that integrates network design and broadcast domination decisions. Given an undirected graph, a feasible broadcast domination is a set of nonnegative integer powers $f_i$ assigned to each node $i$, such that for any node $j$ in the graph, there exists some node $k$ having a positive $f_k$value whose shortest distance to node $j$ is no more than $f_k$. The cost of a broadcast domination solution is the sum of all node power values. The network design problem constructs edges that decrease the minimum broadcast domination cost on the graph. The overall problem we consider minimizes the sum of edge construction costs and broadcast domination costs. We show that this problem is NPhard in the strong sense, even on unweighted graphs. A decomposition strategy will iteratively add valid inequalities based on optimal broadcast domination solutions corresponding to the firststage network design solutions. We demonstrate that our decomposition approach is computationally superior to the solution of a monolithic MIP formulation.
 General Note:
 In the series University of Florida Digital Collections.
 General Note:
 Includes vita.
 Bibliography:
 Includes bibliographical references.
 Source of Description:
 Description based on online resource; title from PDF title page.
 Source of Description:
 This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
 Statement of Responsibility:
 by Siqian Shen.
 Thesis:
 Thesis (Ph.D.)University of Florida, 2011.
 Local:
 Adviser: Smith, Jonathan.
 Electronic Access:
 RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ONCAMPUS USE UNTIL 20120229
Record Information
 Source Institution:
 UFRGP
 Rights Management:
 Applicable rights reserved.
 Classification:
 lcc  LD1780 2011
 System ID:
 UFE0043106:00001
