Reformulation and Cutting-Plane Approaches for Solving Two-Stage Optimization and Network Interdiction Problems


Material Information

Reformulation and Cutting-Plane Approaches for Solving Two-Stage Optimization and Network Interdiction Problems
Physical Description:
1 online resource (211 p.)
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
Smith, Jonathan
Committee Members:
Geunes, Joseph P
Richard, Jean-Philippe
Khargonekar, Pramod


Subjects / Keywords:
algorithms -- decomposition -- domination -- graph -- heuristic -- management -- mathematics -- modelling -- networks -- np -- optimization -- stochastic
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation


This dissertation investigates models and algorithms for solving a class of two-stage optimization problems arising in a variety of practical problems, including network interdiction applications. Traditional decomposition methods for solving large-scale mixed-integer 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 second-stage problems. In the problems we study, there exist a finite set of first-stage feasible solutions and general mixed-integer variables in the second stage. We employ a decomposition strategy and apply the Reformulation-Linearization Technique (RLT) to obtain the convex hull of the formulation in terms of the second-stage 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 sub-optimal polynomial-time dynamic programming (DP) solutions. Our goal is to develop novel formulation and solution methodologies for solving several two-stage (stochastic) MIPs within practical time limits. We first consider a class of two-stage 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 semi-continuous 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 chance-constrained 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 specially-structured 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 polynomial-time DP algorithms for solving these problems with respect to each network-connectivity metric on tree structures and on series-parallel 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 NP-hard in general, and formulate each problem as a mixed-integer 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 NP-hard 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 first-stage 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.
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 (Ph.D.)--University of Florida, 2011.
Adviser: Smith, Jonathan.
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lcc - LD1780 2011
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