A Backward Sampling Framework for Interdiction Problems with Fortification

This paper examines a class of three-stage sequential defender-attacker-defender problems. In these problems the defender first selects a subset of assets to protect, the attacker next damages a subset of unprotected assets in the “interdiction” stage, after which the defender optimizes a “recourse” problem over the surviving assets. These problems are notoriously difficult to optimize, and almost always require the recourse problem to be a convex optimization problem. Our contribution is a new approach to solving defender-attacker-defender problems. We require all variables in the first two stages to be binary-valued, but allow the recourse problem to take any form. The proposed framework focuses on solving the interdiction problem by restricting the defender to select a recourse decision from a sample of feasible vectors. The algorithm then iteratively refines the sample to force finite convergence to an optimal solution. We demonstrate that our algorithm not only solves interdiction problems involving ...

[1]  Hanif D. Sherali,et al.  Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming , 2006, Math. Program..

[2]  Gerald G. Brown,et al.  Interdicting a Nuclear-Weapons Project , 2009, Oper. Res..

[3]  David P. Morton,et al.  Models for nuclear smuggling interdiction , 2007 .

[4]  Richard L. Church,et al.  Production , Manufacturing and Logistics An exact solution approach for the interdiction median problem with fortification , 2008 .

[5]  Churlzu Lim,et al.  Algorithms for Network Interdiction and Fortification Games , 2008 .

[6]  Maria Paola Scaparra,et al.  Optimal Allocation of Protective Resources in Shortest-Path Networks , 2011, Transp. Sci..

[7]  R. Kevin Wood,et al.  Deterministic network interdiction , 1993 .

[8]  Jørgen Tind,et al.  L-shaped decomposition of two-stage stochastic programs with integer recourse , 1998, Math. Program..

[9]  L. N. Vicente,et al.  Discrete linear bilevel programming problem , 1996 .

[10]  J. C. Smith,et al.  A three‐stage procurement optimization problem under uncertainty , 2013 .

[11]  Richard D. Wollmer,et al.  Removing Arcs from a Network , 1964 .

[12]  Alan Washburn,et al.  Two-Person Zero-Sum Games for Network Interdiction , 1995, Oper. Res..

[13]  J. Hooker,et al.  Logic-based Benders decomposition , 2003 .

[14]  John M. Wilson,et al.  The capacitated lot sizing problem: a review of models and algorithms , 2003 .

[15]  J. C. Smith,et al.  Algorithms for discrete and continuous multicommodity flow network interdiction problems , 2007 .

[16]  Richard L. Church,et al.  Identifying Critical Infrastructure: The Median and Covering Facility Interdiction Problems , 2004 .

[17]  Jonathan Cole Smith,et al.  A class of algorithms for mixed-integer bilevel min–max optimization , 2015, Journal of Global Optimization.

[18]  Delbert Ray Fulkerson,et al.  Maximizing the minimum source-sink path subject to a budget constraint , 1977, Math. Program..

[19]  G. D. Eppen,et al.  Solving Multi-Item Capacitated Lot-Sizing Problems Using Variable Redefinition , 1987, Oper. Res..

[20]  R. Kevin Wood,et al.  Shortest‐path network interdiction , 2002, Networks.

[21]  Uday S. Karmarkar,et al.  The Dynamic Lot-Sizing Problem with Startup and Reservation Costs , 1987, Oper. Res..

[22]  Laurence A. Wolsey,et al.  bc -- prod: A Specialized Branch-and-Cut System for Lot-Sizing Problems , 2000 .

[23]  Stephan Dempe,et al.  Foundations of Bilevel Programming , 2002 .

[24]  B. Golden A problem in network interdiction , 1978 .

[25]  Jacques F. Benders,et al.  Partitioning procedures for solving mixed-variables programming problems , 2005, Comput. Manag. Sci..

[26]  David P. Morton,et al.  Stochastic Network Interdiction , 1998, Oper. Res..

[27]  Halil Bayrak,et al.  Shortest path network interdiction with asymmetric information , 2008, Networks.

[28]  Johannes O. Royset,et al.  Solving the Bi-Objective Maximum-Flow Network-Interdiction Problem , 2007, INFORMS J. Comput..

[29]  Gerald G. Brown,et al.  A Two-Sided Optimization for Theater Ballistic Missile Defense , 2005, Oper. Res..

[30]  Matteo Fischetti,et al.  Combinatorial Benders' Cuts for Mixed-Integer Linear Programming , 2006, Oper. Res..

[31]  Matthias Ehrgott,et al.  A comparison of solution strategies for biobjective shortest path problems , 2009, Comput. Oper. Res..

[32]  David L. Woodruff,et al.  Heuristics for Multi-Stage Interdiction of Stochastic Networks , 2005, J. Heuristics.

[33]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[34]  Yuyue Song,et al.  Production , Manufacturing and Logistics Single item lot-sizing problems with backlogging on a single machine at a finite production rate , 2004 .

[35]  J. Salmeron,et al.  Analysis of electric grid security under terrorist threat , 2004, IEEE Transactions on Power Systems.

[36]  David L. Woodruff,et al.  A decomposition algorithm applied to planning the interdiction of stochastic networks , 2005 .

[37]  Richard L. Church,et al.  A bilevel mixed-integer program for critical infrastructure protection planning , 2008, Comput. Oper. Res..

[38]  J. K. Lenstra,et al.  Deterministic Production Planning: Algorithms and Complexity , 1980 .

[39]  Graham K. Rand,et al.  Decision Systems for Inventory Management and Production Planning , 1979 .

[40]  Jonathan Cole Smith,et al.  Survivable network design under optimal and heuristic interdiction scenarios , 2007, J. Glob. Optim..

[41]  Richard L. Church,et al.  Protecting Critical Assets: The r-interdiction median problem with fortification , 2007 .

[42]  J. Salmeron,et al.  Worst-Case Interdiction Analysis of Large-Scale Electric Power Grids , 2009, IEEE Transactions on Power Systems.

[43]  Gerald G. Brown,et al.  Defending Critical Infrastructure , 2006, Interfaces.

[44]  Gerald G. Brown,et al.  Analyzing the Vulnerability of Critical Infrastructure to Attack and Planning Defenses , 2005 .

[45]  J. Cole Smith,et al.  Basic Interdiction Models , 2010 .

[46]  Jonathan F. Bard,et al.  The Mixed Integer Linear Bilevel Programming Problem , 1990, Oper. Res..

[47]  David P. Morton,et al.  A Stochastic Program for Interdicting Smuggled Nuclear Material , 2003 .

[48]  J. Bard,et al.  An algorithm for the discrete bilevel programming problem , 1992 .

[49]  G. Bitran,et al.  Computational Complexity of the Capacitated Lot Size Problem , 1982 .

[50]  Leonardo Lozano,et al.  On an exact method for the constrained shortest path problem , 2013, Comput. Oper. Res..