Reformulation and Sampling to Solve a Stochastic Network Interdiction Problem

Transcription

1 Network Interdiction Stochastic Network Interdiction and to Solve a Stochastic Network Interdiction Problem Udom Janjarassuk Jeff Linderoth ISE Department COR@L Lab Lehigh University jtl3@lehigh.edu informs Annual Meeting San Francisco, CA November 15, 2005

2 Network Interdiction Stochastic Network Interdiction Network Interdiction Problem Elements Capacitated network, good guys, bad guys Good Guy:

3 Network Interdiction Stochastic Network Interdiction Network Interdiction Problem Elements Capacitated network, good guys, bad guys Bad Guy:

4 Network Interdiction Stochastic Network Interdiction Network Interdiction Bad Guy Says: I want to flow as much as possible from r to t r t Drugs, Enemy Supplies, Nuclear Material

5 Network Interdiction Stochastic Network Interdiction Network Interdiction r t Good Guy Says Not So Fast My Friend

6 Network Interdiction Stochastic Network Interdiction Mathematical Formulation Interdiction is a binary decision: { 1 if interdiction occurs on arc (i, j) A, x ij = 0 otherwise. f(x): maximum flow in network if I intervene on arcs x Budget Constraint: X = x {0, 1} A h ij x ij K Network Interdiction Problem min f(x) x X (i,j) A

7 Network Interdiction Stochastic Network Interdiction Stochastic Network Interdiction r t Jeff Is An Idiot Jeff s interdictions are not always successful

8 Network Interdiction Stochastic Network Interdiction Stochastic Network Interdiction S: Set of scenarios ξ ijs : Bernoulli random variable if interdiction on arc (i, j) would be successful in scenario s f s (x): maximum flow if Dudley intervenes on arcs x and scenario s S occurs Stochastic Network Interdiction Problem () min Ef s(x) = min p s f s (x) x X x X s S

9 Duality Linearization Max Flow Formulate Max Flow f s (x) as an LP: A = A {(t, r)} Primal f s (x) = max y R A + y tr y ij u ij (1 ξ ijs x ij ) (i, j) A Ny = 0 Dual min u ij (1 ξ ijs x ij )ρ ij (i,j) A π r π t 1 ρ ij π i + π j 0 (i, j) A ρ ij 0 (i, j) A

10 Duality Linearization Strong Duality For fixed ^x, if Dudley can find (primal) feasible y and (dual) feasible (π, ρ ) such that y tr = u ij (1 ξ ijs^x ij )ρ ij (1) (i,j) A then max flow f s (^x) = y tr Formulation Idea 1 Duplicate (primal and dual) flow variables (y, π, ρ) for each scenario 2 Enforce primal feasibility, dual feasibility, and equality (1) for each scenario

11 Duality Linearization A min min Formulation subject to y trs (i,j) A min s S p s y trs u ij (1 ξ ijs x ij )ρ ijs = 0 s S (i,j) A h ij x ij K y ijs u ij (1 ξ ijs x ij ) 0 (i, j) A, s S Ny s = 0 s S π rs π ts 1 s S ρ ijs π is + π js 0 (i, j) A, s S (x, y, π, ρ) B A R A S + R N S R A S +

12 Duality Linearization Good and Bad Some nice things about the formulation It s a pure minimization problem There are not too many integer variables: (x B A ) If x is fixed, it is decomposible by scenario Integer variables appear only in the first stage! A BAD thing about the formulation y trs u ij (1 ξ ijs x ij )ρ ijs = 0 (i,j) A x ij ρ ijs are nonlinear terms

13 Duality Linearization Linearization Trick Introduce auxiliary variables z ijs Let M be an upper bound on ρ ijs Then z ijs = x ij ρ ijs if and only if 1 z ijs Mx ij 2 z ijs ρ ijs 3 z ijs ρ ijs + M(x ij 1) Lemma In any optimal solution to the dual max flow problem, ρ ijs 1, (i, j) A.

14 Duality Linearization : MILP Formulation min s S p s y trs subject to y trs (i,j) A u ij ρ ijs + (i,j) A u ij ξ ijs z ijs = 0 s S z ijs x ij 0 (i, j) A, s S z ijs ρ ijs 0 (i, j) A, s S ρ ijs z ijs + x ij 1 (i, j) A, s S Primal Feasibility Dual Feasibility (x, y, π, ρ, z) B A R A S + R N S R A S + R A S +

15 Good News and Bad News Problem is just an IP But it s big Name K N A B S 4x x x x x x10 has a mere XX variables and XX constraints

16 Monte Carlo Methods min {f(x) E Pg(x; ξ) g(x; ξ)dp(ξ)} x S Ω Draw ξ 1, ξ 2,... ξ N from P Sample Average Approximation: f N (x) N 1 N j=1 g(x, ξ j ) f N (x) is an unbiased estimator of f(x) (E[ f N (x)] = f(x)). We instead minimize the Sample Average Approximation: min { f N (x)} x S

17 Lower Bound on the Optimal Objective Function Value v = min x S {f(x)} Thm: ^v N = min x S { f N (x)} E[^v N ] v The expected optimal solution value for a sampled problem of size N is the optimal solution value.

18 Estimating E[^v N ] Generate M independent SAA problems of size N. Solve each to get v j N M L N,M 1 M j=1 v j N The estimate L N,M is an unbiased estimate of E[ v N ]. M [LN,M E( v N )] N (0, σ 2 L ) σ 2 L Var( v N) This variance depends on the sample!

19 Confidence Interval s 2 L (M) 1 M 1 M j=1 ( ) 2 v j N L N,M [ L N,M z αs L (M), L N,M + z ] αs L (M) M M These only apply if the v j N are i.i.d. random variables.

20 Upper Bounds f(^x) v ^x S Generate T independent batches of samples of size N E f j N(x) := N 1 N i=1 g(x, ξ i,j ) = f(x), for all x X. T U N,T (^x) := T 1 f j N(^x) j=1

21 More Confidence Intervals T [U N,T (^x) f(^x)] N(0, σ 2 U (^x)), as T, σ 2 U (^x) Var [ f N(^x) ] Estimate σ 2 U (^x) by the sample variance estimator s2 U (^x, T) s 2 1 T ] 2 U (^x, T) [ f T 1 j N (^x) U N,T (^x). j=1 [ U N,T (^x) z αs U (^x; T), U N,T (^x) + z ] αs U (^x; T) T T

22 Solution Times Table of sizes Table of CPLEX times

23 Use a Approach ATR: A parallel solver for two-stage stochastic linear programs, engineered to run a collection of (Condor provided) non-dedicated CPUs Uses MW: Master-Worker framework for parallelization Master: Solves master problem (to determine ^x) Workers: Evaluate f s (^x s S by solving (indepedent) linear programs Since f s (x) is still convex in x, the same cutting plane-based decomposition approach works to solve the problem, but we need just solve the master problem as an integer program For many small instances, the solution of the linear relaxation comes out to be nearly integer

24 1 Solve LP Relaxation of (using ATR) 2 Round (or keep track of relaxed iterations) to get an UB 3 Remove all (most) inactive optimality cuts 4 Solve IP using ATR

25 Computational Details Run an very small configuration of < 50 workstations in lab and Grid lab at Lehigh Run each instance M = 10 times Upper Bound: N =? Did not use any variance reduction techniques in the sampling, which can greatly speed the convergence rate!

26 10x10 K = x10 Example with budget size 10 Upper Bound Lower Bound Value N

27 10x10 K = x10 Example with budget size 15 Upper Bound Lower Bound Value N

28 10x10 K = x10 Example with budget size 20 Upper Bound Lower Bound Value N

29 CPU Times: 10x10, K = 10 Sample size Master (avg) Stdev Wall Clk Stdev

30 Solution Times: 10x10 K = 15 Sample size Master avg Stdev Wall Clk Stdev

31 Solution Times: 10x10 K = 20 Sample size Master avrg Stdev Wall Clk Stdev

32 Conclusions Jeff didn t finish Continuing work: Solve bigger instances! Parallelize (grid-ify) IP master solve Thanks to XPRESS-SP Instances available Acknowledge grants