Stochastic Network Interdiction / October 2001

Transcription

1 Calhoun: The NPS Institutional Archive Faculty and Researcher Publications Faculty and Researcher Publications Stochastic Networ Interdiction / October 2001 Sanchez, Susan M. Monterey, California. Naval Postgraduate School

2 Stochastic Networ Interdiction Susan Sanchez Kevin Wood Operations Research Department Naval Postgraduate School Oct. 2001

3 Purpose of this tal Describe a new, simple solution method for two-stage stochastic integer programs Motivate and illustrate with a particular stochastic, networ-interdiction problem (will consider one other, too) Illustrate two main thrusts of my research, interdiction and SP. 2

4 First: Other wor Interdiction of communications networs: Physical and cyber-attacs General models and solutions for system interdiction and defense General theoretical wor on SPs Applications of SP: Sealift deployments subject to bio-attacs Integer programming 3

5 Generic networ interdiction problem Using limited resources, attac an adversary s networ so as to minimize the functionality of that networ (to the adversary). Networs: Road, pipeline, comm Functionality: Max flow, shortest path, convoy movement, path existence Attacs: Aerial sorties, cruise missiles, special operations Can generalize: system interdiction 4

6 Max-flow interdiction Basic Deterministic Model on G=(N,A) with artificial arc a = (t,s) z * = min max x X y y a s.t. y y = 0 i N FS() i RS() i 0 y u (1 x ) A a A { } where X = x {0,1} Rx r, and 5

7 A Simple Example Suppose we have enough resource to interduc any two arcs u on arcs s t 2 3 6

8 Max-flow interdiction Converts to a MIP (well, IP actually) min min x X π, θ A a u θ 1 if = a s.t. πi π j + x + θ 0 = ( i, j) A a θ 0 A a s t 1 3 7

9 Interdiction under uncertainty Uncertain success or data, SMFI: ( xi ) min Eh, x X ( xi ) where h, max y y 0 a s.t. y y = 0 i N FS i () RS() i 0 y u (1 I x ) A a where I = 1 if interdiction of is successful 0otherwise 8

10 Alternative formulation ( xi ) min Eg, x X ( xi ) where g, max y x I y y 0 a A a s.t. y y = 0 i N FSi () RSi () 0 y u A a Note: Deterministic problems are NPcomplete. It s #P-complete to evaluate Eh or Eg for fixed x: These stochastic problems are really hard. 9

11 SMFI: An instance of a 2SSP ( ) ( x ξ ) min Eg, s.t. D where g x, ξ cx + min fy x D Ax = b 0 x u and integer y 0 s.t. D y = d + B x y 0 (some may be integer) and where ξ vec( f, D, d, B ) D g x, ξ represents the dual of our g( xi, ) ( ) 10

12 Probability of ill Assume p = E[Ĩ ] is nown Weaponeers now this stuff! Well 11

13 Bound on z*, pessimistic New soln methodology needs bounds From Jensen s inequality, obtain a global upper bound given a good ˆx : z* Eh, for any xˆ X ( xi ˆ ) h ˆ, E[ ] ( z ) ( x I ) = max ya s.t. Flow balance in y 0 y u (1 E[ I ] xˆ ) A a Can also use probabilistic bounds 12

14 Bound on z*, pessimistic Actually, because interdictions are binary, and successes/failures are binary in SMFI, we can reformulate the upper-bound problem and minimize that bound via a MIP. * z min u θ + u (1 E[ I ]) x x X, π, θ A a s.t. ( ) 1if = (, t s) πi π j + x + θ 0 = ( i, j) A a θ 0 A a 13

15 Bound on z*, optimistic A lower bounding MIP: ( xi ) z* min Eg, = x X ( ) min g x, E[ I] because g is convex in I x X min x X, π, θ A a u θ 1if = (, t s) s.t. πi π j + EI [ ] x + θ 0 = ( i, j) A a θ 0 A a 14

16 Bounds on z*: Comments Bounds can be improved by expanding in terms of conditional probabilities, e.g., by conditioning on the number of successful interdictions. Can use probabilistic bounds; may be necessary for other 2SSPs. But, eep it simple for now. 15

17 Solution methodology, outline BOUND: Find a global upper bnd ENUMERATE all solns ˆx s.t. call these candidates xˆ X z z* z ( xˆ ) z* z ; SCREEN the candidates (Monte Carlo and statistics) to identify the best, or * the best few x ˆ X X * TEST x ˆ X to determine quality (Or maybe Partially Enumerate, Then Screen: PETS. Or, maybe Bound, Enumerate, Then Screen: BETS. ) 16

18 Fundamental theorem for PE Theorem 1: ˆx can be optimal for SMFI only if g ( xˆ, E[ I ]) z. Proof: Obvious. QED Theorem leads to finding a set of candidate solutions xˆ X using the algorithm on the next slide. For simplicity, assume that the set of feasible interdiction plans defined by X has cardinality constraint: A x = R 17

19 Alg. to find candidate solutions 1. X ; Find a good x ˆ X; 2. Compute ub z given ˆx; (or optimize) 3. Solve z = min g x, E[ I ] for ˆx; x X 4. If ( z > z ) print X and halt; 5. Add ˆx to X; 6. Add constraint ( ) x ˆ = 1 R 1 to constraint set X and go to 2; x 18

20 Alg. to find candidate solutions Find a good x ˆ X z z (Or optimize) ( ˆ ) Compute z = h x, E[ I ] Compute z = min g, E[ ] x X x xˆ x xˆ 1 2 ( x I ) For other 2SSPs, just use other bounds! 19

21 A better enumeration algorithm z Find a good ˆx (Or optimize) ( ˆ ) Compute z = h x, E[ I ] z Use B&B-lie procedure to enumerate all ( ) xˆ X s.t. z = g xˆ, E [ I ] z 20

22 Screening candidate solutions For small R we can compute Eg ( xˆ ), I exactly for each xˆ X : There are only 2 R ways for R attempted interdictions to succeed or fail. Can solve SMFI exactly in this case. Will describe general statistical screening procedures because they are necessary for most applications of BEST, including more complicated interdiction problems (and larger R). * Seeing a near-optimal set X X 21

23 We could do this: ( ) Sample the h xˆ, I for each xˆ X to obtain independent estimates h L = 1 h( ˆ, ˆ ), ( ˆ ) x I Eh x I l L l= 1 h These are distributed with independent t-distributions Reject xˆ that correspond to h being too large (Reject xˆ and xˆ below.) 3 4 h1 h2 h 3 4 h 22

24 But we will do this: Using CRNs, sample the h ( xˆ ), I for each xˆ X to obtain estimates Order: h = 1 h( ˆ, ˆ ), Create difference r.v.s L ( ˆ ) x I Eh x I l L l= 1 h h h 1 2 K These have a joint t-distribution, approximately, and we could exploit that, but let s eep things simple, so = h h 1 23

25 And this: x ˆ Reject as bad if confident that < 0 That is, put xˆ if not confident that X * ˆ ˆ ( ˆ ) ( ) ( ) ( * x, ) I > x1, I x, I Eh Eh Eh s Let be the sample s.d. for estimate α Accept xˆ if the 100( 1 )% confidence K 1 interval on covers 0: K 1 (, ] + z α s 24

26 So: Overall, we ll be 100 (1 α)% confident that we have not rejected a good ˆ X Above procedure depends on Boole- Bonferroni inequality: not very strong. On the other hand, we used CRNs in comparing the xˆ so we have employed a useful variance-reduction technique. (1 or 2 orders of mag. improvement) Many variants/improvements possible x * 25

27 Testing step Not an issue if K* = 1. Will not cover in this tal, except to say that, empirically: All near-optimal solutions in this tal s test problems are within 2% of optimal with 95% confidence. 26

28 Advantages to BEST No large approximating problems with multiple scenarios to solve For the most part, we re solving simple bounding models and using Monte Carlo to evaluate 2SSPs with fixed first-stage variables No complicated decompositions needed 27

29 SMFI, computational results (1) Grid networ s t 100 samples for each u is uniform[10,100], p =0.9 Only resource constraint: Upper bound optimized VR for screening A 1 GHz laptop using GAMS/CPLEX x ˆ x R 28

30 SMFI, computation results (2) 95% confidence Grid A R L X X*

31 Stochastic plant location (SPL) Uncertain demand for a single product x i = 1 if plant i to be built, else 0 ( ) ( xd ) I { x i } i I min Eh, where X = {0,1} x = and x X where h xd, cx+ min{ fy+ gw} y 0 s.t. j J i I y u x plants i I ij i i y + w d customers j J ij j j y 0 i, j; w 0 j J ij j 30

32 SPL, computational results (1) 10 candidate plants, choose 5 20 customer zones (rvs) Demands uniform, ±v% of mean Probabilistic UB, Jensen LB =784.7 v=10: ub=801.9, K=2, K*=1, T=15.6 v=20: ub=840.0, K=5, K*=1, T=17.4 v=40: ub=939.9, K=28, K*=1, T=

33 SPL, computational results (2) 20 candidate plants, choose customer zones (rvs) Demands uniform, ±v% of mean Probabilistic UB, Jensen LB =958.8 v=10: ub= 966.2, K=3, K*=2, T=133 v=20: ub=1006.3, K=72, K*=5, T=255 v=40: ub=1155.1, K=51, K*=17, T=560ª ª LB improved to

34 Extensions BEST (PETS, PEST, BETS?) will wor for any 2SSP provided that First-stage variables are binary or integers of modest magnitude, An optimistic bound is not too hard to compute, and For fixed x, Monte Carlo sampling is efficient. For optimistic bounds, we use Jensen s ineq. and restricted recourse Often, the global, pessimistic bound will be probabilistic 33

35 Generalizations 2 nd -stage integer variables OK Does not depend on distributions: If you can generate the rvs, BEST wors So, dependent rvs OK Probabilistic LBs? Multi-stage??? 34

36 BEST To End

37 Other wor Interdiction of communications networs: Physical attacs and cyberattacs General models for system interdiction and defense General theoretical wor on SPs Applications of SP: Sealift deployments subject to bio-attacs Integer programming 36