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Stochastic Programming on a Grid Jeﬀ Linderoth, Stephen Wright University of Wisconsin-Madison ICCOPT II, August 2007 Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 1 / 50 1 Stochastic Programming Introduction Formulation and Basic Algorithms: Two stages 2 Grid Computing Tools 3 Two-Stage Problems Parallel Algorithms Computational Results: Performance Computational Results: Solution Quality 4 Multistage Problems Formulations Algorithms Implementation Challenges Computational Results Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 2 / 50 In this talk we show how a computation-intensive optimization problem can be solved by “putting it all together:” Good algorithms, matched to the platform Raw power of Grids Programmability of the MW toolkit. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 3 / 50 Outline Stochastic Programming (SP) Formulation and Basic Algorithms for Two-Stage SP Using Condor and MW Asynchronous Trust-Region Algorithm Computational Results: Algorithm Performance Multistage SP: Formulations and Algorithms Multistage SP: Computational Results Collaborators: Alex Shapiro (Georgia Tech), Jierui Shen (Lehigh). Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 4 / 50 Stochastic Programming Optimization of a model with uncertainty. Often formulated mathematically as def min f (x) = Eξ g (x; ξ) = g (x; ξ)p(ξ)dξ, x Ω (p is probability density function) subject to constraints on x ∈ R n . Arises in planning-under-uncertainty applications, where each ξ represents a possible scenario (a possible way in which the model could evolve). Space Ω can contain ﬁnite or inﬁnite scenarios. g (x; ξ) could be the value function of some second level optimization problem parametrized by x. (Recourse.) Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 5 / 50 Example: Network Planning Adding capacity on a telecommunications network for private-line services. (Sen et al., 1994.) Shows nodes and links, Node pair A-B, and a route between A and B. B A Add capacity to some links, to attempt to meet (uncertain) demand for traﬃc between nodes. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 6 / 50 Sample demand proﬁle: Third node pair: Demand 0 (prob .855), Demand 5.39 (prob .095), Demand 75.1 (prob .05). Data: - network topology: n = 89 links. - point-to-point pairs: i = 1, 2, . . . , 86. - demands di for each pair i are random and independent, with 3 to 7 possible scenarios. Total about 1070 scenarios! Decision variables: xj , j = 1, 2, . . . , n: amount of capacity to add on link j. Total new capacity bounded by B. Objective: minimize the expected amount of unmet demand, summed over the m point-to-point pairs. We can’t hope to solve the problem by accounting for all the 1070 possible scenarios exhaustively — it’s much too large. Practical approach: Use sampling to select a subset of N scenarios, randomly. The sample average approximation (SAA) is large, but manageable. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 7 / 50 2-stage stochastic LP with recourse min Q(x) = c T x + EP Q(x; ω) subj. to Ax = b, x ≥ 0, x where P is a probability measure on the space (Ω, F), and Q(x; ω) = min q(ω)T y subject to y Wy = h(ω) − T (ω)x, y ≥ 0. x = ﬁrst-stage vars, y = second-stage vars. Sampled approximation: Sample N points ωj , j = 1, 2, . . . , N from P, and solve N minx Q(x) = c T x + N −1 j=1 Q(x; ωj ) subj. to Ax = b, x ≥ 0. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 8 / 50 Each Q(x; ωj ) is convex, piecewise-linear in x. Q(x) subgradients x Compute subgradients of Q by ﬁnding dual solutions πj of the second-stage LP’s for j = 1, 2, . . . , N (concurrently!) Q(x; ωj ) : minyj q(ωj )T yj , subj. to Wyj = h(ωj ) − T (ωj )x, yj ≥ 0; N summing: c − N −1 T j=1 T (ωj ) πj . Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 9 / 50 “bundle” methods Build up a lower bounding, piecewise linear approximation to Q(x), based on function values Q(x ℓ ) and subgradients g ℓ at iterates x ℓ . Model function Mk (x) after k iterates is Mk (x) = sup Q(x ℓ ) + (g ℓ )T (x − x ℓ ) . ℓ=0,1,...,k Choose next iterate as x k+1 = arg min Mk (x), subj. to Ax = b, x ≥ 0. which can be formulated as: minx,θ θ, subject to Ax = b, x ≥ 0, θ≥ Q(x ℓ ) + (g ℓ )T (x − x ℓ ), ℓ = 0, 1, . . . , k. (Each constraint is called a cut.) Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 10 / 50 Example: After ﬁrst two iterations 0, 1: Q(x) M1 (x) x0 x1 x x 2 is the minimizer of M1 ; add new subgradient to obtain M2 ; take minimizer to obtain x 3 : Q(x) M2 (x) x2 x3 x Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 11 / 50 enhancements trust-region (allows steady progress, exploits good starting point); algorithm that allows deletion of old cuts; group the second-stage problems Q(x; ωj ) into T “chunks” Nt , t = 1, 2, . . . , T , with {1, 2, . . . , N} = ∪t=1,2,...,T Nt . and assign each Nt to a worker processor; multiple cuts at each x (each chunk can return its own subgradients); asynchronous variant is preferred for our target parallel platform. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 12 / 50 trust-region (TR) Choose next iterate as x k+ = arg min Mk (x), subj. to Ax = b, x ≥ 0, x − xk ∞ ≤ ∆k , where ∆k is the trust-region radius. Trivial to modify the LP subproblem: just add the bounds −∆k e ≤ x − x k ≤ ∆k e. If candidate point x k+ is “signiﬁcantly better” (achieves some fraction of the decrease predicted by the model) then set x k+1 ← x k+ . Possibly delete cuts, increase the trust region. Otherwise, set x k+1 ← x k and add subgradient information from x k+ to improve the model. Possibly delete uninteresting cuts, decrease trust region. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 13 / 50 TR properties Denoting the solution set by S, Can delete “irrelevant” cuts liberally, between major iterations; dist(x k , S) → 0; The algorithm may still be too synchronous: requires complete evaluation of Q(x) at a candidate iterate x before proceeding. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 14 / 50 Condor We’ve heard enough about Condor by now. But to recap: Condor pools consist of user workstations, nodes from multiprocessor systems and clusters. Handles scheduling, matching of user requirements to machine characteristics. Checkpointing and migration. Flocking and Glide-in mechanisms allow jobs to execute across multiple pools. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 15 / 50 a challenging environment... The Condor environment is powerful and inexpensive, but challenging to algorithm designers and implementers. dynamic/opportunistic: size and composition of worker pool changes unpredictably during computation heterogeneous: many types of machines, various operating systems, diﬀerent licenses on diﬀerent machines. latency unpredictable, generally slow: workers can be next to each other in a rack, or separated by 6000 miles. Problems that are large and compute-intensive — and algorithms that are asynchronous — are best suited to this platform. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 16 / 50 TR may not be asynchronous enough! The TR approach still synchronizes on the function evaluation at each candidate point x k+ . If there are T chunks of second-stage scenarios, can’t use more than T processors. For many problems of interest, we cannot make T very large (10 – 100) without making the work-per-chunk too small and creating too much contention at the master. May wait for a long time for the last chunk to be evaluated, if its host is suspended or disappears. An asynchronous trust-region (ATR) algorithm increases parallelism and throughput. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 17 / 50 ATR Maintain an incumbent x I : the best point found so far (smallest value of Q). Maintain a basket B of 3 to 20 other x points — possible new incumbents — for which the second-stage LPs are currently being solved. When space becomes available in B, generate a new candidate point by solving a TR subproblem around the current incumbent: x − x I ∞ ≤ ∆. (x I becomes the parent incumbent of the new point.) Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 18 / 50 ATR (continued) When evaluation of a point x ∈ B is completed, accept it as the new incumbent if it is better than the current incumbent x I ; Q(x) gives a signiﬁcant decrease over its parent incumbent. Populate B initially by solving TR subproblems around early incumbents, using partial subgradient information. (Synchronicity parameter σ.) Strategies for cut deletion and adjustment of trust region are adapted from the strategies for the synchronous TR algorithm. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 19 / 50 SSN: computational results ﬁrst stage: 89 variables, 1 constraint; second stage: 706 variables, 175 constraints, 2284 nonzeros. Study the eﬀect of asynchronicity, parallelism on large sampled instances. We report results for N = 104 and N = 105 scenarios, with synch parameter σ = .7. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 20 / 50 ck ll i pa cs wa ﬃc . r i te clo o s pr nk e s/ r. u r n t | i te ch cu av |B ru ATR 186 - 25 50 19 .94 51 ATR 148 - 25 100 15 .89 55 ATR 144 - 50 100 18 .82 47 ATR 79 - 50 200 18 .70 31 ATR 104 3 25 50 9 .88 61 ATR 69 3 25 100 8 .93 47 ATR 67 3 50 100 9 .86 43 ATR 61 3 50 200 6 .90 54 ATR 245 6 25 50 14 .93 91 ATR 197 6 25 100 12 .87 97 ATR 164 6 50 100 13 .81 81 ATR 135 6 50 200 12 .71 80 SSN, N = 10, 000. 1.75M × 7.06M. (Results obtained 8/7/2007.) Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 21 / 50 ) in clo cy (m n cie ck pa cs . r eﬃ i te o s pr nk s/ ll r. u r n t | wa i te ch cu av |B ru ATR 107 - 100 100 75 .17 427 ATR 84 - 100 200 73 .22 275 ATR 123 3 100 100 33 .93 199 ATR 108 3 100 200 32 .77 216 SSN, N = 100, 000. 17.5M × 70.6M. (Results obtained 8/8/2007.) Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 22 / 50 storm: computational results n Cargo ﬂight scheduling problem (Mulvey and Ruszczy`ski). ﬁrst stage: 121 variables; second stage: 1259 variables. For a 250000 scenario sampled approx, LP has size 132, 000, 185 × 314, 750, 121 Started from a solution for a 3000-scenario approximation, whose quality is very good. TR takes a single step and terminates, ATR doesn’t take any steps, just veriﬁes quality of starting point. (For a chunk of 2000 scenarios, task size is about 150 seconds.) Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 23 / 50 ) in clo cy (m n cie ck pa cs r eﬃ i te o s pr nk s/ ll r. u r n t | . wa i te ch cu av |B ru TR 17 - 125 125 106 .55 146 ATR 25 3 125 125 106 .90 116 storm, N = 250, 000. 132M × 315M. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 24 / 50 storm with 107 scenarios LP has size approximately 5.5 × 109 rows and 1.3 × 1010 columns. Used machines at Wisconsin, NCSA (Illinois), New Mexico, Argonne, Italy, Columbia. 800 machines requested, 556 actually used during the run (average of 433 at any one time). Performed in 2001. 600 500 400 #workers 300 200 100 0 0 20000 40000 60000 80000 100000 120000 140000 Sec. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 25 / 50 ) rs y nc (h cie ck pa cs r eﬃ i te clo o s pr nk s/ ll r. u r n t | . wa i te ch cu av |B ru ATR 39 4 1024 1024 433 .67 31.9 storm, N = 107 . Columns: 1.3 × 1010 . Solved 4 × 108 second-stage linear programs during the run (3472 per second). Average task 774 seconds. Total computation time 9014 hours (more than one year). Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 26 / 50 solution quality Can we get useful estimates for the optimal objective values of the true problem from the sampled problem? Can we get conﬁdence intervals on these estimates? How do the solutions of the sampled approximation relate to those of the real problem? Using ATR, along with relevant theory (some recent), we have performed computational and statistical studies of these issues for some diﬃcult problems from the literature. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 27 / 50 lower bound for Z ∗ It’s well known that EZN ≤ Z ∗ . This is true for any unbiased estimator. In particular can use certain variance reduction techniques (e.g. Latin hypercube) to select the sample {ω1 , ω2 , . . . , ωN }. Generate M batches — each a sampled approximation of size N of the (i ) (i ) (i ) form {ω1 , ω2 , . . . , ωN }, i = 1, 2, . . . , M — and solve the M SAAs to obtain optimal values (1) (2) (M) ZN , ZN , . . . , ZN . Then estimate EZN by M (i ) LM = M −1 ZN . i =1 Use sample variance, central limit theorem to get a conﬁdence interval. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 28 / 50 upper bound for Z ∗ ˆ Given any feasible point x , we have Q(ˆ ) ≥ Q(x ∗ ). x Choose an x that appears to be nearly optimal, e.g. min of some QN . ˆ ¯ Choose T i.i.d. samples, each of size N (using MC or LH): (i ) (i ) (i ) {¯ 1 , ω2 , . . . , ωN }, i = 1, 2, . . . , T , ω ¯ ¯¯ Deﬁning ¯ N (i ) (i ) QN (ˆ ) ¯ x ˆ ¯ = c x + N −1 T Q(ˆ ; ωj ). x ¯ j=1 we get an unbiased estimator: T (i ) UN,T = T −1 ¯ QN (ˆ). ¯ x i =1 Again, use sample variance to get conﬁdence interval. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 29 / 50 results on bounds: SSN Results of Mak, Morton, Wood (1999). lower upper batch/sample 30 × 1000 1 × 100000 estimate 9.22 9.98 95% conﬁdence ±0.21 ±0.11 ˆ Using diﬀerent techniques, Mak et al, generate an approximate solution x using N = 2000, and obtain upper bound 10.06 ± 0.12 (95% conﬁdence interval); with 95% likelihood, the optimal Z ∗ is within 0.77 of this value. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 30 / 50 solution estimates for SSN (95% conﬁdence intervals) Monte Carlo: N Lower Upper 50 4.11 ± 1.23 12.88 ± 0.12 100 7.66 ± 1.31 11.31 ± 0.12 500 8.54 ± 0.34 10.42 ± 0.12 1000 9.31 ± 0.23 10.20 ± 0.06 5000 9.98 ± 0.21 10.01 ± 0.09 Latin Hypercube: N Lower Upper 50 10.10 ± 0.81 11.39 ± 0.02 100 8.90 ± 0.36 10.52 ± 0.03 500 9.87 ± 0.22 10.05 ± 0.02 1000 9.83 ± 0.29 9.97 ± 0.03 5000 9.84 ± 0.10 9.90 ± 0.03 Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 31 / 50 SSN Monte Carlo 14 Lower Bound Upper Bound 12 10 Value 8 6 4 10 100 1000 10000 N Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 32 / 50 SSN Latin Hypercube 14 Lower Bound Upper Bound 12 10 Value 8 6 4 10 100 1000 10000 N Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 33 / 50 Multistage Decision Making Random vectors ξ1 ξ2 ξ3 ξT ξ1 ∈ Rn1 , ξ2 ∈ Rn2 , . . . , ξT ∈ RnT x1 x2 xT −1 xT Make sequence of decisions x1 ∈ X1 , x2 ∈ X 2 , . . . , xT ∈ X T . Risk Neutral: We always aim to optimize the expected value of our current decision xt Linear: Assume Xt are polyhedra Discrete: Assume ξt are drawn from a discrete distribution. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 34 / 50 Scenario Tree N: Set of nodes in the tree ρ(n): Unique predecessor of node n in the tree S(n): Set of successor nodes of n x0 qn : Probability that the sequence ˆ ξ1 of events leading to node n occurs ˆ ξ2 xn xn : Decision taken at node n xρ(n) Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 35 / 50 Multistage Stochastic Programming Deterministic Equivalent T zSP = min qn cn xn Tn xρ(n) + Wn xn = hn ∀n ∈ N n∈N Value Function of node n def T Qn (xρ(n) ) = min cn xn + qmn Qm (xn ) | Wn xn = hn − Tn xρ(n) ˆ xn m∈S(n) ˆ qmn : conditional probability of node n given node m Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 36 / 50 Nested Decomposition 0: Root node of the scenario tree x0 : Initial state of the system Recursive Formulation zSP = Q0 (x0 ) def Cost to go: Gn (x) = m∈S(n) qmn Qm (x) ˆ k Mn (x): Lower bound on Gn (x) in iteration k T k Qn (xρ(n) ) ≥ min cn xn + Mn (xn ) Wn xn = hn − Tn xρ(n) ((MLPn )) xn Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 37 / 50 Action Pictures x0 Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 38 / 50 MWImplementation MWTask—Work Collection of nodes (going the same direction) from the same stage The xρ(n) from these nodes MWTask—Result (Forward): xn (Backwards): Cut(s) for Gρ(n) act on completed task() is responsible for updating node state and deciding which nodes to evaluate next Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 39 / 50 Synchronicity is Bad! All processors waiting for this node to ﬁnish! Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 40 / 50 MW Implementation—Asynchronous Don’t wait for all children (Sn ) to report before starting an evaluation of k Mn . Management of cuts is important! We may require lots of memory to store the cuts Ex.: 27,000 nodes in period T -1, each node contains 20 cuts, xn ∈ R100 ⇒ ≥ 400MB just to store cuts Grid: Since we don’t have guarantees about worker processors, we cannot store cuts on the workers. Grid: All cuts (must) be stored on the master processor Leads to memory overload of master Leads to increased “service time” of the master for worker requests. (contention) Grid: ⇒ We must do what we can to compress and reduce the number of cuts Don’t record duplicates Aggregate nodes Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 41 / 50 Cut Management—Aggregation Form the deterministic equivalent of a group of nodes, and treat this as one larger “supernode” Node subproblems get larger Fewer cuts Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 42 / 50 MWAND MW Asychronous Nested Decomposition Our “Monster Solver” Magic WAND? :-) Uses the COIN Osi Interface to build MLPn Uses the COIN Clp (simplex) solver to solve MLPn Does not use the COIN-Smi to manipulate stochastic program Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 43 / 50 sutil: A Stochastic Programming Utility Library Czyzyk, Linderoth, and Shen Reads SMPS ﬁles Creates (implicity) sampled scenario trees Creates deterministic equivalents Aggregates nodes Passes stochastic information between processors sutil is available at COR@L Computational Optimization Reseach @ Lehigh: http://coral.ie.lehigh.edu http://coral.ie.lehigh.edu/sutil/ Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 44 / 50 Computational Environment—Multistage Right now, we are using a “baby grid” All Linux machines Helpful Hint: Don’t unleash your code onto a big grid unless you are reasonable sure it is working well. Location Processor Number Wisconsin Vary in speed by factor of roughly 20 785 NCSA Intel Xeon 3.2 GHz 1280 Argonne/U of C Intel Xeon 2.4 GHz 288 Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 45 / 50 A small Multistage SSN A Set of stages T Set J of links Sets It of demands B C D Random demand dt (ξ) ∈ R|It | Budget each period Install capacity on links each E F period to minimize the total expected unserved demand Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 46 / 50 Some (Limited) Computational Results T =5 Aggregate last three periods together Clustering/Tasking: Set by K N DE Size hand for each instance 30 0.81M 18M * 31M α1 = 0.8, α2 = 0.1 50 6.25M 140M * 236M K : Realizations/Period 60 12.9M 290M * 488M N: Number of scenarios DE: Size of deterministic equivalent Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 47 / 50 Computational Results It: Number of iterations (Times MLP0 was solved) E: Parallel utilization Time machines solving MLPn Time machines available K It Avg Workers Wall Time CPU Time E 30 9 62 2:34:21 6:15:15:10 67% 50 7 75 1:12:49:27 85:20:24:15 77% 60 11 162 3:16:51:00 431:12:15:37 73% Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 48 / 50 Workers in Solving ssn5-60 Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 49 / 50 Conclusions Grid computing has made it possible to solve many interesting instances of stochastic optimization that were intractable for uniprocessor computing. Has allowed computational experimentation on issues that were previously investigated theoretically, e.g. quality of solutions, sharpness of solutions, convergence of sampled solution to true solution as sample size grows. Need to pull together good algorithms, smart implementations, MW infrastructure, Condor grids. Jeﬀ Linderoth, Stephen Wright (UW-Madison) Stochastic Programming on a Grid ICCOPT II, August 2007 50 / 50

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