A Scalable Decomposition Approach for Stochastic Mixed-Integer - - PowerPoint PPT Presentation

SLIDE 1

Kibaek Kim Jointly work with Brian Dandurand (ANL), Cosmin Petra (LLNL), and Victor Zavala (UW-Madison) Mathematics and Computer Science Division Argonne National Laboratory UG Workshop, Berlin, Germany January 15, 2019

A Scalable Decomposition Approach for Stochastic Mixed-Integer Programs

1

SLIDE 2

Argonne: Vital part of DOE National Laboratory System

2

Argonne: Vital part of DOE National Laboratory System

SLIDE 3

Today’s Talk is about …

§ Stochastic Programming

– Integer variables – Parallel algorithms – Software implementation

§ Dual Decomposition

– New parallel algorithms to accelerate solutions – Asynchronous parallel computation – New branch-and-bound for global optimality

§ Numerical Results

– Open-source software package – SIPLIB test instances – Test problems for stochastic unit commitment – Argonne’s high performance computing clusters

3

Ax0 = b0 T1x0 +W1x1 = b1 . . . ... . . . TNx0 +WNxN = bN y0A +y1T1 · · · +yNTN = π0 y1W1 = π1 ... . . . +yNWN = πN

SLIDE 4

Parallel Decomposition Methods for Stochastic MIP

4

Kim, Kibaek and Victor M. Zavala. "Algorithmic innovations and software for the dual decomposition method applied to stochastic mixed-integer programs." Mathematical Programming Computation (2017): 1-42.

SLIDE 5

Stochastic Mixed Integer Programming (SMIP)

5

Make here-and-now decision x

• Operational decisions
• Logical decisions
• Countable items

General formulation of SMIP: Recourse function of an integer program: Observe stochastic event ω

• System Failure
• Demand and supply
• Cost and price
• Weather

Make wait-and-see decision y for given first- stage decision x and event ω

• Recourse action to event realization

(e.g. re-scheduling, system restoration)

• Time-dependent decisions

min cT x + E[Q(x, ω)] s.t. Ax ≥ b, x ∈ Rn1−p1

+

× Zp1

+

Q(x, ω) = min q(ω)T y s.t. W(ω)y ≥ h(ω) − T(ω)x, y ∈ Rn2−p2

+

× Zp2

+

SLIDE 6

1 2 3 4 5 1 2 3 4 5 −55 −50 −45 −40 −35 −30 x1 x2 cT x + E[Q(x,ω)]

Dual Decomposition for SMIP

§ Becomes More Challenging!

– The recourse function is nonconvex and discontinuous. – Benders Decomposition cannot be used.

§ Dual Decomposition

– Lagrangian relaxation of the nonanticipativity constraints – Seek for the best lower bound by solving

Decomposed in each scenario j z ≥ zLD := max

λ N

X

j=1

Dj(λ)

6

min cT x(ω1) +q(ω1)T y(ω1) + . . . +cT x(ω1) +q(ωN)T y(ωN) s.t. Ax(ω1) ≥ b, T(ω1)x(ω1) +W(ω1)y(ω1) ≥ h(ω1) . . . ... . . . Ax(ωN) ≥ b T(ωN)x(ωN) +W(ωN)y(ωN) ≥ h(ωN) x(ω1) . . . = x(ωN) x(ω1), . . . x(ωN) ∈ Rn1−p1

+

× Zp1

+

y(ω1), . . . y(ωN) ∈ Rn2−p2

+

× Zp2

+

Nonanticipaticity constraints

z = min

x0,xj,yj N

X

j=1

pj

• cT xj + qT

j yj

• s.t.

(xj, yj) ∈ Gj, j = 1, . . . , N,

N

X

j=1

Hjxj = 0 (λ)

Nonanticipaticity constraints

D(λ) := min

xj,yj N

X

j=1

[pj(cT xj + qT

j yj) − λT Hjxj]

s.t. (xj, yj) ∈ Gj, j = 1, . . . , N

SLIDE 7

Dual-Search Method 1: Subgradient Method

§ WIDELY USED in non-differentiable

• ptimization

§ The dual variable is updated as where the step size rule [1] is given by § No way of proving optimality

7

Subgradient of D(λ) at λk

START SOLVE the Lagrangian dual D(λ) for given λk D(λ) > zLB? zLB improved for the last m iterations? UPDATE λk+1 and zUB REDUCE βk+1 by a half min{Duality gap, βk+1} < ε? END YES UPDATE zLB YES YES NO NO NO Increment k

λk+1 = λk − αk X

s∈S

Hsxk

s

αk := βk zUB D(λk) k P

s∈S Hsxk sk2 2

1. Fisher, Marshall L. "The Lagrangian relaxation method for solving integer programming problems." Management science 50.12_supplement (2004): 1861-1871.

SLIDE 8

Dual-Search Method 2: Cutting Plane Method

§ Outer approximation to solve § Piecewise linear concave in λ [1] § The Master problem is given by § Finite convergence with Optimality

8

max

λ

X

s∈S

Ds(λ)

Ds(λ) = min

xs,ys

ps

• cT xs + qT

s ys

• + λT (Hsxs)

s.t. (xs, ys) ∈ conv.hull(Gs) mk = max

θs,λ

X

s∈S

θs s.t. θs ≤ Ds(λl) + (Hsxl

s)T (λ − λl),

s ∈ S, l = 0, 1, . . . , k

START SOLVE the Lagrangian dual D(λ) for given λ0 SOLVE the Master mk- <= D(λk+1)? END YES UPDATE zLB if D(λk+1) > zLB NO SOLVE the Lagrangian dual D(λ) for given λk+1 ADD linear inequalities to the Master Increment k

1. Fisher, Marshall L. "The Lagrangian relaxation method for solving integer programming problems." Management science 50.12_supplement (2004): 1861-1871.

SLIDE 9

Dual-Search Method 2: Cutting Plane Method

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λ Ds(λ) λ Ds(λ)

SLIDE 10

λ Ds(λ)

New Dual Decomposition Method

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10 20 30 40 50 60 70 80 90 100 5 10 15 Number of Iterations ||λk+1 − λk||2 Subgradient CPM IPM

§ The cutting-plane solutions of the Master problem can

– oscillate significantly; – suffer from degeneracy.

§ Interior-Point Cutting-Plane Method

– Solve the master by using interior-point method to find feasible solutions (vs. extreme optimal points) – Also use the best known upper bound for early termination of IPM iterations

§ Adding a set of valid inequalities:

– Benders-type feasibility cuts – Benders-type optimality cuts – Other cuts can also be added (e.g., cover cuts) – Further cut away the suboptimal solution space

Best known upper bound

SLIDE 11

DSP: Scalable Decomposition Solver

§ Decomposition methods for Structured Programming

– Exploiting block-angular structures – Dantzig-Wolfe decomposition + (Parallel) Branch-and-Bound – Benders decomposition – Dual decompositioParallel

§ Parallel computing via MPI library

11

Ax0 = b0 T1x0 +W1x1 = b1 . . . ... . . . TNx0 +WNxN = bN y0A +y1T1 · · · +yNTN = π0 y1W1 = π1 ... . . . +yNWN = πN

SLIDE 12

12

SLIDE 13

DSP reads models from Julia

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min ( −1.5 x1 − 4 x2 +

3

X

s=1

psQ(x1, x2, ξs

1, ξs 2) : x1, x2 ∈ {0, . . . , 5}

) , where Q(x1, x2, ξs

1, ξs 2) =

min

y1,y2,y3,y4

− 16y1 + 19y2 + 23y3 + 28y4 s.t. 2y1 + 3y2 + 4y3 + 5y4 ≤ ξs

1 − x1

6y1 + y2 + 3y3 + 2y4 ≤ ξs

2 − x2

y1, y2, y3, y4 ∈ {0, 1} and (ξs

1, ξs 2) ∈ {(7, 7), (11, 11), (13, 13)} with probability 1/3.

Only 15 lines of Julia script!

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using Dsp, MPI # Load packages

2

MPI.Init() # Initialize MPI

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m = Model(3) # Create a Model object with three scenarios

4

xi = [[7,7] [11,11] [13,13]] # random parameter

5

@variable(m, 0 <= x[i=1:2] <= 5, Int)

6

@objective(m, Min, -1.5*x[1]-4*x[2])

7

for s in 1:3

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q = Model(m, s, 1/3);

9

@variable(q, y[j=1:4], Bin)

10

@objective(q, Min, -16*y[1]+19*y[2]+23*y[3]+28*y[4])

11

@constraint(q, 2*y[1]+3*y[2]+4*y[3]+5*y[4]<=xi[1,s]-x[1])

12

@constraint(q, 6*y[1]+1*y[2]+3*y[3]+2*y[4]<=xi[2,s]-x[2])

13

end

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solve(m, solve_type=:Dual, param="myparams.txt")

15

MPI.Finalize() # Finalize MPI

SLIDE 14

SIPLIB Test Instances

§ SIPLIB: publically available test library for stochastic integer programming

– http://www2.isye.gatech.edu/~sahmed/siplib/ – DCAP: complete recourse, first-stage mixed-integer and second-stage pure integer

• Number of scenarios: 200, 300 and 500

– SSLP: complete recourse, first-stage pure integer and second-stage mixed-integer

• Number of scenarios: 5, 10, 15, 50 and 100

14

Name # Rows # Columns # Integers dcap233 200 3006 5412 5406 dcap233 300 4506 8112 8106 dcap233 500 7506 13512 13506 dcap243 200 3606 7212 7206 dcap243 300 5406 10812 10806 dcap243 500 9006 18012 18006 dcap332 200 2406 4812 4806 dcap332 300 3606 7212 7206 dcap332 500 6006 12012 12006 dcap342 200 2806 6412 6406 dcap342 300 4206 9612 9606 dcap342 500 7006 16012 16006 Name # Rows # Columns # Integers sslp 5 25 50 1501 6505 6255 sslp 5 25 100 3001 13005 12505 sslp 15 45 5 301 3465 3390 sslp 15 45 10 601 6915 6765 sslp 15 45 15 901 10365 10135 sslp 10 50 50 3001 25510 25010 sslp 10 50 100 6001 51010 50010

SLIDE 15

Computational Results

15

Instance Scen (r) Method Iter UB LB Gap (%) Time (s) dcap233 200 DDSub 10000 1835.34 1799.08 1.97 3853 DDCP 85 1835.34 1833.38 0.11 935 DSP 36 1835.34 1833.36 0.11 584 300 DDSub 1503 1645.22 1633.88 0.68 5564 DDCP 93 1645.22 1642.73 0.15 1705 DSP 43 1645.22 1642.74 0.15 1145 500 DDSub 2628 1737.94 1729.49 0.48 16791 DDCP 117 1738.47 1736.66 0.10 9451 DSP 44 1737.94 1736.66 0.10 2155 dcap243 200 DDSub 1995 2322.50 2311.37 0.47 3157 DDCP 52 2322.50 2321.18 0.05 843 DSP 37 2322.50 2321.18 0.05 694 300 DDSub 10000 2559.48 2546.91 0.49 8869 DDCP 53 2559.92 2556.66 0.12 1313 DSP 36 2559.92 2556.67 0.12 1167 500 DDSub 126 2167.97 2110.37 2.65 1992 DDCP 64 2168.38 2165.47 0.13 2648 DSP 39 2168.38 2165.46 0.13 2290 dcap332 200 DDSub 126 1068.24 983.23 7.95 443 DDCP 85 1065.87 1059.08 0.64 708 DSP 43 1063.52 1059.08 0.41 512 300 DDSub 126 1260.42 1165.89 7.52 733 DDCP 80 1257.01 1250.90 0.49 1298 DSP 48 1257.01 1250.91 0.49 986 500 DDSub 126 1596.49 1541.24 3.46 1333 DDCP 76 1593.00 1587.06 0.37 2000 DSP 43 1592.02 1587.06 0.31 1496 dcap342 200 DDSub 126 1620.19 1582.10 2.35 505 DDCP 76 1620.76 1618.07 0.16 734 DSP 37 1620.18 1618.07 0.13 577 300 DDSub 126 2069.00 2020.11 2.36 881 DDCP 73 2067.76 2065.43 0.11 1271 DSP 37 2067.96 2065.42 0.12 930 500 DDSub 126 1906.18 1861.25 2.36 1716 DDCP 87 1905.38 1902.98 0.12 2400 DSP 44 1905.36 1902.97 0.12 1788

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• Smaller gaps in shorter time
• Less iterations
• Time reduced by a factor of 8
• Solution times: 9 ~ 37 mins.
• Still positive gaps

SLIDE 16

Computational Results

16

Instance Scen (r) Method Iter UB LB Gap (%) Time (s) sslp 5 25 50 DDSub 645

• 121.60
• 123.76

1.77 244 DDCP 26

• 121.60
• 121.60

0.00 14 DSP 5

• 121.60
• 121.60

0.00 6 100 DDSub 995

• 127.37
• 128.94

1.23 643 DDCP 28

• 127.37
• 127.37

0.00 32 DSP 5

• 127.37
• 127.37

0.00 19 sslp 10 50 50 DDSub 774

• 364.64
• 369.21

1.25 3105 DDCP 62

• 364.64
• 364.64

0.00 402 DSP 11

• 364.64
• 364.64

0.00 174 100 DDSub 1224

• 354.19
• 364.64

2.95 9591 DDCP 72

• 354.19
• 354.19

0.00 1545 DSP 12

• 354.19
• 354.19

0.00 777 500 DDSub 344

• 349.13
• 356.35

2.06 > 21600 DDCP 107

• 349.13
• 349.13

0.00 10771 DSP 17

• 349.13
• 349.13

0.00 1614 1000 DDSub 177

• 351.71
• 358.88

2.03 > 21600 DDCP 66

• 351.71
• 351.99

0.07 > 21600 DSP 11

• 351.71
• 351.71

0.00 3251 2000 DDSub 115

• 347.26
• 354.03

1.94 > 21600 DDCP 54

• 347.26
• 349.60

0.67 > 21600 DSP 10

• 347.26
• 347.26

0.00 4762 sslp 15 45 5 DDSub 914

• 262.40
• 262.42

0.01 2490 DDCP 21

• 262.40
• 262.40

0.00 164 DSP 5

• 262.40
• 262.40

0.00 32 10 DDSub 427

• 260.50
• 268.15

2.94 8867 DDCP 88

• 260.50
• 260.50

0.00 1988 DSP 17

• 260.50
• 260.50

0.00 515 15 DDSub 541

• 253.60
• 265.30

4.61 > 21600 DDCP 89

• 253.60
• 253.60

0.00 14917 DSP 17

• 253.60
• 253.60

0.00 3092

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SLIDE 17

10

−1

10 10

1

10

2

10

3

dcap233_200 dcap233_300 dcap233_500 dcap243_200 dcap243_300 dcap243_500 dcap332_200 dcap332_300 dcap332_500 dcap342_200 dcap342_300 dcap342_500 sslp_5_25_50 sslp_5_25_100 sslp_15_45_5 sslp_15_45_10 sslp_15_45_15 sslp_10_50_50 sslp_10_50_100 DDSub (Solution Time) DDSub (# of Iterations) DDCP (Solution Time) DDCP (# of Iterations)

DSP vs. Existing Methods

§ Benchmark with

– DSP – Subgradient method (DDSub) – Standard cutting-plane method (DDCP)

17

1 2 3 4 5 6 7 8 9 10 Optimality Gap (%) dcap233_200 dcap233_300 dcap233_500 dcap243_200 dcap243_300 dcap243_500 dcap332_200 dcap332_300 dcap332_500 dcap342_200 dcap342_300 dcap342_500 sslp_5_25_50 sslp_5_25_100 sslp_15_45_5 sslp_15_45_10 sslp_15_45_15 sslp_10_50_50 sslp_10_50_100 DDSub DDCP DSP

Number of Iterations and Solution Time Optimality Gap

SLIDE 18

DSP vs. Existing Methods - Convergence

18

20 40 60 80 −300 −295 −290 −285 −280 −275 −270 −265 −260 −255 Number of Iterations Objective Value 100 200 300 400 500 Number of Iterations DDSub (Lower Bound) DDSub (Best Lower Bound) DDCP (Lower Bound) DDCP (Best Lower Bound) DSP (Lower Bound) DSP (Best Lower Bound)

SLIDE 19

Oscillating Dual Variable Values

19

10 20 30 40 50 60 70 80 90 100 5 10 15 Number of Iterations ||λk+1 − λk||2 DDSub DDCP DSP

SLIDE 20

Strong Scaling Results of DSP

20

5 10 25 50 100 250 10

−1

10 10

1

10

2

Cores Speedup Linear Speedup sslp_10_50_2000 sslp_10_50_1000 sslp_10_50_500 sslp_10_50_100 sslp_10_50_50

SLIDE 21

An Asynchronous Variant

21

Kim, Kibaek, Cosmin G. Petra, and Victor M. Zavala. "An Asynchronous Bundle-Trust-Region Method for Dual Decomposition of Stochastic Mixed- Integer Programming." SIAM Journal on Optimization (accepted).

SLIDE 22

1 2 3 4 5 1 2 3 4 5 −55 −50 −45 −40 −35 −30 x1 x2 cT x + E[Q(x,ω)]

Dual Decomposition for SMIP

§ Becomes More Challenging!

– The recourse function is nonconvex and discontinuous. – Benders Decomposition cannot be used.

§ Dual Decomposition

– Lagrangian relaxation of the nonanticipativity constraints – Seek for the best lower bound by solving

Decomposed in each scenario j z ≥ zLD := max

λ N

X

j=1

Dj(λ)

22

min cT x(ω1) +q(ω1)T y(ω1) + . . . +cT x(ω1) +q(ωN)T y(ωN) s.t. Ax(ω1) ≥ b, T(ω1)x(ω1) +W(ω1)y(ω1) ≥ h(ω1) . . . ... . . . Ax(ωN) ≥ b T(ωN)x(ωN) +W(ωN)y(ωN) ≥ h(ωN) x(ω1) . . . = x(ωN) x(ω1), . . . x(ωN) ∈ Rn1−p1

+

× Zp1

+

y(ω1), . . . y(ωN) ∈ Rn2−p2

+

× Zp2

+

Nonanticipaticity constraints

z = min

x0,xj,yj N

X

j=1

pj

• cT xj + qT

j yj

• s.t.

(xj, yj) ∈ Gj, j = 1, . . . , N,

N

X

j=1

Hjxj = 0 (λ)

Nonanticipaticity constraints

D(λ) := min

xj,yj N

X

j=1

[pj(cT xj + qT

j yj) − λT Hjxj]

s.t. (xj, yj) ∈ Gj, j = 1, . . . , N

SLIDE 23

Synchronous Bundle Method

§ Processors have to wait for the slowest one. § Causing Load-Imbalancing § Resulting in low parallel efficiency § Potentially increasing solution time

23 1 2 3 4 5 5 10 15 20 25 30 35 40 45 50 Scenario Subproblem Total Solution Time (sec)

Solution Time Synchronization Synchronization Synchronization Synchronization

SLIDE 24

24

§ Inexact Approximation of the Lagrangian Dual

§ Ingredients are not really new!

– Incremental Subgradient [Kiwiel 2004, Bertsekas 2001, 2011, 2015] – Incremental-Like Bundle [Emiel and Sagastizabal 2010] – Asynchronous Subgradient [Nedic et al. 2001] – Asynchronous Bundle [Fischer and Helmber 2014] – Asynchronous Benders [Linderoth and Wright 2003]

§ Trial point queue (for synchronization points)

– Necessary to update the trust region – Size of the queue – FIFO vs. LIFO

§ Minimum number of worker processors to wait for updating the bundle § Static vs. dynamic scenario allocations

˜ mk,l(λ) := max X

j∈J

θj s.t. θj ≤ Dj(λi) + (Hjxi

j)T (λ − λi), i ∈ Bk,l, j ∈ Ji

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Update the Bundle with a subset of scenarios Master Lower Bounding Lower Bounding Lower Bounding

Trial Point Queue

Asynchronous Bundle Method

Algorithm should be carefully designed!

SLIDE 25

Solution Time Synchronization Synchronization Synchronization

max

θj,λ

θ1 + θ2 + θ3 s.t. θ1 ≤ ¯ D1

1 + (H1¯

x1

1)T (λ − λ1)

θ3 ≤ ¯ D1

3 + (H3¯

x1

3)T (λ − λ1)

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θ1 ≤ ¯ D2

1 + (H1¯

x2

1)T (λ − λ2)

θ2 ≤ ¯ D1

1 + (H1¯

x1

1)T (λ − λ1)

θ3 ≤ ¯ D2

3 + (H3¯

x2

3)T (λ − λ2)

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θ2 ≤ ¯ D2

1 + (H1¯

x2

1)T (λ − λ2)

θ3 ≤ ¯ D3

3 + (H3¯

x3

3)T (λ − λ3)

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Asynchronous Bundle Method

25

SLIDE 26

Computational Experiment Settings

§ DSP: an open-source software framework for parallel decomposition methods

– Written in C++ with MPI library – Uses CPLEX for solving the master (by barrier method) and sub-MIP problems with default parameter settings

§ All computations were performed on the Blues cluster.

– 630-node computing cluster at Argonne National Laboratory – Intel Sandy Bridge Xeon E5-2670 2.6GHz – 16 cores per compute node – 64 GB of memory on each node

26

SLIDE 27

Problem Instances – Stochastic Unit Commitment

§ Scheduling power generators and dispatching power from generators to loads

– Over electric grid network – For 24-hour time horizon – Under supply uncertainty (wind power generation)

§ A test system data for the Western Electricity Coordinating Council

– 8 load profiles – 10 sets of wind power scenarios – Created 80 problem instances

§ Each SMIP has 340,244 variables and 352,840 constraints.

27

SLIDE 28

Load Imbalance

§ Percent imbalance metric: – How much the maximum time is deviated from the mean time. § Distribution of the average percent imbalance metrics resulting from the synchronous BTR – 27% ~ 84% – 18 highly imbalanced instances (> 50%)

28

νpk := ✓tmax

pk

¯ tpk − 1 ◆ × 100%,

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Highly imbalanced

SLIDE 29

29

Load Imbalance – What you really see…

20 40 60 80 Iteration 2 4 6 8 10 12 14 Process 10 20 30 40 50 Seconds 20 40 60 80 Iteration 2 4 6 8 10 12 14 Process 10 20 30 40 50 Seconds

Low Imbalanced Instance Highly Imbalanced Instance

SLIDE 30

Asynchronous Computing Results

30

Highly imbalanced Highly imbalanced

Async-Q1P1 is the fastest for 70% of the instances. Sync is slow for all the instances. Sync is slow by a factor 3.4.

SLIDE 31

Computational Experiments – Other Variants

§ Static vs. Dynamic allocation of the scenario subproblems

– Dynamic allocation improves parallel efficiency in general. – However, the dynamic allocation loses the MIP warm-start.

§ FIFO vs. LIFO for evaluating trial points

– FIFO shows marginally better results.

§ All these variations do not affect the convergence results of the method.

31

Static allocation wins! FIFO wins!

SLIDE 32

Strong Scaling of the Asynchronous Method

32

SLIDE 33

Scalable Branching on Dual Decomposition

33

Kim, Kibaek and Brian Dandurand. "Scalable Branching on Dual Decomposition of Stochastic Mixed-Integer Programming Problems." Mathematical Programming Computation (submittted).

SLIDE 34

Scalable Branching Approaches

§ Dual decomposition is a bounding method. § Branch-and-bound is the most natural approach to proving

• ptimality for MIP.

– We already have a very tight bound from DD. – Let's branch!

§ Branching rules to consider:

– at which solution point (or value) – in which direction

34

z ≥ zLD := max

λ N

X

j=1

Dj(λ)

SLIDE 35

Branch-and-Price Method

§ A natural choice for branching is on fractional integer variables in the original space.

35

DWD is the dual of the dual decomposition.

1. Lulli, Guglielmo, and Suvrajeet Sen. "A branch-and-price algorithm for multistage stochastic integer programming with application to stochastic batch-sizing problems." Management Science 50.6 (2004): 786- 796.

min

αk

n,x0

N

X

n=1 K

X

k=1

⇥ Dn(λk

n) − (xk j )T λk n

⇤ αk

n

s.t.

K

X

k=1

xk

nαk n = x0,

n = 1, . . . , N,

K

X

k=1

αk

n = 1,

n = 1, . . . , N, αk

n ≥ 0,

n = 1, . . . , N, k = 1, . . . , K

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max

θn,λn N

X

n=1

θn s.t.

N

X

n=1

λn = 0 (x0) θn ≤ Dn(λk

n) + (xk n)T (λn − λk n),

n = 1, . . . , N, k = 1, . . . , K (αk

n)

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Dualizing

SLIDE 36

Branch-and-Price Method

§ At an optimum (or feasible) of DWD, § Branching is of the form:

36

K

X

k=1

xk

nˆ

αk

n = ˆ

x0,

K

X

k=1

yk

nˆ

αk

n = ˆ

yn, n = 1, . . . , N,

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Fractional? Fractional? K

X

k=1

xk

njαk n  bˆ

x0jc _

K

X

k=1

xk

njαk n bˆ

x0jc + 1

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ynjc _

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njαk n bˆ

ynjc + 1

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The second-stage integer variables need branched!

SLIDE 37

Caroe-and-Schultz Branching

§ Interesting idea to avoid branching on the second-stage integer variables

– At the last iteration K, they used the average for the variable value to branch: – For fractional integer variables, – For continuous variables,

37

˜ x0 =

N

X

n=1

pnxK

n ,

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xnj  b˜ x0jc _ xnj b˜ x0jc + 1

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sha1_base64="+9poqrbxizev2ijFQx379x6nIWI=">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</latexit><latexit sha1_base64="+9poqrbxizev2ijFQx379x6nIWI=">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</latexit><latexit sha1_base64="+9poqrbxizev2ijFQx379x6nIWI=">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</latexit>

xnj ≤ ˜ x0j − ✏ _ xnj ≥ ˜ x0j + ✏

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Great, the branchings are applied to ALL scenarios!

SLIDE 38

Can we do better?

38

Observations

• 1. Carøe-and-Schultz (CS) branching uses an arbitrary point (i.e., average

solution) to branch.

• 2. DWD solution point provides the Lagrangian dual bound.
• Let ˆ

x0 be the DW solution point (vs. the average point ˜ x0 by CS) at some node.

• Assume that ˆ

x0 6= ˜ x0.

• When branching on continuous variables at the CS point, one of the child nodes

would still contain the DW solution point.

• Best lower bound in the tree would not be improved in the CS branching.

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SLIDE 39

New Branching Approach

39

We use the DWD solution point to branch: ˆ x0 =

K

X

k=1

xk

nˆ

↵k

n

v.s. ˜ x0 =

N

X

n=1

pnxK

n

! (1)

• For fractional integer variables,

xnj  bˆ x0jc _ xnj bˆ x0jc + 1 (2)

• For continuous variables,

xnj  ˆ x0j ✏ _ xnj ˆ x0j + ✏ (3)

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Finite termination has been proved for zero epsilon!

SLIDE 40

Some Details of Implementation

§ Algorithmic Details:

– Upper bounds are evaluated at a given point from each method, after rounding any fractional integer variable values. – No warm- or hot-start for node subproblem solutions – Gap tolerance: 0.01% – 2-hour wall clock time limit

§ Computation Settings:

– Implemented in DSP with Coin-ALPS – The master was solved by CPLEX-12.7.0. – Subproblems were solved in parallel using MPICH. – Ran on Argonne's Bebop cluster (Intel Broadwell 36 cores)

40

SLIDE 41

SIPLIB Test Instances

§ SIPLIB: publically available test library for stochastic integer programming

– http://www2.isye.gatech.edu/~sahmed/siplib/ – DCAP: complete recourse, first-stage mixed-integer and second-stage pure integer

• Number of scenarios: 200, 300 and 500

– SSLP: complete recourse, first-stage pure integer and second-stage mixed-integer

• Number of scenarios: 5, 10, 15, 50 and 100

41

Name # Rows # Columns # Integers dcap233 200 3006 5412 5406 dcap233 300 4506 8112 8106 dcap233 500 7506 13512 13506 dcap243 200 3606 7212 7206 dcap243 300 5406 10812 10806 dcap243 500 9006 18012 18006 dcap332 200 2406 4812 4806 dcap332 300 3606 7212 7206 dcap332 500 6006 12012 12006 dcap342 200 2806 6412 6406 dcap342 300 4206 9612 9606 dcap342 500 7006 16012 16006 Name # Rows # Columns # Integers sslp 5 25 50 1501 6505 6255 sslp 5 25 100 3001 13005 12505 sslp 15 45 5 301 3465 3390 sslp 15 45 10 601 6915 6765 sslp 15 45 15 901 10365 10135 sslp 10 50 50 3001 25510 25010 sslp 10 50 100 6001 51010 50010

SLIDE 42

Numerical Results - DCAP (Easy)

§ BNP cannot solve any instance in 2-hour time limit. § New approach tends to solve fewer node subproblems and thus faster than CS.

42

Nodes Nodes Instance Method UB LB Gap (%) Solved Left Time dcap233 200 BNP 1842.09 1833.43 0.47 1630 1629 TO CS 1834.71 1834.53 OPT 29 99 CS+DW 1834.58 1834.54 OPT 31 93 dcap233 300 BNP 1671.87 1642.76 1.74 377 376 TO CS 1644.35 1644.19 OPT 57 751 CS+DW 1644.25 1644.19 OPT 25 253 dcap233 500 BNP 1775.76 1736.69 2.20 165 164 TO CS 1737.61 1737.50 OPT 51 589 CS+DW 1737.52 1737.52 OPT 29 446 dcap243 200 BNP 2336.11 2321.44 0.62 711 684 TO CS 2322.67 2322.48 OPT 137 1155 CS+DW 2322.50 2322.47 OPT 33 178 dcap243 300 BNP 2574.20 2556.59 0.68 520 519 TO CS 2559.41 2559.12 OPT 29 195 CS+DW 2559.19 2559.19 OPT 31 196 dcap243 500 BNP 2193.46 2165.40 1.27 241 240 TO CS 2167.41 2167.21 OPT 33 394 CS+DW 2167.39 2167.31 OPT 43 660

<latexit sha1_base64="(nul)">(nul)</latexit><latexit sha1_base64="(nul)">(nul)</latexit><latexit sha1_base64="(nul)">(nul)</latexit><latexit sha1_base64="(nul)">(nul)</latexit>

SLIDE 43

Numerical Results - DCAP (Hard)

§ CS also failed to solve some instances in 2-hour time limit. § Clear to see that the new approach outperformed CS

43

Nodes Nodes Instance Method UB LB Gap (%) Solved Left Time dcap332 200 BNP 1098.54 1059.09 3.59 626 627 TO CS 1060.75 1060.66 OPT 395 1661 CS+DW 1060.70 1060.59 OPT 89 349 dcap332 300 BNP 1313.85 1250.84 4.79 277 274 TO CS 1252.81 1252.68 OPT 395 4481 CS+DW 1252.76 1252.63 OPT 89 832 dcap332 500 BNP 1695.66 1586.94 6.41 135 134 TO CS 1589.27 1588.32 0.05 232 213 TO CS+DW 1588.91 1588.66 0.01 223 166 TO dcap342 200 BNP 1691.89 1618.09 4.36 524 523 TO CS 1619.64 1619.48 OPT 443 2496 CS+DW 1619.56 1619.47 OPT 177 793 dcap342 300 BNP 2136.25 2065.51 3.31 301 300 TO CS 2067.61 2067.41 OPT 151 2437 CS+DW 2067.52 2067.43 OPT 125 773 dcap342 500 BNP 2028.81 1902.89 6.20 145 144 TO CS 1904.92 1904.61 0.01 99 72 TO CS+DW 1904.73 1904.53 OPT 95 1699

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SLIDE 44

Numerical Results - SSLP

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Nodes Nodes Instance Method UB LB Gap (%) Solved Left Time sslp 5 25 50 BNP

• 121.6
• 121.6

OPT 1 1 CS

• 121.6
• 121.6

OPT 1 1 CS+DW

• 121.6
• 121.6

OPT 1 1 sslp 5 25 100 BNP

• 127.37
• 127.37

OPT 1 1 CS

• 127.37
• 127.37

OPT 1 1 CS+DW

• 127.37
• 127.37

OPT 1 1 sslp 10 50 50 BNP

• 364.64
• 364.64

OPT 1 31 CS

• 364.64
• 364.64

OPT 1 31 CS+DW

• 364.64
• 364.64

OPT 1 30 sslp 10 50 100 BNP

• 354.19
• 354.19

OPT 1 36 CS

• 354.19
• 354.19

OPT 1 37 CS+DW

• 354.19
• 354.19

OPT 1 36 sslp 10 50 500 BNP

• 349.136
• 349.136

OPT 1 792 CS

• 349.136
• 349.136

OPT 1 786 CS+DW

• 349.136
• 349.136

OPT 1 788 sslp 10 50 1000 BNP

• 351.711
• 351.711

OPT 1 1964 CS

• 351.711
• 351.711

OPT 1 2275 CS+DW

• 351.711
• 351.711

OPT 1 2544 sslp 10 50 2000 BNP

• 347.263
• 347.263

OPT 1 2024 CS

• 347.263
• 347.263

OPT 1 2006 CS+DW

• 347.263
• 347.263

OPT 1 2008 sslp 15 45 5 BNP

• 262.4
• 262.4

OPT 1 4 CS

• 262.4
• 262.4

OPT 1 4 CS+DW

• 262.4
• 262.4

OPT 1 4 sslp 15 45 10 BNP

• 260.5
• 260.5

OPT 1 101 CS

• 260.5
• 260.5

OPT 1 101 CS+DW

• 260.5
• 260.5

OPT 1 101 sslp 15 45 15 BNP

• 253.601
• 253.601

OPT 1 267 CS

• 253.601
• 253.601

OPT 1 267 CS+DW

• 253.601
• 253.601

OPT 1 267

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SLIDE 45

Summary and Future Research Questions

§ Dual decomposition is the Lagrangian relaxation applied to SMIP

– Not just that.

§ Allowing to exploit special structures embedded in SMIP

– Benders-type cuts – Inexact interior point methods and other bundle methods – Any better dual search methods, customized to this setting?

§ Parallel computing

– Asynchronous communication significantly reduces the solution time.

§ Hope for global optimality

– Specialized scalable branching method! – Other branching ideas? Generalized disjunctions, strong branching, etc.

§ Other questions

– Scenario partitioning/grouping – Primal heuristics – Extended to multi-stage stochastic programs

45

SLIDE 46

Questions?

§ Contact:

– Kibaek Kim, kimk@anl.gov

§ Acknowledgements:

– U.S. Department of Energy, Office of Science, ECRP – U.S. Department of Energy, Office of Science, MMICCS – U.S. Department of Energy, Office of Electricity, GMLC

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