Scalable Stochastic Programming

Scalable Stochastic Programming

Cosmin G. Petra

Mathematics and Computer Science DivisionArgonne National Laboratory

[email protected]

Joint work with Mihai Anitescu, Miles Lubin and Victor Zavala

Motivation

Sources of uncertainty in complex energy systems– Weather– Consumer Demand– Market prices

Applications @Argonne – Anitescu, Constantinescu, Zavala– Stochastic Unit Commitment with Wind Power Generation– Energy management of Co-generation– Economic Optimization of a Building Energy System

2

Stochastic Unit Commitment with Wind Power

Wind Forecast – WRF(Weather Research and Forecasting) Model– Real-time grid-nested 24h simulation – 30 samples require 1h on 500 CPUs (Jazz@Argonne)

3

1min COST

s.t. , ,

, ,

ramping constr., min. up/down constr.

wind

wind

p u dsjk jk jk

s j ks

sjk kj

windsjk

j

wik ksj

ndsk

jjk

j

c c cN

p D s k

p D R s k

p

p

S N T

N

N

N

N

S T

S T

Slide courtesy of V. Zavala & E. Constantinescu

Wind farmThermal generator

Thermal Units Schedule? Minimize Cost

Satisfy Demand/Adopt wind power

Have a Reserve

Technological constraints

Optimization under Uncertainty Two-stage stochastic programming with recourse (“here-and-now”)

4

0

0 0 0)( , )( ,x x

Min f x Mi f xn x E_

subj. to.

0 1 2,0 0

,1

, ,(

1)) (

S

S

x xi

i ix x

xS

Min f x f

0 0 0

0

0 1, .. , ..

,

0, 0,k k k k

k Sk

A b

A B b

x

x

x x

x

subj. to.1 2, , , S

( ) : ( ( ), ( ), ( ), ( ), ( ))A B b Q c

continuous discrete

Sampling

Statistical Inference

M batches

Sample average approximation (SAA)

0) ( ) (( )

0

x b AB x

x

00

0

0

0

A b

x

x

subj. to.

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5

Solving the SAA problem – PIPS solver

Interior-point methods (IPMs)– Polynomial iteration complexity: (in theory)– IPMs perform better in practice (infeasible primal-dual path-following)– No more than 30-50 iterations have been observed for n less than 10 million– We can confirm that this is still true for n being hundred times larger

– Two linear systems solved at each iteration

– Direct solvers needs to be used because IPMs linear systems are ill-conditioned and needs to be solved accurately

– We solve the SAA problems with a standard IPM (Mehrotra’s predictor-corrector) and specialized linear algebra

– PIPS solver

)(O nL

Linear Algebra of Primal-Dual Interior-Point Methods

6

1

2T Tx Qx c x

subj. to.

Min

0

Ax b

x

Convex quadratic problem

0

T xQ Arhs

yA

IPM Linear System

1 1

1 1

2 2

2 2

01 2 0

0

0 00 0

0 00 0

0 00 0

0 0 00 0 0 0 0 0 0

T

T

TS S

S ST T T T

S

H BB A

H BB A

H BB A

A A A H AA

Two-stage SP

arrow-shaped linear system(modulo a permutation)

Multi-stage SP

nested

S is the number of scenarios

7

The Direct Schur Complement Method (DSC) Uses the arrow shape of H

1.Implicit factorization 2. Solving Hz=r

2.1. Backward substitution 2.2. Diagonal Solve

20

1 11 1 1 10

2 22 2 2

0

10 20 01 2 0

T T T

T T

T TS NS S S S

S c cS

T

Tc

T

L DH G L L

L DH G L L

L DH G L L

L L L L DG G G H L

0

0

1

1

1 1,

,

, ,

,

.

Ti i i i

Ti i i i

S

i

Ti

Tc c c

i i

i S

H H G

D L

L D L H

L G L D

C G

L C

1

10 0

10

, 1, ,i i i

c i i

S

i

Lw r i S

w L r wL

1 , 0,...,i i iv iD Sw

1

0 0

0 0 1,, .

c

T Ti i i i i S

L v

L vz L z

z

2.3. Forward substitution

Parallelizing DSC – 1. Factorization phase

8

1

10 1

11

, , ,T Ti i i i i i

i

i

T

ii

i

iS

SL D L H L G L D

GC H

i

G

2

10

1

2

2

, , ,T Ti i i i i i

i

i

T

ii

i

iS

SL D L H L G L D

GC H

i

G

10

1

, , ,

p

Ti i

T Ti i i i i i i i p

pi

iS

L D L H L G L D i

C G

S

H G

0

1,i

p

Ci

C H

cT

c cD LL C 2. Triangular solves

Process 1

Process 2

Process p

Process 1

Factorization of the 1st stage Schur complement matrix = BOTTLENECK

Sparse linear algebraMA57

Dense linear algebraLAPACK

9

Parallelizing DSC – 2. Triangular solvesProcess 1

Process 2

Process p

1

1

1

1 0

,i i i

i iSi

Lw i

L

S

w

r

r

2

1

1

2 0

,i i i

i iSi

Lw i

L

S

w

r

r

1

1

0

, p

p

i i i

ii

iS

L S

w

w r i

r L

1

1

0 0

,i i i

T Ti i i i

D wv i

z

S

L v L z

1 1 10 0

1

0 0 0, ,

p

ii

c c c

r r

w r vL z vwD L

Process 1

Process 1

1

0 0

2,i i i

T Ti i i i

D wv i

z

S

L v L z

Process 2

1

1

0 0

,i i i

T Ti i i i

D wv i

z

S

L v L z

Process p

1st stage backsolve = BOTTLENECK

1.Fa

ctor

izatio

n

Sparse linear algebra Sparse linear algebra

Dense linear algebra


10

Implementation of DSC

Fact Backsolves

Fact Backsolves

Fact Backsolves

Comm

Dense fact backsolve

MPI_Allreduce

1

p

ii

C

Comm

MPI_Allreduce

1

p

ii

r

forw.subst.Dense solve





Factorization Triangular solves

Proc 1

Proc 2

Proc p

Computations are replicated on each process.

Scalability of DSC

11

Unit commitment 76.7% efficiency

but not always the case

Large number of 1st stage variables: 38.6% efficiency

on Fusion @ Argonne

12

BOTTLENECK SOLUTION 1: STOCHASTIC PRECONDITIONER

The Stochastic Preconditioner The exact structure of C is

IID subset of n scenarios:

The stochastic preconditioner (P. & Anitescu, in COAP 2011)

For C use the constraint preconditioner (Keller et. al., 2000)

13

1

11

00

0

1

.

0

T T Ti i i i

S

iiQ A B Q B A A

C

A

S

SS

1 2{ , , , }nk k k K

1

1

1

0

1.

i i i ii

n

k k k kki

T TnS Q A B B AQ

n

0

0

.0

TnS A

MA

Implementation of PSC

14

Comm

MPI_Reduce(to proc p+1)

iiC

K

F. B.

Fact Backsolves

Dense fact of prcnd.

Comm

1

p

ii

C

MPI_Allreduce

backsolve

backsolve

backsolve

Backsolve

Comm

1

p

ii

r

MPI_Reduce(to proc 1)

Krylov solve

Prcnd tri. slv. comm

forw.subst.Comm

forw.subst.

forw.subst.

forw.subst.

MPI_Bcast

Proc 1

Proc 2

Proc p

Proc p+1

Proc 1

Proc 2

Proc p

Proc p+1

REMOVES the factorization bottleneckSlightly larger solve bottleneck

F. B.

F. B.

F. B.

Fact Backsolves

Fact Backsolves

idle

idle

The “Ugly” Unit Commitment Problem

15

DSC on P processes vs PSC on P+1 processOptimal use of PSC – linear scaling

Factorization of the preconditioner can not behidden anymore.

• 120 scenarios

Quality of the Stochastic Preconditioner

“Exponentially” better preconditioning (P. & Anitescu, 2011)

Proof: Hoeffding inequality

Assumptions on the problem’s random data1. Boundedness2. Uniform full rank of and

16

21

24 24Pr(| ( exp) 1| ) 2

||2 ||n SS max

n

p L SS pS

)(A )(B

1

11

0

1i i i ii

T Tn k

n

k k k ki

S Q A B BQ An

1

01

11 S

S i i i iT T

iiS Q A B BQ A

S

not restrictive

Quality of the Constraint Preconditioner

has an eigenvalue 1 with order of multiplicity .

The rest of the eigenvalues satisfy

Proof: based on Bergamaschi et. al., 2004.

17

0

0 0

TnS A

MA

0

0 0

TSS A

CA

p r

1M C 2r

m1

ax1 1( ) ( ) ( )0 .min n S n SS S M C S S

The Krylov Methods Used for

BiCGStab using constraint preconditioner M

Preconditioned Projected CG (PPCG) (Gould et. al., 2001)– Preconditioned projection onto the

– Does not compute the basis for Instead,

–

18

1

0 0 0 0T T

nP S ZZ Z Z

0.KerA

1 10 0 0 0 0 0( )T

Ny xA rA A S

0

0

Pr .00

Tn g rS A

guA

is computed from

100 0

200 00

TS

xS A r

yA r

0 0Cz =r

0Z 0 .KerA

Performance of the preconditioner

Eigenvalues clustering & Krylov iterations

Affected by the well-known ill-conditioning of IPMs.

19

1

0

1

0

( ) , where and S

( ( )) ( ) ) ) )( ( ( )(

n N N

T T

S

S Q A D

S

Q A

S

BD B

E

20

SOLUTION 2: PARALELLIZATION OF STAGE 1 LINEAR ALGEBRA

21

Parallelizing the 1st stage linear algebra

We distribute the 1st stage Schur complement system.

C is treated as dense.

Alternative to PSC for problems with large number of 1st stage variables.

Removes the memory bottleneck of PSC and DSC.

We investigated ScaLapack, Elemental (successor of PLAPACK)– None have a solver for symmetric indefinite matrices (Bunch-Kaufman);– LU or Cholesky only.– So we had to think of modifying either.

0

0

,0

TAQC

A

Q dense symm. pos. def., 0A sparse full rank.

22

Cholesky-based -like factorization

Can be viewed as an “implicit” normal equations approach.

In-place implementation inside Elemental: no extra memory needed.

Idea: modify the Cholesky factorization, by changing the sign after processing p columns.

It is much easier to do in Elemental, since this distributes elements, not blocks.

Twice as fast as LU

Works for more general saddle-point linear systems, i.e., pos. semi-def. (2,2) block.

110

, where 00

,T TT

T T TT T

QQ LL AQ

L I LA

L AALL

A I LLA L

TLDL

23

Distributing the 1st stage Schur complement matrix

All processors contribute to all of the elements of the (1,1) dense block

A large amount of inter-process communication occurs. Each term is too big to fit in a node’s memory.

Possibly more costly than the factorization itself.

Solution: collective MPI_Reduce_scatter calls• Reduce (sum) terms, then partition and send to destination (scatter) • Need to reorder (pack) elements to match matrix distribution• Columns of the Schur complement matrix are distributed as they are

calculated

11

10

1 T Ti i i i

ii

S

Q A B BQ QS

A


24

DSC with distributed first-stage

Fact BacksolvesComm

backsolve

MPI_Reduce_scatter

1

)(:,p

i bi

C B

Comm

MPI_Allreduce

1

p

ii

r

forw.subst.

backsolve

backsolve

forw.subst.

forw.subst.

Proc 1

Proc 2

Proc p

ELEMENTAL

ELEMENTAL

Fact Backsolves

Fact Backsolves

Schur complement matrix is computed and reduced block-wise. (B blocks of columns )

For each b=1:B

1 2, , , BB B B

25

Reduce operations

Streamlined copying procedure - Lubin and Petra (2010) Loop over continuous memory and copy

elements in send buffer Avoids divisions and modulus ops needed to

compute the positions

“Symmetric” reduce for Only lower triangle is reduced

Fixed buffer size A variable number of columns reduced.

Effectively halves the communication (both data & # of MPI calls).

TLDL

26

Large-scale performance

First-stage linear algebra: ScaLapack (LU), Elemental(LU), and

Strong scaling of PIPS with and 90.1% from 64 to 1024 cores 75.4% from 64 to 2048 cores > 4,000 scenarios On Fusion Lubin, P., Anitescu, in OMS 2011

TLDL

SAA problem:1st stage variables: 82,000

Total #: 189 millionThermal units: 1,000Wind farms: 1,200

TLDLLU

27

Towards real-life models – Economic dispatch with transmission constraints

Current status: ISOs (Independent system operator) use– deterministic wind profiles, market prices and demand– network (transmission) constraints– Outer 1-h timestep 24 horizon simulation– Inner 5-min timestep 1h horizon corrections

Stochastic ED with transmission constraints (V. Zavala et. al. 2010)– Stochastic wind profiles & transmission constraints– Deterministic market prices and demand– 24 horizon with 1h timestep– Kirchoff’s laws are part of the constraints– The problem is huge: KKT systems are 1.8 Bil x 1.8 Bil

Generator Load node (bus)

28

Solving ED with transmission constraints on Intrepid BG/P 32k wind scenarios (k=1024) 32k nodes (131,072 cores) on Intrepid BG/P Hybrid programming model: SMP inside MPI

– Sparse 2nd-stage linear algebra: WSMP (IBM) – Dense 1st-stage linear algebra: Elemental with SMP BLAS + OpenMP for

packing/unpacking buffer. For a 4h Horizon problem very good strong scaling Lubin, P., Anitescu, Zavala – in proceedings of SC 11.


29

Stochastic programming – a scalable computation pattern

Scenario parallelization in a hybrid programming model MPI+SMP– DSC, PSC (1st stage < 10,000 variables)

Hybrid MPI/SMP running on Blue Gene/P– 131k cores (96% strong scaling) for Illinois ED problem with grid constraints. 2B

variables, maybe largest ever solved?

Close to real-time solutions (24 hr horizon in 1 hr wallclock)– Further development needed, since users aim for

• More uncertainty, more detail (x 10)• Faster Dynamics Shorter Decision Window (x 10)• Longer Horizons (California == 72 hours) (x 3)

Thank you for your attention!

Questions?

30

Documents

Scalable Stochastic Programming