Scalable Stochastic Programming Cosmin Petra and Mihai Anitescu Mathematics and Computer Science Division Argonne National Laboratory Informs Computing Society Conference Monterey, California January, 2011

Motivation  Sources of uncertainty in complex energy systems –Weather –Consumer Demand –Market prices  – 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 3 Slide courtesy of V. Zavala & E. Constantinescu Zavala’s SA2 talk

Optimization under Uncertainty  Two-stage stochastic programming with recourse (“here-and-now”)  4 subj. to. continuous discrete Sampling Inference Analysis M samples Sample average approximation (SAA) subj. to.

Linear Algebra of Primal-Dual Interior-Point Methods 5 subj. to. Min Convex quadratic problem IPM Linear System Two-stage SP arrow-shaped linear system (via a permutation) Multi-stage SP nested

6 The Direct Schur Complement Method (DSC)  Uses the arrow shape of H 1.Implicit factorization 2. Solving Hz=r 2.1. Back substitution 2.2. Diagonal Solve 2.3. Forward substitution

Parallelizing DSC – 1. Factorization phase 7 2. Backsolve Process 1 Process 2 Process p Process 1 Factorization of the 1 st stage Schur complement matrix = BOTTLENECK

Parallelizing DSC – 2. Backsolve 8 Process 1 Process 2 Process p Process 1 Process 2 Process p 1 st stage backsolve = BOTTLENECK 1.Factorization

Scalability of DSC 9 Unit commitment 76.7% efficiency but not always the case Large number of 1 st stage variables: 38.6% efficiency on Argonne

BOTTLENECK SOLUTION 1: STOCHASTIC PRECONDITIONER 10

Preconditioned Schur Complement (PSC) 11 (separate process) REMOVES the factorization bottleneck Slightly larger backsolve bottleneck

The Stochastic Preconditioner  The exact structure of C is  IID subset of n scenarios:  The stochastic preconditioner (Petra & Anitescu, 2010)  For C use the constraint preconditioner (Keller et. al., 2000) 12

The “Ugly” Unit Commitment Problem 13  DSC on P processes vs PSC on P+1 process Optimal use of PSC – linear scaling Factorization of the preconditioner can not be hidden anymore. 120 scenarios

Quality of the Stochastic Preconditioner  “Exponentially” better preconditioning (Petra & Anitescu 2010)  Proof: Hoeffding inequality  Assumptions on the problem’s random data 1.Boundedness 2.Uniform full rank of and 14 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.,

Performance of the preconditioner  Eigenvalues clustering & Krylov iterations  Affected by the well-known ill-conditioning of IPMs. 16

SOLUTION 2: PARALELLIZATION OF STAGE 1 LINEAR ALGEBRA 17

Parallelizing the 1 st stage linear algebra  We distribute the 1 st stage Schur complement system.  C is treated as dense.  Alternative to PSC for problems with large number of 1 st 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. 18 dense symm. pos. def., sparse full rank.

ScaLapack (ORNL) 19  Classical block distribution of the matrix  Blocked “down-looking” Cholesky - algorithmic blocks  Size of algorithmic block = size of distribution block!  For cache-performance - large algorithmic blocks  For good load balancing - small distribution blocks  Must trade off cache-performance for load balancing  Communication: basic MPI calls  Inflexible in working with sub-blocks

Elemental (UT Austin) 20  Unconventional “elemental” distribution: blocks of size 1.  Size of algorithmic block size of distribution block  Both cache-performance (large alg. blocks) and load balancing (distrib. blocks of size 1)  Communication  More sophisticated MPI calls  Overhead O(log(sqrt(p))), p is the number of processors.  Sub-blocks friendly  Better performance in a hybrid approach, MPI+SMP, than ScaLapack

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. 21

Distributing the 1 st 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.  Possibly more costly than the factorization itself.  Solution: use buffer to reduce the number of messages when doing a Reduce_scatter.  approach also reduces the communication by half – only need to send lower triangle. 22

Reduce operations 23  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).

Large-scale performance 24  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 SAA problem: 1 st stage variables: 82,000 Total #: 189 million Thermal units: 1,000 Wind farms: 1,200

Concluding remarks  PIPS – parallel interior-point solver for stochastic SAA problems –Largest SAA prob. 189 Mil vars = 82k 1 st -stage vars + 4k scens * 47k 2 nd -stage vars 2048 cores  Specialized linear algebra layer –Small-sized 1 st -stage subproblems  DSC –Medium-sized 1 st -stage  PSC –Large-sized 1 st -stage  Distributed SC  Current work: Scenario parallelization in a hybrid programming model MPI+SMP 25

Thank you for your attention! Questions? 26