1. Scalable Multi-Stage Stochastic Programming
Cosmin Petra and Mihai Anitescu
Mathematics and Computer Science Division
Argonne National Laboratory
LANS Informal Seminar
March 2010
Work sponsored by U.S. Department of Energy Office of Nuclear Energy, Science & Technology

2. Problem formulation
Multi-stage stochastic programming (SP) problem with recourse
subj. to.

3. Deterministic approximation
Discrete and finite random variables
Stage t=1
0.4
0.3
0.3
Stage t=2 50 45 49
Scenario Tree
0.4
0.6
0.5
0.5
0.35
0.35
0.3
Stage t=3 6 7 4 5 5 3 3
Sample average approximation (SAA) is used for continuous or discrete infinitely supported random variables.

4. The Deterministic SAA SP Problem
subj. to.

5. Two-Stage SP Problem subj. to. Block separable obj. fcn.
Half-arrow shaped Jacobian

6. Multistage SP Problems
Depth-first traversal of the scenario tree
Nested half-arrow shaped Jacobian
Block separable obj. func. 1 4 2 3 5 6 7
s.t.

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

8. Linear Algebra for Multistage SP Problems
Two stages SP -> arrow shape
T stages (T>2)
Nested arrow shape: depth-first traversal of the scenario tree
Each block diagonal has the structure 2-stage H has.
For each non-leaf node a two-stage problem is solved.

9. The Direct Schur Complement Method (DSC)
Uses the arrow shape of H
Solving Hz=r
Implicit factorization
Back substitution
Diagonal solve
Forward substitution

10. Parallelizing the Schur Complement
Scenario-based parallelization
Bottlenecks from the Schur Complement
Implicit factorization
Back substitution
Diagonal solve
Forward substitution
Gondzio OOPS, Zavala et.al., 2007 (in IPOPT)
Solving S scenarios each with cost w, cost of SC is c, using P processes 10

11. The Preconditioned Schur Complement (PSC)
Goal: “Hide” the term from the parallel execution flow.
How? Krylov subspace iterative methods for the solution of
Solve iteratively
A each iteration and are needed.
A preconditioner M for C should
be cheap to invert (or cheap to compute )
cluster the eigenvalues of

12. The Stochastic Preconditioner
The exact structure of C is
IID subset of n scenarios:
The stochastic preconditioner of is
For C use the constraint preconditioner (Keller et. all., 2000)

13. But I said that … it has to be cheap to solve with the preconditioner
Solve with the factors of M
Factorization of M done before C is computed
Cost of the Krylov solve is slightly larger than before.
Even when one process is dedicated to M
(separate process)

14. Quality of the Stochastic Preconditioner
“Exponentially” better preconditioning
Proof: Hoeffding inequality
Assumptions on the problem’s random data
Boundedness
Uniform full rank of and

15. 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

16. The Krylov Methods Used for
BiCGStab using constraint preconditioner M
Projected CG (PCG) (Gould et. al., 2001)
Preconditioned projection onto the
Does not compute the basis for Instead,

17. Observations Real-life performance on linear SAA SP ( )
few Krylov iterations for more than half of IPM iterations
several tenths of inner iterations as IPM approaches the solution
PCG takes fewer iterations than BiCGStab
Affected by the well-known ill-conditioning of IPMs.
For convex quadratic SP the performance should improve.

18. A Parallel Interior-Point Solver for Stochastic Programming (PIPS)
Convex QP SAA SP problems
Input: users specify the scenario tree
Object-oriented design based on OOQP
Linear algebra: tree vectors, tree matrices, tree linear systems
Scenario based parallelism
tree nodes (scenarios) are distributed across processors
inter-process communication based on MPI
dynamic load balancing
Mehrotra predictor-corrector IPM

19. Tree Linear Algebra – Data, Operations & Linear Systems
Tree vector: b, c, x, etc
Tree symmetric matrix: Q
Tree general matrix: A
Operations
Linear systems: for each non-leaf node a two-stage problem is solved via Schur complement methods as previously described.
Min
subj. to. 1 4 2 3 5 6 7

20. Parallelization – Tree Distribution
The tree is distributed across processes.
Example: 3 processes
Dynamic load balancing of the tree
Number partitioning problem --> graph partitioning --> METIS
CPU 1
CPU 2
CPU 3 1 4 4 5 6 7 2 3

21. Numerical Experiments
Experiments on two 2-stage SP problems
Economic Optimization of a Building Energy System
Unit Commitment with Wind Power Generation
The Building Energy System Problem
The size of the problem was artificially increased
There is no benefit of solving with that many CPUs
The Unit Commitment Problem
Original problem is using the Illinois wind farms grid
Solved problems = 3 x Original pb. + up to 13 x more sampling data
Strong scaling investigated

22. Economic Optimization of a Building Energy System
Zavala et. al., 2009
1.2 mil variables, few degrees of freedom
Size of Schur complement = 357
Uncertainty (in RHS of the SP only)

23. PIPS on the Building Energy System Problem
The Direct Schur Complement – parallel scaling
97.6% parallel efficiency from 5 to 50 processors

24. Stochastic Unit Commitment with Wind Power Generation
Constantinescu et. al., 2009
MILP (relaxation solved)
Largest instance:
664k variables
1.4M constraints
400 scenarios
2.2k Schur Complement Matrix.
70% efficiency from 10 to 200 CPUs
Uncertainty is also only in RHS

25. PIPS on a Unit Commitment Problem
DSC on P processes vs PSC on P+1 process
120 scenarios
Optimal use of PSC
Factorization of the preconditioner can not be
hidden anymore.

26. Conclusions
The DSC method offers a good parallelism in an IPM framework.
The PSC method improves the scalability.
PIPS – solver for SP problems.
PIPS seems to be ready for larger problems.

27. Future work New scalable methods for a more efficient software
Linear algebra/Optimization methods that take into account the characteristics of the particular sampling method.
SAA error estimate.
Looking for applications
parallelize the 2nd stage sub-problems in a 2-stage setup
use multi-stage SP with slim nodes
PIPS
Continuous case: IPM hot-start, other stochastic preconditioners, etc.
Use with MILP/MINLP solvers.
A ton of other small enhancements.

28. Thank you for your attention!
Questions?