1. On parallelizing dual decomposition in stochastic integer programming
Cosmin Petra
Mathematics and Computer Science Division
Argonne National Laboratory, USA
Joint work with Miles Lubin (MIT), Burhaneddin Sandikçi, Kipp Martin (U Chicago)
INFORMS Annual Meeting
Oct 2012

2. Overview Block-angular structure
Motivation: stochastic optimization of the power grid
Revisiting dual decomposition algorithm of Carøe and Schultz
Parallelizing the solution to Lagrangian dual
Parallel numerical experiments

3. Stochastic Formulation
Discrete distribution leads to block-angular (MI)LP

4. Large-scale (dual) block-angular LPs
Extensive form
In terminology of stochastic LPs:
First-stage variables (decision now): x0
Second-stage variables (recourse decision): x1, …, xN
Each diagonal block is a realization of a random variable (scenario)

5. Stochastic Optimization and the Power Grid
Unit Commitment: Determine optimal on/off schedule of thermal (coal, natural gas, nuclear) generators. Day-ahead market prices. (hourly)
Mixed-integer
Economic Dispatch: Set real-time market prices. (every 5-10 min.)
Continuous Linear/Quadratic
Challenge: Integrate energy produced by highly variable renewable sources into these control systems.
Minimize operating costs, subject to:
Physical generation and transmission constraints
Reserve levels
Demand …

6. Stochastic unit commitment with wind power
Scenarios obtained using numerical weather prediction codes
Real-time grid-nested 24h parallel simulation using WRF
Thermal generator
Wind farm
Slide courtesy of V. Zavala & E. Constantinescu

7. Computational challenges and difficulties in power grid
May require many scenarios (100s, 1,000s, 10,000s …) to accurately model uncertainty
“Large” scenarios (Wi up to 100,000 x 100,000)
“Large” 1st stage (1,000s, 10,000s of variables)
Easy to build a practical instance that requires 100+ GB of RAM to solve
 Requires distributed memory
Integer constraints
Real-time solution needed in our applications

8. Dual decomposition - formulation
Feasibility sets
Extensive form in this notation is
Split-variable formulation
Non-anticipativity constraints are explicitly enforced

9. Dual decomposition (Carøe and Schultz, 1999)
Apply Lagrangian relaxation (LR) to non-anticipativity constraints
For mixed-integer problems, LR provides a lower bound on the optimal value
Branch and bound on non-anticipativity variables

10. Dual decomposition – computational appeal
Relaxing non-anticipativity decouples the scenarios
Typically better lower bound than from LP relaxation
Lagrangian dual has similar computational pattern to Benders
At each iteration, solve a continuous master problem and an independent MILP for each scenario
Trivially parallel?
Not quite, there are interesting algorithmic and computational questions, as we'll see.
To our knowledge, no previously published parallel implementations, although scope for parallelism has been observed

11. “State variable” formulation of non-anticipativity
Caroe and Schultz consider r-1 constraints in the form
We use “state variable” formulation with r constraints (Sen, 2005)
Lagrangian dual problem can be stated as
Will see later why this formulation is useful for parallel computations

12. Characterization of the solution
Proposition (Carøe and Schultz)
Optimal objective of Lagrangian dual is that of a partially convexified LP relaxation
Theoretical equivalence with Lulli and Sen’s (2004) branch-and-price (column generation)
However, solving the Lagrangian dual also gives a solution to (1) (useful in branch-and-bound)
The optimal value equals the optimal value of the linear program
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13. Optimization of the Lagrangian Dual
Each is concave and non-differentiable
For a fixed , is a subgradient of , where
By solving/evaluating (mixed-integer LP), we get at least one subgradient for free
Carøe and Schultz suggests using proximal bundle black-box solvers.
Alternatives are other variants of cutting-plane algorithms (boxstep or level regularization)
- Use subgradients to form a piecewise linear model of the objective function.
- Find optimum of model to determine next trial point. (Master problem)
- Evaluate trial point (MILP subproblems) to check objective value for convergence. If not converged, update model with new subgradient(s) and repeat.
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14. Let’s open the proximal bundle black box
Proximal bundle QP master is the trial point, the regularization parameter
The dual, below, is typically advantageous for computation
The proximal bundle master QP has a (dual) block-angular structure
- Use subgradients to form a piecewise linear model of the objective function.
- Find optimum of model to determine next trial point. (Master problem)
- Evaluate trial point (MILP subproblems) to check objective value for convergence. If not converged, update model with new subgradient(s) and repeat.
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15. Dual of Proximal Bundle QP Master
Block-angular structure
Direct result of equality-constrained formulation, doesn't hold for other formulations of non-anticipativity.
But also applies to other forms of regularization.

16. Computational significance
The master QP is overall sparse (but may be block dense).
Dense QP solvers, which are typically used in black-box proximal bundle codes, can't efficiently solve it.
Master QP’s structure can be exploited: PIPS (Petra) or OOPS(Gondzio)
With the ability to solve the master in parallel, we address a serial bottleneck of execution. Now, both MILP subproblems and QP master can be solved in parallel.
Greater potential for parallel speedup (Amdahl's law)

17. Numerical experiments
Implementation in C++ using MPI.
Looking at solving Lagrangian dual, no branching implemented
SCIP used for MILP subproblems.
No “compression of bundle” (removing cuts/subgradients)
Serial experiments
Cutting plane vs. black-box proximal bundle vs. our implementation
Parallel experiments
Scalability of full proximal bundle algorithm
More detailed look at scalability of parallel QP solver

18. Test instances
Stochastic mixed-integer instances dcap and sslp from SIPLIB by Shabbir Ahmed
Stochastic LP product instance from Huseyin Topaloglu

19. Test architecture “Fusion” high-performance cluster at Argonne
320 nodes
InfiniBand QDR interconnect
Two 2.6 Ghz Xeon processors per node (total 8 cores)
Most nodes have 36 GB of RAM, some have 96 GB
We use 1 MPI process (= parallel process) per core

20. Serial experiments OOQP - General sparsity-exploiting QP IPM solver
Time limit 7200 seconds
OOQP - General sparsity-exploiting QP IPM solver
PIPS - Specialized IPM solver for block-angular structure
ConicBundle - Open-source off-the-shelf proximal bundle code

21. Parallel experiments – serial master, parallel subproblems

22. Parallel experiments – parallel master, parallel subproblems
- Overall 10x speedup vs. 2x speedup over on dcap332 by solving master in parallel
- Master has little eect for sslp Also little speedup by solving subproblems in parallel. Imbalance in time to solve MILP subproblems.
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23. Scalability of master QP
Scope for parallelism in solving master QP depends on number of linking variables (= number of first-stage variables) being small relative to diagonal blocks (= subgradient cuts per scenario).
Stochastic integer test problems have very small first stage (order of 10). Important to consider performance on larger first stage for practical problems.
Stochastic LP product instances used for this (1,000 scenarios)
We look only at QP solves, not proximal bundle convergence.

24. Small first stage

25. Medium first stage

26. Large first stage

27. Energy application – stochastic unit commitment
State of Illinois power grid
12-hour horizon
64 scenarios
3,621,180 vars.
3,744,468 cons.
3,132 binary
LP Relaxation objective: 939,208
LP Relaxation + CglProbing cuts: 939,626
Feasible solution (rounding): 942,237
Optimality gap: 0.27% (0.5% is acceptable in industry practice)
Lagrangian relaxation: 941,176
Feasible solution (rounding): 943,351
Optimality gap: 0.23% (combined: 0.11%)

28. Conclusions and future work
Revisited dual decomposition from the perspective of parallel computation, addressed bottleneck of solving master
Dual decomposition promising approach for parallel solution of stochastic mixed-integer programs
More work needed to address load imbalance, perhaps asynchronism as in Linderoth and Wright
Branch and bound implementation
Large scale computational study for stochastic unit commitment

29. Weather System Operator ON/OFF Generation Levels Demand Distribution
Supplier
Weather
Consumer