# Decomposition strategy for the stochastic pooling problem

## Author

Listed:
• Xiang Li
• Asgeir Tomasgard
• Paul Barton

()

## Abstract

The stochastic pooling problem is a type of stochastic mixed-integer bilinear program arising in the integrated design and operation of various important industrial networks, such as gasoline blending, natural gas production and transportation, water treatment, etc. This paper presents a rigorous decomposition method for the stochastic pooling problem, which guarantees finding an $${\epsilon}$$ -optimal solution with a finite number of iterations. By convexification of the bilinear terms, the stochastic pooling problem is relaxed into a lower bounding problem that is a potentially large-scale mixed-integer linear program (MILP). Solution of this lower bounding problem is then decomposed into a sequence of relaxed master problems, which are MILPs with much smaller sizes, and primal bounding problems, which are linear programs. The solutions of the relaxed master problems yield a sequence of nondecreasing lower bounds on the optimal objective value, and they also generate a sequence of integer realizations defining the primal problems which yield a sequence of nonincreasing upper bounds on the optimal objective value. The decomposition algorithm terminates finitely when the lower and upper bounds coincide (or are close enough), or infeasibility of the problem is indicated. Case studies involving two example problems and two industrial problems demonstrate the dramatic computational advantage of the proposed decomposition method over both a state-of-the-art branch-and-reduce global optimization method and explicit enumeration of integer realizations, particularly for large-scale problems. Copyright Springer Science+Business Media, LLC. 2012

## Suggested Citation

• Xiang Li & Asgeir Tomasgard & Paul Barton, 2012. "Decomposition strategy for the stochastic pooling problem," Journal of Global Optimization, Springer, vol. 54(4), pages 765-790, December.
• Handle: RePEc:spr:jglopt:v:54:y:2012:i:4:p:765-790
DOI: 10.1007/s10898-011-9792-0
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File URL: http://hdl.handle.net/10.1007/s10898-011-9792-0

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## References listed on IDEAS

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1. Arthur M. Geoffrion, 1970. "Elements of Large-Scale Mathematical Programming Part I: Concepts," Management Science, INFORMS, vol. 16(11), pages 652-675, July.
2. Arthur M. Geoffrion, 1970. "Elements of Large Scale Mathematical Programming Part II: Synthesis of Algorithms and Bibliography," Management Science, INFORMS, vol. 16(11), pages 676-691, July.
3. John R. Birge, 1985. "Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs," Operations Research, INFORMS, vol. 33(5), pages 989-1007, October.
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## Citations

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Cited by:

1. Frank, Stephen M. & Rebennack, Steffen, 2015. "Optimal design of mixed AC–DC distribution systems for commercial buildings: A Nonconvex Generalized Benders Decomposition approach," European Journal of Operational Research, Elsevier, vol. 242(3), pages 710-729.
2. Akshay Gupte & Shabbir Ahmed & Santanu S. Dey & Myun Seok Cheon, 2017. "Relaxations and discretizations for the pooling problem," Journal of Global Optimization, Springer, vol. 67(3), pages 631-669, March.
3. Boukouvala, Fani & Misener, Ruth & Floudas, Christodoulos A., 2016. "Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO," European Journal of Operational Research, Elsevier, vol. 252(3), pages 701-727.
4. Ríos-Mercado, Roger Z. & Borraz-Sánchez, Conrado, 2015. "Optimization problems in natural gas transportation systems: A state-of-the-art review," Applied Energy, Elsevier, vol. 147(C), pages 536-555.

### Keywords

Nonconvex mixed-integer nonlinear programming; Stochastic programming; Stochastic pooling problem; Decomposition; Large-scale; 90C26; 90C15;

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