Abstract
Nonconvex optimization problems, such as those often seen in chemical engineering applications due to integer decisions and inherent system physics, are known to scale poorly with problem size. Decomposition methods, such as Benders or Lagrangean decomposition, offer much promise for improving solution efficiency. However, automatically identifying subproblems for use with these solution methods remains an open problem. Herein, a general algorithmic framework entitled Detection of Communities for Optimization Decomposition (DeCODe) is presented. DeCODe uses community detection, a concept originating from network theory, to generate optimization subproblems which are strongly interacting within individual subproblems but weakly interacting between different ones. The importance of communities in solving problems using decomposition solution methods is showcased via a least squares regression problem. The ability of DeCODe to identify nontrivial decompositions of optimization problems is demonstrated through a large renewable energy and chemical production optimal design problem and two mixed integer nonlinear program test problems.
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References
Allman A, Daoutidis P (2017) Optimal design of synergistic distributed renewable fuel and power systems. Renew Energy 100:78–89
Allman A, Daoutidis P (2018) Optimal scheduling for wind-powered ammonia generation: effects of key design parameters. Chem Eng Res Des 131:5–15
Allman A, Tiffany D, Kelley S, Daoutidis P (2017) A framework for ammonia supply chain optimization incorporating conventional and renewable generation. AIChE J 63:4390–4402
Allman A, Tang W, Daoutidis P (2018a) Towards a generic algorithm for identifying high-quality decompositions of optimization problems. Comput Aided Chem Eng 44:943–948
Allman A, Palys MJ, Daoutidis P (2018b) Scheduling-informed optimal design of systems with time-varying operation: a wind-powered ammonia case study. AIChE J. https://doi.org/10.1002/aic.16434. (in press)
Allman A, Tang W, Daoutidis P (2018c) Community detection method for decomposition for optimization problems. http://license.umn.edu/technologies/20180354_community-detection-method-for-decomposition-for-optimization-problems. Accessed June 2019
Alshamsi A, Diabat A (2018) Large-scale reverse supply chain network design: an accelerated benders decomposition algorithm. Comput Chem Eng 124:545–559
Benders JF (1962) Partitioning procedures for solving mixed-variables programming problems. Numer Math 4:238–252
Blondel VD, Guillaume JL, Lambiotte R, Lefebvre E (2008) Fast unfolding of communities in large networks. J Stat Mech Theory Exper 2008:P10008
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 3:1–122
Brunaud B, Lainez-Aguirre JM, Pinto JM, Grossmann IE (2018) Inventory policies and safety stock optimization for supply chain planning. AIChE J. https://doi.org/10.1002/aic.16421. (in press)
Chu Y, You F, Wassick JM, Agarwal A (2015) Integrated planning and scheduling under production uncertainties: bi-level model formulation and hybrid solution method. Comput Chem Eng 72:255–272
Conejo AJ, Castillo E, Minguez R, Garcia-Bertrand R (2006) Decomposition techniques in mathematical programming: engineering and science applications. Springer, Berlin
Dantzig GB, Wolfe P (1960) Decomposition principle for linear programs. Oper Res 8:101–111
Daoutidis P, Jogwar SS, Tang W (2018) Decomposing complex plants for distributed control: perspectives from network theory. Comput Chem Eng 114:43–51
del Rio-Chanona EA, Fiorelli F, Vassiliadis VS (2016) Automated structure detection for distributed process optimization. Comput Chem Eng 89:135–148
Florez-Quiroz A, Palma-Behnke R, Zakeri G, Moreno R (2016) A column generation approach for solving generation expansion planning problems with high renewable energy penetration. Electr Power Syst Res 136:232–241
Fortunato S, Hric D (2016) Community detection in networks: a user guide. Phys Rep 659:1–44
Geoffrion AM (1972) Generalized benders decomposition. J Opt Theory Appl 10:237–260
Girvan M, Newman MEJ (2002) Community structure in social and biological networks. Proc Natl Acad Sci 99:7821–7826
Gondzio J, Grothey A (2007) Parallel interior-point solver for structured quadratic programs: application to financial planning problems. Ann Oper Res 152:319–339
Grossmann IE (2012) Advances in mathematical programming models for enterprise-wide optimization. Comput Chem Eng 47:2–18
Guignard M, Kim S (1987) Lagrangean decomposition: a model yielding stronger Lagrangean bounds. Math Program 39:215–228
Jalving J, Cao Y, Zavala VM (2019) Graph-based modeling and simulation of complex systems. Comput Chem Eng 125:134–154
Jiang Y, Rodriguez MA, Harjunkoski I, Grossmann IE (2014) Optimal supply chain design and management over a multi-period horizon under demand uncertainty. Part II: a lagrangean decomposition algorithm. Comput Chem Eng 62:211–224
Jogwar SS, Daoutidis P (2017) Community-based synthesis of distributed control architectures for integrated process networks. Chem Eng Sci 172:434–443
Karypis G, Kumar V (1998) Multilevel k-way partitioning scheme for irregular graphs. J Parallel Distrib Comput 48:96–129
Khaniyev T, Elhedlhi S, Erenay FS (2018) Structure detection in mixed integer programs. INFORMS J Comput 30:570–587
Lara CL, Mallapragada DS, Papageorgiou DJ, Venkatesh A, Grossmann IE (2018) Deterministic electric power infrastructure planning: mixed-integer programming model and nested decomposition algorithm. Eur J Oper Res 271:1037–1054
Marti R, Lucia S, Sarabia D, Paulen R, Engall S, de Prada C (2015) Improving scenario decomposition algorithms for robust nonlinear model predictive control. Comput Chem Eng 79:30–45
MINLP World: MINLP library. www.gamsworld.org/minlp/minlplib.htm
Moharir M, Kang L, Daoutidis P, Almansoori A (2017) Graph representation and decomposition of ODE/hyperbolic PDE systems. Comput Chem Eng 106:532–543
Moharir M, Pourkargar DB, Almansoori A, Daoutidis P (2018) Distributed model predictive control of an amine gas sweetening plant. Indus Eng Chem Res 57:13103–13115
Newman MEJ (2006) Modularity and community structure in networks. Proc Natl Acad Sci 103:8577–8582
Pourkargar DB, Almansoori A, Daoutidis P (2018) Impact of decomposition on distributed model predictive control. Ind Eng Chem Res 56:9606–9616
Schweidtmann AM, Mitsos A (2018) Deterministic global optimization with artificial neural networks embedded. J Opt Theory Appl. https://doi.org/10.1007/s10957-018-1396-0. (in press)
Tang W, Daoutidis P (2018) Network decomposition for distributed control through community detection in input–output bipartite graphs. J Process Control 64:7–14
Tang W, Allman A, Pourkargar DB, Daoutidis P (2018a) Optimal decomposition for distributed optimization in nonlinear model predictive control through community detection. Comput Chem Eng 111:43–54
Tang W, Pourkargar DB, Daoutidis P (2018b) Relative time averaged gain array (RTAGA) for distributed control-oriented network decomposition. AIChE J 64:1682–1690
Tawarmalani M, Sahinidis NV (2005) A polyhedral branch-and-cut approach to global optimization. Math Program 103(2):225–249
Zavala VM, Laird CD, Biegler LT (2008) Interior-point decomposition approaches for parallel solution of large-scale nonlinear parameter estimation problems. Chem Eng Sci 63:4834–4845
Zhang J, Chmielewski DJ (2017) Value-optimal sensor network design for steady-state and closed-loop systems using the generalized benders decomposition. Ind Eng Chem Res 56:11860–11869
Zhang Q, Grossmann IE (2016) Enterprise-wide optimization for industrial demand side management: fundamentals, advances, and perspectives. Chem Eng Res Des 116:114–131
Zhang Q, Grossmann IE, Sundaramoorthy A, Pinto JM (2016) Data-driven construction of convex region surrogate models. Optim Eng 17:289–332
Zhang Y, Wang J, Zeng B, Hu Z (2017) Chance-constrained two-stage unit commitment under uncertain load and wind power output using bilinear Benders decomposition. IEEE Trans Power Syst 32:3637–3647
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Financial support from the National Science Foundation and the University of Minnesota Initiative for Renewable Energy and the Environment (IREE) is gratefully acknowledged.
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Allman, A., Tang, W. & Daoutidis, P. DeCODe: a community-based algorithm for generating high-quality decompositions of optimization problems. Optim Eng 20, 1067–1084 (2019). https://doi.org/10.1007/s11081-019-09450-5
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DOI: https://doi.org/10.1007/s11081-019-09450-5