Piece-by-piece approach to solving large nonlinear water resources management models

Most large nonlinear optimization models are composed of “pieces”—subsets of decision variables—and constraints whose union is the entire model. Each piece represents an additional aspect of the situation being modeled. This opens the possibility of solving the simplest piece first, adding the constraints and variables of another piece, and solving this submodel from a starting point provided by the first solution. This process is repeated until the original model is solved. This “piece-by-piece” approach provides each submodel with a good starting point, which greatly increases the probability that a good nonlinear solver will find an optimal solution. We apply it to a large multiperiod nonlinear programming (NLP) model with 13,700 variables, 10,000 equations, and a high degree of nonlinearity (54.3% of the nonzero Jacobian elements are nonconstant), arising from water resources planning and operation in a river basin. Using the GAMS modeling language and the CONOPT2 NLP solver, the piece-by-piece method...

[1]  Pedro J. Restrepo,et al.  Dual dynamic programing: A note on implementation , 1999 .

[2]  D. McKinney,et al.  Decomposition methods for water resources optimization models with fixed costs , 1998 .

[3]  M. Pereira,et al.  Stochastic Optimization of a Multireservoir Hydroelectric System: A Decomposition Approach , 1985 .

[4]  K. D. W. Nandalal,et al.  Optimal operation of a reservoir for quality control using inflows and outflows , 1995 .

[5]  Ignacio E. Grossmann,et al.  Systematic Methods of Chemical Process Design , 1997 .

[6]  Ismail Serageldin.  Water Resources Management: A New Policy for a Sustainable Future , 1995 .

[7]  Ximing Cai,et al.  A Modeling Framework for Sustainable Water Resources Management , 1999 .

[8]  A. Ben-Tal,et al.  Optimal design of water distribution networks , 1994 .

[9]  Guna N. Paudyal,et al.  Irrigation planning by multilevel optimization , 1990 .

[10]  Bruce A. McCarl,et al.  A Nonlinear Mixed Integer Program Model for Evaluating Runoff Impoundments for Supplemental Irrigation , 1995 .

[11]  Prabhata K. Swamee,et al.  Decomposition of Large Water Distribution Systems , 1990 .

[12]  R. Young,et al.  Modeling Intrastate and Interstate Markets for Colorado River Water Resources , 1994 .

[13]  Augustine O. Esogbue,et al.  Dynamic Programming for Optimal Water Resource Systems Analysis , 1989 .

[14]  Yacov Y. Haimes,et al.  Optimal maintenance-related decision making for deteriorating water distribution systems: 2. Multilevel decomposition approach , 1992 .

[15]  Arne Stolbjerg Drud,et al.  CONOPT - A Large-Scale GRG Code , 1994, INFORMS J. Comput..

[16]  Steven M. Gorelick,et al.  Benefits of an irrigation water rental market in a saline stream-aquifer system , 1990 .

[17]  Yacov Y. Haimes,et al.  Hierarchical analyses of water resources systems : modeling and optimization of large-scale systems , 1977 .

[18]  Leon S. Lasdon,et al.  Solving Large Nonconvex Water Resources Management Models Using Generalized Benders Decomposition , 2001, Oper. Res..

[19]  Leon S. Lasdon,et al.  Optimization Theory of Large Systems , 1970 .

[20]  André Turgeon.  A decomposition method for the long-term scheduling of reservoirs in series , 1981 .

[21]  Michael A. Saunders,et al.  A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints , 1982 .

[22]  A. M. Geoffrion.  Generalized Benders decomposition , 1972 .

[23]  David Kendrick,et al.  GAMS, a user's guide , 1988, SGNM.

[24]  S. Gorelick,et al.  Simulating physical processes and economic behavior in saline, irrigated agriculture: model development , 1990 .