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 is able to solve this model, while all attempts to solve the complete model from various starting points fail to find a feasible solution.