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An efficient method for nonlinearly constrained networks
Institution:1. Department of Computer Science, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, Belo Horizonte 31270-901, Minas Gerais, Brazil;2. Department of Computer Science, Universidade Federal Fluminense, R. Passo da Pátria 156, Niterói, 24210-240 Rio de Janeiro, Brazil;3. Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland;1. Department of Mathematics and Calculus Institute, FCEN, University of Buenos Aires, Ciudad Universitaria, C1428EGA CABA, Argentina;2. Department of Business and Management Science, NHH Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway;3. Department of Industrial Engineering, FCFM, University of Chile, Beauchef 851, Santiago, Chile;4. CONICET, Godoy Cruz 2280 C1425FOB, CABA, Argentina;1. SICS, P.O. Box 1263, Kista SE-164 29, Sweden;2. Automatic Control Lab, KTH, Osquldas väg 10, Stockholm SE-100 44, Sweden;3. MCS Division, Argonne National Laboratory, Lemont, IL 60439, USA;1. Department of Mathematical Sciences “G. L. Lagrange”, Dipartimento di Eccellenza 2018-2022, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy;2. Department of Mathematics “G. Peano”, Via Carlo Alberto 10, Torino 10123, Italy;3. Optiflow Company, 160 Chemin de la Madrague-Ville, Marseille 13015, France
Abstract:Many nonlinear network flow problems (in addition to the balance constraints in the nodes and capacity constraints on the arc flows) have nonlinear side constraints, which specify a flow relationship between several of the arcs in the network flow model. The short-term hydrothermal coordination of electric power generation is an example of this type. In this work we solve this kind of problem using an approach in which the efficiency of the well-known techniques for network flow can be preserved. It lies in relaxing the side constraints in an augmented Lagrangian function, and minimizing a sequence of these functions subject only to the network constraints for different estimates of the Lagrange multipliers of the side constraints. This method gives rise to an algorithm, which combines first- and superlinear-order multiplier methods to estimate these multipliers. When the number of free variables is very high we can obtain a superlinear-order estimate by means of the limited memory BFGS method fitted to our problem. An extensive computational comparison with other methods has been performed. The numerical results reported indicate that the algorithm described may be employed advantageously to solve large-scale network flow problems with nonlinear side constraints.
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