Algoritmos para problemas de programação não-linear com variáveis inteiras e contínuas [thesis]

Rafael Durbano Lobato
Agradecimentos Agradeço ao meu pai eà minha mãe por todo o apoio que recebi durante toda a minha vida. Quaisquer palavras que eu usasse aqui não seriam suficientes para expressar todo o meu agradecimento. Ao meu orientador, professor Ernesto Birgin, por sua atenção em todas as nossas reuniões, por sua motivação, por seu profundo interesse e dedicação a este trabalho. Também agradeço aos professores Carlos Eduardo Ferreira, Marcelo Gomes de Queiroz e Francisco Gomes pelos comentários valiosos
more » ... ntários valiosos que contribuíram para o aperfeiçoamento desta dissertação. Aos meus amigos do BCC 2003, pelas diversas partidas de boliche e de futebol no Cepê, e a todo o pessoal da "salinha", pela companhia e amizade ao longo desses anos. i ii Palavras-chave: programação não-linear, programação não-linear inteira mista, Lagrangianos Aumentados, variáveis inteiras. iii iv Abstract Many optimization problems contain both integer and continuous variables and can be modeled as mixed-integer nonlinear programming problems. Problems of this nature appear frequently in chemical engineering and include, for instance, process synthesis, design of distillation columns, heat exchanger network synthesis and oil and gas production. In this work, we present algorithms based on Augmented Lagrangians and branch and bound for solving mixed-integer nonlinear programming problems. Two approaches are considered. In the first one, an Augmented Lagrangian algorithm is used for solving nonlinear programming problems that appear at each node in the branch and bound method. In the second approach, we use a branch and bound method for solving boxconstrained problems with integer variables that appear as subproblems of the Augmented Lagrangian algorithm. Both algorithms guarantee to find an optimal solution for convex problems and have appropriate strategies to deal with non-convex problems, although there is no guarantee of optimality in this case. We present a problem of packing rectangles within an arbitrary convex region and propose models for this problem that result in nonlinear programs with integer and continuous variables. We have performed some numerical experiments and compared the results reached by the method described in this work and the results obtained by other methods. We have also performed experiments with mixed-integer nonlinear programming problems found in the literature and compared the performance of our method to that of other method publicly available.
doi:10.11606/d.45.2009.tde-06072009-130912 fatcat:ykzd5vujunh6vmgxfu7urln3a4