Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-09-04T12:20:42Z No. of bitstreams: 1 erickmariodonascimentooliveira.pdf: 3492871 bytes, checksum: 845fa85f6d95efe2e7ad13563f342bc3 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-09-04T13:21:49Z (GMT) No. of bitstreams: 1 erickmariodonascimentooliveira.pdf: 3492871 bytes, checksum: 845fa85f6d95efe2e7ad13563f342bc3 (MD5) / Made available in DSpace on 2018-09-04T13:21:49Z (GMT). No. of bitstreams: 1 erickmariodonascimentooliveira.pdf: 3492871 bytes, checksum: 845fa85f6d95efe2e7ad13563f342bc3 (MD5) Previous issue date: 2018-06-26 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho é apresentado o comportamento do algoritmo IED quando aplicado a problemas de programação em dois níveis. Para isso, o problema do seguidor é substituído pelas condições necessárias de primeira ordem de Karush-Kuhn-Tucker e, dessa maneira, o problema de programação em dois níveis é transformado em um problema de otimização com restrições não lineares. Dessa forma, as condições necessárias para utilização do algoritmo IED (Interior Epigraph Directions) são satisfeitas. Esse método tem como característica resolver problemas de otimização não convexa e não diferenciáveis via utilização da técnica de dualidade Lagrangiana, onde as funções de restrições são introduzidas na função objetivo para formar a função Lagrangiana. Além disso, o método considera o problema dual induzido por um esquema generalizado da dualidade Lagrangiana aumentada e obtém a solução primal produzindo uma sequência de pontos no interior do epígrafo da função dual. Dessa forma, o valor da função dual, em algum ponto do espaço dual, é dado pela minimização da Lagrangiana. Por fim, experimentos numéricos são apresentados em relação à utilização do algoritmo IED em problemas de programação em dois níveis encontrados na literatura. / This work presents the behavior of the IED algorithm when applied to bilevel programming problems. For this, the follower problem is replaced by the first-order necessary Karush-Kuhn-Tucker’s conditions and thus, the problem of bilevel programming turns into an optimization problem with non-linear constraints. Thus, the conditions required for use of the IED (Interior Epigraph Directions) algorithm are satisfied. This method has the characteristic of solving non-convex and non-differentiable optimization problems using the Lagrangian duality technique, where the constraint functions are introduced into the objective function for formulation of the Lagrangian. Furthermore, the method considers the dual problem induced by a generalized scheme of augmented Lagrangian duality and obtains the primal solution by producing a sequence of points inside the dual function epigraph. Then the value of the dual function, at some point in the dual space, is given by Lagrangian minimization. Finally, numerical experiments are presented showing the use of the IED algorithm in bilevel programming problems found in the literature.

This thesis presents the mixed integer bilevel programming problems where some optimality conditions and solution algorithms are derived. Bilevel programming problems are optimization problems which are partly constrained by another optimization problem. The theoretical part of this dissertation is mainly based on the investigation of optimality conditions of mixed integer bilevel program. Taking into account both approaches (optimistic and pessimistic) which have been developed in the literature to deal with this type of problem, we derive some conditions for the existence of solutions. After that, we are able to discuss local optimality conditions using tools of variational analysis for each different approach. Moreover, bilevel optimization problems with semidefinite programming in the lower level are considered in order to formulate more optimality conditions for the mixed integer bilevel program. We end the thesis by developing some algorithms based on the theory presented

This thesis presents the mixed integer bilevel programming problems where some optimality conditions and solution algorithms are derived. Bilevel programming problems are optimization problems which are partly constrained by another optimization problem. The theoretical part of this dissertation is mainly based on the investigation of optimality conditions of mixed integer bilevel program. Taking into account both approaches (optimistic and pessimistic) which have been developed in the literature to deal with this type of problem, we derive some conditions for the existence of solutions. After that, we are able to discuss local optimality conditions using tools of variational analysis for each different approach. Moreover, bilevel optimization problems with semidefinite programming in the lower level are considered in order to formulate more optimality conditions for the mixed integer bilevel program. We end the thesis by developing some algorithms based on the theory presented