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Semidefinite relaxations and lagrangean duality in truss topology design problem

Adelaide Cerveira, Fernando Bastos

Abstract


In this paper we consider the classical problem of finding the stiffest truss under a given load and with a volume constraint. This is a well-studied problem for continuous cross sectional areas. Generally, the optimal solutions obtained contain bars with many different cross sectional areas. In real life we have only a finite set of possible values for those areas, so it is important to consider discrete constraints. In this paper we consider the design problem for a single cross sectional area value case i.e., if the bar belongs to the structure its cross-sectional area has a given value. There is no loss of generality since the discrete problem can be formulated as a binary problem. We derive semidefinite relaxations and lagrangean relaxations for this problem. The aim of this paper is to improve the bound obtained from linear relaxation in order to be used in a branch and bound framework. As we will see, the best bound is provided by lagrangean relaxations.

Keywords


truss topology design (TTD), stiffness, semidefinite programming (SDP), relaxation, duality

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