The paper considers the classical problem of finding a truss design
with minimal compliance subject to a given external force and a volume
bound. Feasible structures are defined through the ground structure
approach. While this problem is well-studied for continuous bar areas,
we treat here the case of discrete areas. This problem is of big
practical relevance if the truss must be built from pre-produced bars
with given areas. As a special case, we treat the design problem for a
single bar area, i.e., a 0/1-problem.
In contrast to heuristic methods considered in other approaches,
this paper together with Part I presents an algorithmic framework for
the calculation of a global optimizer of the underlying large-scaled
mixed integer design problem. This framework is given by a convergent
branch-and-bound algorithm which is based on straightforward continuous
relaxations of the bar areas. The main issue of the paper and of the
approach lies in the fact that the relaxed nonlinear optimization problem
can be formulated as a quadratic programming problem QP. Although the
Hessian of this QP is indefinite, it is possible to circumvent non-convexity
and to calculate global optimizers. Moreover, the QPs to be solved in the
branch-and-bound search tree differ from each other just in the objective
function. Therefore, very good starting points are available. This makes
the resulting branch-and-bound methodology very efficient.
In Part II we use the theory devolped in Part I for the setup of the
branch-and-bound method tailored to the original problem. We focus on
implementation details and finite convergence. The paper closes with several
large-scale numerical examples. To the knowledge of the authors, these
examples are by far the largest ones treated by means of global optimization.