On the trajectories of the epsilon-relaxation approach for stress-constrained truss topology optimization

We consider the nonconvex problem of minimizing the weight of a linearly elastic truss structure subject to stress constraints under multiple load conditions. The design variables are the cross-sectional areas of the elements, and the stress constraints are imposed only on elements with strictly positive areas. To avoid degenerate feasible domains, it has been suggested that the stress constraints of the original problem should be relaxed by a positive scalar ε, leading to the so-called ε-relaxed problem. In this paper, the trajectories associated with optimal solutions of the ε-relaxed problems, for continuously decreasing values of ε, are studied in detail on some carefully chosen examples. The global trajectory is defined as the path followed by the global optimal solution to the ε-relaxed problem, and we present two parameterized examples for which the global trajectory is discontinuous for arbitrarily small values of ε>0. From that we conclude that, in practice, a sequence of solutions to the ε-relaxed problem for decreasing values on ε may not converge to the global optimal solution of the original problem, even if the starting point is on the global trajectory.