Global weight optimization of frame structures under free-vibration eigenvalue constraints

Minimizing the weight in topology optimization of frame structures under free-vibration eigenvalue constraints constitutes a challenging nonconvex polynomial optimization problem with strong singularities in the feasible set. Here, we adopt a nonlinear semidefinite programming formulation, which consists of a minimization of a linear function over a basic semi-algebraic feasible set, and provide its bilevel reformulation. This bilevel program maintains a special structure: The lower level is univariate and quasiconvex, and the upper level is enumerative. After deriving the sufficient and necessary conditions for the solvability of the lower-level problem, we provide a way to construct feasible points to the original semidefinite program, and using such a feasible point, we show that the conditions for convergence of the Lasserre hierarchy are met. Moreover, we show how to construct lower and upper bounds for each level of the Lasserre hierarchy. Using these bounds, we develop a simple sufficient condition of global -optimality. Finally, we prove that the optimality gap converges to zero in the limit if the set of global minimizers is convex. We demonstrate these results with three representative problems, for which the hierarchy indeed converges in a finite number of steps.