Global Weight Optimization of Frame Structures under Free-Vibration Eigenvalue Constraints
把框架结构在振动特征值约束下的重量优化建模为多项式半定规划，用Lasserre层级求全局最优，并给出上下界和最优性判定，适合结构工程和优化算法研究者。
Abstract. Topology optimization of frame structures under free-vibration eigenvalue constraints constitutes a challenging nonconvex polynomial optimization problem with disconnected feasible sets. In this article, we first formulate it as a polynomial semidefinite programming problem (SDP) of minimizing a linear function over a basic semi-algebraic feasible set. We then propose to solve this problem by the Lasserre hierarchy of linear semidefinite relaxations providing a sequence of increasing lower bounds. To obtain also a sequence of upper bounds and thus conditions on global [Formula: see text]-optimality, we provide a bilevel reformulation that exhibits a special structure: The lower level is quasi-convex univariate, and it has a nonempty interior if the constraints of the upper-level problem are satisfied. After deriving the conditions for the solvability of the lower-level problem, we thus provide a way to construct feasible points to the original SDP. Using such a feasible point, we modify the original nonlinear SDP to satisfy the conditions for the deployment of the Lasserre hierarchy. Solving arbitrary degree relaxation of the hierarchy, we prove that scaled first-order moments associated with the problem variables satisfy feasibility conditions for the lower-level problem and thus provide guaranteed upper and lower bounds on the objective function. Using these bounds, we develop a simple sufficient condition for global [Formula: see text]-optimality and prove that the optimality gap [Formula: see text] converges to zero if the set of global minimizers is convex. Finally, we illustrate these results with four representative problems for which the hierarchy converges in at most five relaxation degrees.