The discrete topology and sizing optimization of frame structures with compliance constraints is studied using a novel approach, which is capable of finding the theoretical lower bounds and high-quality discrete solutions in an efficient manner. The proposed approach works by reformulating the discrete problem as a relaxed semidefinite programming (SDP) problem. This reformulation is made possible by a linear relaxation of the original discrete space and the elimination of the nonconvex equilibrium equation using a semidefinite constraint. A continuous global optimum is first derived using existing solvers and then the discrete solution is discovered by the neighborhood search. Numerical examples are presented, including the sizing optimization of 2-Bay 6-Story frame and 3-Bay 10-Story frame, the topology and sizing optimization of 2-Bay 6-Story braced frame. A topology and sizing example with multiple load cases is also provided. The proposed approach and three other metaheuristic algorithms are used to solve these examples. Theoretical lower bounds for these examples can be efficiently discovered by the proposed approach. For the sizing problems, the discrete solutions by the proposed approach are all better than the other algorithms. For the topology and sizing problems, the proposed approach achieves discrete solutions better than genetic algorithm, but worse than the other metaheuristics. The computational superiority of the proposed approach is validated in all the examples.

利用一种新颖的方法研究了具有柔顺性约束的框架结构的离散拓扑和尺寸优化，该方法能够有效地找到理论下界和高质量离散解决方案。所提出的方法通过将离散问题重新构造为宽松的半定规划（SDP）问题而起作用。通过对原始离散空间进行线性弛豫并使用半定约束消除非凸平衡方程，可以实现这种重构。首先使用现有的求解器得出连续的全局最优解，然后通过邻域搜索发现离散解。给出了数值示例，包括2层6层框架和3层10层框架的尺寸优化，2-Bay 6层支撑框架的拓扑和尺寸优化。还提供了具有多个负载工况的拓扑和大小调整示例。所提出的方法和其他三种元启发式算法用于解决这些示例。通过所提出的方法可以有效地发现这些例子的理论下界。对于尺寸问题，所提方法的离散解均优于其他算法。对于拓扑和大小问题，所提出的方法比遗传算法要好，但比其他元启发法要差。在所有示例中都验证了所提出方法的计算优势。所提出的方法和其他三种元启发式算法用于解决这些示例。通过所提出的方法可以有效地发现这些例子的理论下界。对于尺寸问题，所提方法的离散解均优于其他算法。对于拓扑和大小问题，所提出的方法比遗传算法要好，但比其他元启发法要差。在所有示例中都验证了所提出方法的计算优势。所提出的方法和其他三种元启发式算法用于解决这些示例。通过所提出的方法可以有效地发现这些例子的理论下界。对于尺寸问题，所提方法的离散解均优于其他算法。对于拓扑和大小问题，所提出的方法比遗传算法要好，但比其他元启发法要差。在所有示例中都验证了所提出方法的计算优势。所提出的方法比遗传算法更好地实现离散解，但比其他元启发式方法差。在所有示例中都验证了所提出方法的计算优势。所提出的方法比遗传算法更好地实现离散解，但比其他元启发式方法差。在所有示例中都验证了所提出方法的计算优势。