钢框架结构离散优化问题的理论下界

Theoretical Lower Bound on Discrete Optimization Problem of Steel Frame

针对两种典型的钢框架结构离散优化问题,即柔度约束的最小体积问题和体积约束的最小柔度问题,提出了基于凸组合的线性松弛方法,将关联离散变量进行线性松弛,进而将非线性、非凸的离散优化问题转化为松弛的凸规划问题.其中,体积约束的最小柔度问题可松弛为二阶锥规划问题,柔度约束的最小体积问题可松弛为半定规划问题.采用成熟的优化求解器,就可以得到两类凸规划问题的全局最优解,也就是原离散优化问题的理论下界.以一跨四层钢框架的离散优化问题为例,用所提出方法进行求解,并用枚举法和遗传算法对优化结果进行验证.数值结果证明,所提出方法可以快速得到离散优化问题的理论下界.

Aiming at two typical discrete optimization problems of steel frame,namely,the volume minimization with compliance constraint and the compliance minimization with volume constraint,a linear relaxation approach based on convex combination is proposed.Meanwhile,the linked discreteness of design variables is also relaxed lin⁃early,and the original nonlinear and nonconvex problems are recast as relaxed convex programming problems.Spe⁃cifically,the compliance minimization with volume constraint is reestablished as a second-order cone programming,and the volume minimization with compliance constraint is reformulated as a semidefinite programming.The global optimum solutions of two types of convex programming problems can be readily derived using existing mature optimi⁃zation solvers.These global optimum solutions are also the theoretical lower bound for the discrete optimization prob⁃lems.An example of a one-bay four-story frame is presented,and the results by the proposed approach are compared with the solutions by complete enumeration and genetic algorithm.The comparison demonstrates that the proposed approach is capable of achieving the theoretical lower bound in an efficient manner.