求解一类可分离凸规划的对偶显式模型DP-EM方法

发布时间：2018-07-25 09:38

【摘要】：推导对偶目标函数的精确显式表达式,可选用更多成熟高效的求解方法,从而进一步提高了非线性规划对偶理论求解结构拓扑优化问题的效率.研究工作来源于非线性凸规划同其对偶规划的间隙为零,可以等价转化为对偶问题求解,通常可以大大地缩小问题的规模,可是二者不具有显式关系却影响了对偶解法的应用.所幸的是,结构优化当中一大类问题包括连续体结构拓扑优化问题,不仅具有凸性,而且具有变量可分离性,于是原变量和对偶变量之间有了显式关系,因此,对偶解法成了38年来被应用的有效方法之一.然而长期以来,对偶问题的目标函数并不是显式,这缘于含参数的极小化问题导致目标函数为隐式表达,常见的显式化方法是进行二阶近似.本文突破了对偶问题难以显式化只能采用近似显式的定势,将我们提出的"对偶规划-显式模型"(DP-EM)方法应用于连续体结构拓扑优化,并与对偶序列二次规划(DSQP)算法及移动渐近线(MMA)算法为求解器的方法进行计算效率对比,结果显示:(1)MMA算法比DP-EM算法和DSQP算法的外部迭代次数均多;(2)DP-EM算法与DSQP算法外循环次数相同,而内循环数显著减少.说明了DP-EM算法具有显式对偶函数的优势.
[Abstract]:The exact explicit expression of dual objective function can be derived and more mature and efficient methods can be used to solve the structural topology optimization problem. Thus the efficiency of nonlinear dual programming theory in solving structural topology optimization problems is further improved. The research work comes from the fact that the gap between the nonlinear convex programming and its dual programming is zero, which can be equivalent to solving the dual problem, and usually can greatly reduce the scale of the problem. However, the application of the dual solution is affected by the fact that the two problems have no explicit relation. Fortunately, one of the major problems in structural optimization includes continuum topology optimization, which is not only convexity but also variable separability, so there is an explicit relationship between the original variable and the dual variable. Dual solution has become one of the effective methods used in 38 years. However, for a long time, the objective function of the dual problem is not explicit, which is caused by the parameter minimization problem, which leads to the implicit expression of the objective function. The common explicit method is the second-order approximation. In this paper, we break through the fact that the dual problem is difficult to be explicit and can only adopt the approximate explicit setting. We apply the dual program-explicit model (DP-EM) method to the topology optimization of continuum structure. Compared with dual sequence quadratic programming (DSQP) algorithm and moving asymptote (MMA) algorithm as solver, the results show that: (1) MMA algorithm has more external iterations than DP-EM algorithm and DSQP algorithm; (2) DP-EM algorithm has the same number of outer cycles as DSQP algorithm. The number of internal cycles decreased significantly. It is shown that the DP-EM algorithm has the advantage of explicit dual function.
【作者单位】：北京工业大学工程数值模拟中心;湖南城市学院土木工程学院;
【基金】：国家自然科学基金(11672103) 湖南省自然科学基金(2016JJ6016)资助项目
【分类号】：O221

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