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 引用本文: 隋允康, 彭细荣. 求解一类可分离凸规划的对偶显式模型DP-EM方法[J]. 力学学报, 2017, 49(5): 1135-1144.
Sui Yunkang, Peng Xirong. A DUAL EXPLICIT MODEL BASED DP-EM METHOD FOR SOLVING A CLASS OF SEPARABLE CONVEX PROGRAMMING[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(5): 1135-1144.
 Citation: Sui Yunkang, Peng Xirong. A DUAL EXPLICIT MODEL BASED DP-EM METHOD FOR SOLVING A CLASS OF SEPARABLE CONVEX PROGRAMMING[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(5): 1135-1144.

## A DUAL EXPLICIT MODEL BASED DP-EM METHOD FOR SOLVING A CLASS OF SEPARABLE CONVEX PROGRAMMING

• 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: An explicit exact formula is derived for the objective function of the dual model of a class of separable convex programming problems. It makes more mature and efficient methods can be chose to solve the dual model. Therefore, the advantage of applying the duality theory of nonlinear programming to efficiently solve structural topology optimization problems is fully exploited. The research work is rooted in that the gap of a nonlinear convex programming with its dual programming is zero. Solving original programming can be equivalently transformed into solving its dual programming. The scale of the solved programming can usually be reduced greatly. But an explicit relationship is not existed between the original programming and dual programming has affected the application of the dual solution algorithm. Fortunately, the programming models of a large class of structural optimization problems, including the continuum topology optimization, are convex and separable. And an explicit relationship between the original variables and their dual variables is existed; therefore, the dual solution algorithm has become one of the effective methods for 38 years. However, the objective function of the dual problem is not explicit for a long time. It is because the dual problem is a parametric minimization problem which leads to the objective function is expressed as an implicit expression. The common explicit expression for the dual objective function is a two-order approximation. The regular thinking tendency that the dual problem is too difficult to be expressed explicitly and can only be expressed approximately is breakthrough. A dual programming explicit model (DP-EM) method is put forward for the topology optimization of continuum structures. Comparison of computational efficiency among the DP-EM method, the dual sequential quadratic program (DSQP) method and the method of moving asymptotes (MMA) is presented. The results showed that:(1) more external iterations are needed for the MMA algorithm than the DP-EM algorithm and DSQP algorithm; (2) same external iterations are needed for the DP-EM algorithm and DSQP algorithm, but internal iterations is less for the DP-EM method. It shows the advantage of the DP-EM algorithm due to its explicit dual function.

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