RESERCH ON UNIVERSAL SOLUTION OF TRANSFORMING SEPARABLE CONVEX PROGRAMMING TO DUAL PROGRAMMING WITH EXPLICIT MODEL

This paper aims to improve the modeling and solving level of DP-EM (dual programming-explicit model) method. Based on the characteristics of a class of convex programming with separable variables, the DP-EM model breaks through the usual way of using second-order approximation for the dual objective function, and derives an explicit dual objective function. The DP-EM method is more efficient than the dual sequential quadratic programming (DSQP) and the method of moving asymptotes (MMA) when it is applied to the ICM method solving the continuum topology optimization problems. In this paper, the common explicit models are abstracted into universal separable convex programming, and then converted into DP-EM models under certain conditions. Four processing methods are proposed: (1) The approximate solution of iterative approximation of dual variables; (2) The solution of objective and constraint functions with the exponential function form; (3) The solution of objective and constraint functions with the power function form; (4) Accurate solution based on variable transformation. In order to conduct numerical verification, extensive calculations have been carried out. Limited by paper space, five representative examples among them are listed. Example 1 is a pure mathematical problem, which is used to compare the efficiency of the processing method 1 and the processing method 4. The remaining four examples are all continuum topology optimization problems modeled and solved by the ICM (independent continuous and mapping) method, including displacement, stress, fatigue constraint problems and fail-safe optimization. Those four examples are illustrations of the processing method 3. All the results show the universality of the proposed method and the higher solving efficiency. The proposed method can used for different penalty functions in the variable density method and filtering functions in the ICM method. And the proposed method is more efficient than the MMA method. The contribution of the work is as follows: (1) In depth, it deepens the research on the dual solution of structural optimization; (2) In breadth, it makes a contribution to the dual theory of mathematical programming.