非光滑优化的双层规划模型及内点算法

AN INTERIOR-POINT BILEVEL PROGRAMMING FOR CONTACT OPTIMIZATION

对于包含接触约束的非光滑结构优化问题，其非光滑性体现在状态函数并不是处处可微的，针对含有应力约束及接触约束的非光滑结构优化问题，建立了一种双层规划模型，避免了求解时非光滑性所带来的问题，同时提出了一种迭代算法，用对偶内点二次规划进行分析，线性规划进行优化，算例表明这种方法十分有效．

A distinguishing characteristic of ,the problems of nonsmooth mechanics is the fact that the state as a function of the design or control is not everywhere differentiable. If the nondifferentiable points is not dense, the algorithm for smooth problem, by fairly moderate modified, can be utilized, while if the optimal point can be nondifferentiable, it will be hard to solve using smooth methods. Previous attempts to solve nonsmooth mechanics problems have been based on penalty method, however, the method has the drawback of sensitivity with respect to the penalty parameter. Someone also put much effort into finding generalized gradients at the nondifferentiable points. In this paper, our aim is to introduce simple and efficient method to solve nonsmooth mechanics problems. Firstly, a two-level programming model for some nonsmooth structural optimization problems is presented. When an optimization problem has both stress and contact displacement constraints, we combine the structural optimization model with contact analysis model to formulate a twolevel model. Quadratic programming (QP) is used in analysis level and linear programming used in optimization level. Secondly, considering the special form of constraints in QP, a surrogate dual interior-point algorithm is introduced for solving QP in this paper. An explicit surrogate dual formulation is derived and an interior-point algorithm is developed. At first, the idea of surrogate constraint is adopted to convert a QP to a surrogate problem and Karush-Kuhn-Tucker conditions are used to derive the explicit dual form. The transformation used in Karmarkar's algorithm for linear programming is then utilized to develop an interior-point algorithm for solving this explicit surrogate dual problem. Finally, several examples with unilateral or bilateral contact constraints are provided to verify the proposed algorithm.