MA对偶-信赖域算法在非线性不等式约束优化问题中的应用研究

The application of MA dual-trust region method for nonlinear inequality constraint optimization

针对含有非线性不等式约束条件的优化问题,提出了MA对偶-信赖域算法。在每次迭代过程中,基于信赖域方法和问题的逼近属性,构造了原优化问题中目标函数和约束函数的移动渐进线函数,由此建立简单的子优化问题。运用对偶方法求解子问题得到原优化问题的下降方向,再用线搜索方法取得搜索步长,最后得到下一步的迭代点。应用数学推理证明了该算法的全局收敛性。以悬臂梁最小柔度问题为例,应用MA对偶-信赖域算法对优化问题进行了求解,数值算例的结果表明,MA对偶-信赖域算法在求解非线性约束优化问题时比MMA和GCMMA算法的迭代次数少,收敛速度快。

In allusion to the nonlinear optimization problem of inequality constraints, MA(moving asymptotes) dual-trust region method is presented. In each time of iteration, based on the approximation properties and the trust region method, the moving asymptotes functions of object and constraint functions of the original optimization are constructed. The simple subproblems of the original optimization problem are built up. By using the MA dual-trust region method to solve the subproblem of original optimization, the down direction of the original optimization is got, and the length of search step of original optimization is obtained by the linear search method. Finally, the next iteration point of the original optimization is gained. Appling the mathematical reasoning, the paper proves that the convergence of MA dual-trust region algorithm is global. The results of numerical example show that the number of iteration MA dual-trust region method in solving nonlinear inequality constrained optimization is less than that of MMA and GCMMA algorithms, and the speed of MA dual-trust region method is faster than that of MMA and GCMMA algorithms. Meantime, as an example, the MA dual-trust region method is applied to the minimum compliance of the cantilever beam.