自适应的信赖域算法研究

The trust region method is an important class of numerical methods for solving nonlinear optimization problems. In the traditional trust region method, the trial step generated by the trust region subproblem may not be able to be accepted at some iteration point, it may need to solve several to get the next iteration point; Secondly, most of the adjustment strategy of the trust region radius has blindness. At each iteration, we hope to generate a suitable trust region radius automatically based on the current iterative information，so the new trust region model is more consistent with the objective function. In addition, the numerical experiments are mostly for low-dimensional optimization problems，however, we often need solve large-dimensional optimization problem in practical applications. This project will study the model of adaptive trust region method; Both the global convergence and superlinear convergence of the new algorithm are analyzed under reasonable assumptions, which can give the theoretical support for solving practical problems; Numerical experiments will show that the new method is feasible and effective for solving engineering problems.

信赖域法是求解非线性优化问题的一类重要的数值计算方法。在基本信赖域方法中，由信赖域子问题产生的试探步在某些迭代点处不一定能够被接受，可能需要求解多次才能得到下一个迭代点；其次，现在讨论的信赖域半径的调整策略大部分具有盲目性。在当前迭代点处我们希望利用当前迭代的信息自动产生一个合适的信赖域半径,此时模型函数和目标函数具有良好的一致性；另外，现在的数值试验大多针对低维优化问题，然而在实际应用中，往往需要解决高维优化问题。因此，本项目将研究自适应的信赖域算法模型；在合理的假设条件下，对新算法的全局收敛性和超线性收敛性进行理论分析，为实际问题的解决提供理论支撑基础；通过数值模拟试验，明确新方法对解决工程问题是可行和有效的。

信赖域法是求解非线性优化问题的一类重要的数值计算方法。本项目研究了自适应的信赖域算法模型，即：在当前迭代点处利用已知迭代点的信息构造合适的信赖域半径，使得模型函数和目标函数具有更好的一致性；在合理的假设条件下，证明了新算法的全局收敛性和超线性收敛性理论；通过数值模拟试验，表明新方法对于求解无约束优化问题是可行和有效的。项目组全体成员根据申请书中所规定的研究内容，基本上完成了本项目研究的全部目标。在这一年的时间里，我们共发表论文2篇，在投论文1篇，1篇论文准备中。