Projected Hessian Updating Methods

这个项目将研究解决非线性规划问题的数值方法。该问题涉及最小化非线性函数，该函数受一组非线性等式和不等式约束。序列二次规划（SQP）方法对于解决这种性质的问题特别有效。假设目标函数和约束函数的一阶导数是可用的，但二阶导数可能太昂贵而无法计算。相反，该方法通常在每次迭代中更新一个合适的矩阵，该矩阵近似于二阶导数信息。我们感兴趣的是开发SQP方法，它保持对投影到约束的切空间上的二阶导数信息的近似。这项工作的主要动机是只有投影矩阵才进入非线性问题的最优性条件。我们将研究与算法的全局和局部收敛性质有关的问题。其中包括拟牛顿更新、主动集策略和价值函数的选择。主要目标有两个：生产高效可靠的软件，并为这些方法提供理论框架。

This project will investigate numerical methods for solving nonlinear programming problems. The problems involve minimizing a nonlinear function subject to a set of nonlinear equality and inequality constraints. Sequential quadratic programming (SQP) methods are particularly effective for solving problems of this nature. It is assumed that first derivatives of the objective and constraint functions are available, but that second derivatives may be too expensive to compute. Instead the methods typically update a suitable matrix which approximates second derivative information each iteration. We are interested in developing SQP methods which maintain an approximation to second derivative information projected onto the tangent space of the constraints. The main motivation for the work is that only the projected matrix enters into the optimality conditions for the nonlinear problem. We will investigate issues relating to both the global and local convergence properties of algorithms. These include the choice of quasi-Newton update, active set strategy and merit function. The main goal is two-fold: to produce efficient and reliable software and to provide a theoretical framework for these methods.