New Methods for Nonlinear Optimization

伯德和施纳贝尔教授将继续他们在无约束和约束优化方面的研究。研究了四个主题。首先，无约束优化的张量方法和约束优化方法的初步发展。其次，约束优化的信赖域方法相对容易实现，即使存在线性相关的约束梯度，也具有令人满意的局部和全局收敛理论，并且在实践中具有高效和鲁棒性。第三，解决隐式最小二乘问题的算法的发展，这是在曲线拟合或没有因变量的情况下出现的最小二乘版本。第四，深入分析了几种非线性约束优化的准牛顿方法，以保证任意正定初始Hessian近似的局部超线性收敛为目标，推广了Powell方法的结果。本研究解决了许多应用中出现的重要优化问题。

Professors Byrd and Schnabel will continue their research in unconstrained and constrained optimization. Four topics are investigated. First, tensor methods for unconstrained optimization and initial development of such methods for constrained optimization. Second, trust region methods for constrained optimization that are relatively easy to implement, have satisfactory local and global convergence theory even in the presence of linearly dependent constraint gradients and perform efficiently and robustly in practice. Third, the development of algorithms for solving implicit least squares problems, a version of least squares that arises in curve fitting or where there are no dependent variables. Fourth, a thorough analysis of several quasi-Newton methods for nonlinearly constrained optimization, with the goal of guaranteeing local superlinear convergence with an arbitrary positive definite initial Hessian approximation, extending Powell's results for the BFGS methods. This research addresses important optimization problems that occur in many applications.