Developing and Understanding Methods for Nonlinear Optimization

该项目致力于研究无约束和约束 优化. 本课题共分五个部分：第一，张量方法 将开发约束优化和大型，稀疏 非线性方程组 这些方法显示出巨大的前景， 产生通用算法， 非奇异和奇异问题。 第二，信赖域方法将 非线性不等式和等式约束 具有令人满意的局部和全局优化问题 收敛理论，即使存在线性相关 约束梯度，并在以下方面有效和稳健地执行 实践 第三，将对几个问题进行深入分析。 非线性约束优化的正割方法， 目标是保证局部和超线性收敛， 任意正定初始Hessian逼近 在 特别是，方法将调查，利用充分的海森 的拉格朗日，包括一些新的，有前途的增广拉格朗日 这些方法也将通过计算进行研究。 第四， 将开发算法来解决隐式非线性最小 平方问题，曲线拟合中出现的优化问题 或者在没有因变量的情况下进行数据拟合。 这个问题 导致非线性等式约束优化问题， 期望用信赖域方法求解。 五是一些 并行和顺序的方法将研究全球 优化. 这些是随机算法， 划分可行域，并导致两者的改进 并行和顺序计算。

This project is devoted to research in unconstrained and constrained optimization. The project has five parts: First, tensor methods will be developed for constrained optimization and for large, sparse systems of nonlinear equations. These methods show great promise of yielding general purpose algorithms that are highly efficient on nonsingular and singular problems. Second, trust region methods will be developed for nonlinearly inequality and equality constrained optimization problems that have satisfactory local and global convergence theory even in the presence of linearly dependent constraint gradients, and perform efficiently and robustly in practice. Third, a thorough analysis will be performed of several secant methods for nonlinearly constrained optimization, with the goal of guaranteeing local and superlinear convergence with an arbitrary positive definite initial Hessian approximation. In particular, methods will be investigated that utilize the full Hessian of the Lagrangian, including some new, promising augmented Lagrangian methods that will also be investigated computationally. Fourth, algorithms will be developed to solve the implicit nonlinear least squares problem, an optimization problem that arises in curve fitting or in data fitting when there is no dependent variable. This problem results in a nonlinear equality constrained optimization problem which is expected to be solved by trust region methods. Fifth, some parallel and sequential methods will be investigated for global optimization. These are stochastic algorithms which adaptively partition the feasible region, and lead to improvements in both parallel and sequential computation.