Developing and Understanding Methods for Nonlinear Optimization

本项目致力于无约束优化和约束优化的研究。该项目有五个部分：首先，张量方法将被开发用于约束优化和大型稀疏非线性方程组。这些方法显示出很有希望产生对非奇异和奇异问题高效的通用算法。其次，发展了求解非线性不等式和等式约束最优化问题的信赖域方法，这些问题即使在线性相关约束梯度的情况下也具有令人满意的局部和全局收敛理论，并且在实际应用中具有高效和健壮的性能。第三，对非线性约束最优化的几种割线方法进行了深入的分析，目标是在任意正定的初始Hessian逼近下保证局部和超线性收敛。特别是，将研究利用拉格朗日的全部海森的方法，包括一些新的、有希望的增广拉格朗日方法，这些方法也将被计算研究。第四，将开发算法来解决隐式非线性最小二乘问题，这是在没有因变量的情况下在曲线拟合或数据拟合中出现的优化问题。该问题归结为一个非线性等式约束优化问题，有望用信赖域方法求解。第五，对全局优化的并行和顺序方法进行了研究。这些都是随机算法，它们自适应地划分可行域，并导致并行和顺序计算的改进。

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.