Mathematical Sciences: Continuation and Bifurcations Investigations In Constrained Optimization

约束优化问题渗透到应用数学、工程和科学的许多领域，从线性和非线性规划、变分演算和最优控制到参数识别的变分公式和逆问题、不适定问题和连续介质力学。本研究计划的主要目标是开发一类新的鲁棒和非常快速的算法来解决这些问题。这些算法是基于使用快速数值延拓技术遵循平滑惩罚路径的最优性。这些算法发展的关键是一个扩展的拉格朗日方程组，它包含惩罚路径作为解，是一阶必要条件的摄动，有时，但并不总是，为寻找多个最优点提供基础。问题表述和延拓方法为吉洪诺夫正则化技术的发展和应用提供了一个自然的框架，以解决由模型误差或不确定性引起的敏感、病态或非约束优化问题，并为参数优化问题的研究提供了一个自然的框架。此外，在这种情况下，传统的与顺序优化算法相关的病态调节被消除了三个光滑惩罚函数：二次惩罚函数的等式约束和对数障碍和二次损失函数的不等式约束。在线性规划中，其中一个问题的表述和对应算法与Karmarkar算法非常相似，在速度上也有相似的期望；然而，这些快速连续惩罚路径算法也适用于上述具有类似速度期望的约束优化问题。这类研究的目的是开发新的计算机软件，以最有效和鲁棒的方式优化具有约束的多变量函数。这种类型的软件有大量潜在的应用，从资源管理到航班调度和机器设计。

Constrained optimization problems permeate much of applied mathematics, engineering and the sciences ranging from linear and nonlinear programming, calculus of variations and optimal control to variational formulations of parameter identification and inverse problems, ill-posed problems, and continuum mechanics. The primary objective of this research program is the development of a new class of robust and very fast algorithms for the solution of these problems. These algorithms are based on the use of fast numerical continuation techniques to follow smooth penalty paths to optimality. The key to the development of these algorithms is an expanded Lagrangian system of equations which contains the penalty path as a solution, is a perturbation of the first order necessary conditions, and sometimes, but not always, provides a basis for finding multiple optima. The problem formulation and continuation methodology provide a natural framework for the development and application of Tikhonov regularization techniques to the solution of sensitive, ill- conditioned, or ill-constrained optimization problems arising from model error or uncertainty and for the investigation of the parametric optimization problem. Also, in this setting the traditional ill-conditioning associated with a sequential optimization algorithm is removed for three smooth penalty functions: the quadratic penalty function for equality constraints and the logarithmic barrier and quadratic loss function for inequality constraints. In linear programming one of the problem formulations and correspondence algorithms is very similar to the Karmarkar algorithm with similar expectations in speed; however, these fast continuation-penalty path algorithms are also applicable to the aforementioned constrained optimization problems with similar expectations in speed. Research of this type aims at developing new computer software for optimization of functions of many variables with constraints in the most efficient and robust way. Software of this type has a large number of potential applications, ranging from management of resources to airline scheduling and machine design.