Mathematical Sciences: Algorithms for Large Scale Optimization

将研究最小化的算法 多元非光滑复合函数 的实例 这样的函数是可积的和有界的数据拟合，并且 非线性规划的罚函数 总 一种方法是使用信任域框架， 方法，在梯度投影方法的精神，解决 每次迭代的子问题。 意图是本 算法将最适合于非线性问题， 约束是部分可分离的（即约束 导数矩阵具有块对角结构）。 重要 这些问题的例子包括约束离散最优 控制和多商品网络流。 特别注意 将给出算法的并行实现 计算机体系结构 大规模线性方程组的迭代解法 还将研究编程问题。 这些方法， 它使用一些成熟的技术作为框架， 一般约束优化，具有的优点是， 从来不需要对约束矩阵进行因式分解。 相反，迭代方法（如连续超松弛或 梯度投影/共轭梯度技术）用于 求解凸二次规划子问题， 迭代 这些不同的算法在 将执行向量/并行处理器，以确定两者 基本方法的有效性，以及 可以利用哪些计算机体系结构。

Research will be carried out into algorithms for minimizing nonsmooth composite functions of many variables. Instances of such functions are integrable and bounded data fitting, and penalty functions for nonlinear programming. The general approach is to use a trust-region framework, and iterative methods, in the spirit of gradient projection methods, to solve the subproblem at each iteration. It is intended that the algorithms will be most suitable for nonlinear problems in which the constraints are partially separable (i.e. the contraint derivative matrix has a block-diagonal structure). Important examples of such problems include constrained discrete optimal control, and multicommodity network flow. Particular attention will be given to implementation of the algorithms on parallel computer architectures. Iterative solution methods for large-scale linear programming problems will also be investigated. These methods, which use as a framework some well-established techniques for general constrained optimization, have the advantage that factorization of the constraint matrix is never required. Instead, iterative methods (such as successive over-relaxation or a gradient projection/conjugate gradient technique) are used to solve the convex quadratic programming subproblems at each iteration. Implementation of these various algorithms on vector/parallel processors will be carried out, to determine both the effectiveness of the underlying approach, and the extent to which computer architectures can be utilized.