AF: Small: Approximate Augmented Lagrangians: First-Order and Parallel Optimization Methods, with Applications to Stochastic Programming

持续优化是一门数学学科，在工程设计和商业/物流规划中有着广泛的应用。目前最常见的解决方案技术很难适应由成千上万个并行处理元素组成的新发展的计算机体系结构。结合几种现有的技术和一些最近的主要研究者的结果，这个项目探索了一种解决持续优化问题的方法，这种方法应该比目前的标准解决方案更容易适应并行计算机体系结构，允许体系结构的全部能力被带到大型，耗时的问题上。如果没有这些新的解决方法，关键设计和规划问题的解决方案可能无法从未来十年预计的计算能力的大多数进步中受益。该项目还将涉及与巴西研究界的合作工作。技术方法是利用拉格朗日算法和共轭梯度算法的最新进展，产生一种新的模块化并行连续约束优化求解器。该求解器由经典的增广拉格朗日外环组成，子问题由最近发展的相对误差准则终止的状态盒约束共轭梯度法求解。研究包括三个阶段：第一阶段的目标是创建一个面向对象的、模块化的串行实现，对其进行广泛的测试，并解决一些理论问题。第二阶段的目标是将第一阶段的子策略发展成一个并行求解器，用户可以明确地指定如何将问题结构映射到多个处理元素。第三阶段的目标是使结构检测和映射过程自动化。第二和第三阶段将使用随机编程问题作为测试用例。

Continuous optimization is a mathematical discipline with extensiveapplications in engineering design and business/logistical planning.Its currently most common solution techniques are difficult to adaptto newly evolving computer architectures comprising dozens tothousands of processing elements working in parallel. Combiningseveral existing techniques with some recent results of the principalinvestigator, this project explores a means of solving continuousoptimization problems that should adapt more readily to parallelcomputer architectures than present standard solvers, allowing thearchitectures' full power to be brought to bear on large,time-consuming problems. Without such new solution approaches,solution of critical design and planning problems may not benefit frommost of the advances in computing power anticipated for the nextdecade. The project will also involve cooperative work with theBrazilian research community.The technical approach is to capitalize on recent advances inaugmented Lagrangian and conjugate gradient algorithms to produce anew kind of modular parallel continuous constrained optimizationsolver. The solver consists of a classical augmented Lagrangian outerloop, with subproblems solved by the a state-of-the artbox-constrained conjugate gradient method terminated by a recentlydeveloped relative error criterion. The research consists of threestages: the goal of stage one is to create an object-oriented, modularserial implementation, test it extensively, and address sometheoretical issues. Stage two aims to evolve the stage-one substrateinto a parallel solver for which the user explicitly specifies how tomap the problem structure to multiple processing elements. Stagethree's goal is to automate the structure detection and mappingprocess. Stages two and three will use stochastic programmingproblems as test cases.