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

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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.