Physical systems whose states evolve in response to variable externally applied controlling forces, voltages, temperatures, interest rate policies, resource allocation strategies, and the like, are commonly modeled by differential equations or approximating difference equations. Related optimization problems arise naturally when some choice is permitted in the way the control inputs are applied in time or space. These optimal control problems have classic precursors in the Calculus of Variations, and may also be viewed as specially structured nonlinear programs in function spaces or finite-dimensional sequence spaces. The most difficult optimal control problems enforce strict pointwise bounds or other inequality constraints on the control and state variables. Run-of-the-mill problems in this category can easily entail hundreds or thousands of variables with comparably many constraints, and when cast in the currently accepted mathematical format, are often so badly scaled that the simplest and most readily implemented iterative optimization methods are effectively incapacitated. The more sophisticated Newtonian and quasi-Newtonian algorithms do provide computational countermeasures for bad scaling near almost-singular local minimizers; however, Newtonian scaling procedures are costly and do not guarantee good nonlocal convergence properties while the iterates are still far from a local minimizer.
The proposed investigation would address these issues with theoretical and computational evaluations of standard optimization schemes implemented in a new nonstandard mathematical framework for optimal control problems with pointwise state and control constraints. The alternative framework replaces local differential or difference forms of the state evolution equations by nonlocal integrated forms, and replaces state variables in the primal variable set by new artificial variables that coincide with the state when all constraints are met. Recent experiments with hybrid augmented Lagrangian projection methods outlined in this proposal indicate real and important analytical and computational advantages in the new formulation. The goal of the proposed study is to gain more experience with algorithm implementations in this setting, and to develop sharpened optimality conditions and other mathematical tools needed to achieve a deeper understanding of the observed behavior and improve the computational methods. Improvements in this area must have an immediate and significant practical impact, since large-scale computationally challenging optimal control problems arise in many physical settings.
