@article{dunn_2000, title={On state constraint representations and mesh-dependent gradient projection convergence rates for optimal control problems}, volume={39}, ISSN={["0363-0129"]}, DOI={10.1137/S0363012999351656}, abstractNote={Two distinct nonlinear programming formulations are investigated for ODE optimal control problems with pointwise state and control constraints. The first formulation treats the differential equations of state as an equality constraint in the conventional manner. The second formulation employs a different equality constraint entailing the integrated state transition map. Related convergence rate estimates are developed for augmented gradient projection methods and discrete-time approximations to a large representative class of ODE control problems. In the first formulation, the rate estimates are mesh-dependent, and the predicted number of inner loop gradient projection iterations needed to achieve a fixed small deviation from the optimal value of the augmented Lagrangian is inversely proportional to the square of the mesh width. In the second formulation, the convergence rates and predicted iteration counts are mesh-invariant. The computational costs-per-iteration in the two formulations are comparable. These estimates elucidate previously published numerical experiments with augmented gradient projection% AGP methods and constrained regulator problems.}, number={4}, journal={SIAM JOURNAL ON CONTROL AND OPTIMIZATION}, author={Dunn, JC}, year={2000}, month={Dec}, pages={1082–1111} }
@inbook{dunn_1998, title={Augmented gradient projection calculations for regulator problems with pointwise state and control constraints}, DOI={10.1007/978-1-4757-6095-8_7}, abstractNote={A new implementation of the augmented gradient projection (AGP) scheme is described for discrete-time approximations to continuous-time Bolza optimal control problems with pointwise bounds on control and state variables. In the conventional implementation, control and state vectors are the primal variables, and local forms of the control problem’s state equations are treated as equality constraints incorporated in an augmented Lagrangian with a penalty parameter c. In the new implementation, the original control vectors and new artificial control vectors are the primal variables, and an integrated form of the state equations replaces the usual local form in the augmented Lagrangian. The resulting relaxed nonlinear program for the augmented Lagrangian amounts to a Bolza problem with pure pointwise control constraints, hence the associated gradient and Newtonian direction vectors can be computed efficiently with adjoint equations and dynamic programming techniques. For unscaled AGP methods and prototype regulator problems with bound constraints on control and state vectors, numerical experiments indicate rapid deterioration in the convergence properties of the conventional implementation as the discrete-time mesh is refined with the penalty constant fixed. In contrast, the new implementation of the unscaled AGP scheme exhibits mesh-independent convergence behavior. The new formulation also offers certain additional computational advantages for control problems with separated control and state constraints.}, booktitle={Optimal control: Theory, algorithms, and applications}, publisher={Boston: Kluwer Academic Pubs.}, author={Dunn, J. C.}, editor={W. W. Hager and Pardalos, P. M.Editors}, year={1998}, pages={130–153} }
@article{dunn_1998, title={L-2 sufficient conditions for end-constrained optimal control problems with inputs in a polyhedron}, volume={36}, ISSN={["0363-0129"]}, DOI={10.1137/S0363012995288513}, abstractNote={An $\hbox{{\bbb L}}^2$-local optimality sufficiency theorem is proved for a class of structured infinite-dimensional nonconvex programs with constraints of the form $u\in\Omega$ and h(u)=0, where $\Omega$ is a set of Lebesgue measurable essentially bounded vector-valued functions $u(\cdot ): [0,1]\rightarrow \hbox{{\bbb R}}^m$ with range in a polyhedron U, and h is a smooth map of the space of essentially bounded functions $u(\cdot )$ into ${\bbb R}^k$. The sufficiency theorem is based on formal counterparts of the finite-dimensional Karush--Kuhn--Tucker sufficient conditions in a Cartesian product of polyhedra, a strengthened variant of Pontryagin's necessary condition, and structure and continuity conditions on the first and second differentials of the objective function and equality constraint functions. The new sufficient conditions are directly applicable to nonconvex continuous-time Bolza optimal control problems with control-quadratic Hamiltonians, unqualified affine inequality constraints on vector-valued control inputs, and equality constraints on the terminal state vector or equivalent isoperimetric constraints on integrals of functions depending on the state and control variables.}, number={5}, journal={SIAM JOURNAL ON CONTROL AND OPTIMIZATION}, author={Dunn, JC}, year={1998}, month={Sep}, pages={1833–1851} }
@inproceedings{dunn_1998, title={On L-2 sufficient conditions for end-constrained optimal control problems with inputs in a polyhedron}, DOI={10.1109/cdc.1998.758679}, abstractNote={An L/sup 2/-local optimality sufficiency theorem proved for constrained optimal control problems is described here. This theorem applies to a class of structured infinite-dimensional nonconvex programs with constraints of the form, u/spl isin//spl Omega/ and h(u)=0, where /spl Omega/ is a set of Lebesgue measurable essentially bounded vector-valued functions u(/spl middot/):[0, 1]/spl rarr/R/sup m/ with range in a polyhedron U, and h is a smooth map of the space of essentially bounded functions u(/spl middot/) into R/sup k/. The theorem is based on formal counterparts of the finite-dimensional Karush-Kuhn-Tucker sufficient conditions in a Cartesian product of polyhedra, a strengthened variant of Pontryagin's necessary condition, and structure/continuity conditions on the first and second differentials of the objective function and equality constraint functions. Its sufficient conditions are directly applicable to nonconvex continuous-time Bolza optimal control problems with control-quadratic Hamiltonians, affine inequality constraints on the control inputs, and equality constraints on the terminal state vector, or equivalent isoperimetric constraints.}, booktitle={Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, December 16-18, 1998}, author={Dunn, J. C.}, year={1998} }
@inbook{dunn_1998, title={On second order sufficient conditions for structured nonlinear programs in infinite-dimensional function spaces}, booktitle={Mathematical programming with data perturbations}, publisher={New York: Marcel Dekker}, author={Dunn, J. C.}, year={1998}, pages={83–108} }