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Advanced Computing Research Aiping Liao Algorithm applied to fDTOC applying Procedure dNewton automatic differentiation automatic techniques backward recursion bisection method calculate Newton's direction calculate the Newton Coleman and Liao Computing Research Institute control variable convergence Cornell Theory Center discrete time optimal discrete-time optimal control dNewtm DTOC format DTOC problems DTOC setting dummy variable elementary functions f(xk Gaussian elimination gDTOC given by Procedure given Xi Griewank Hessian H identity matrix intermediate variables involved real computations j-th component Lemma Minimization modified to handle Newton step nonzero element number of operations number of stages objective function optimal control problems Pn+i positive definite Procedure fDTOC procedure for calculating properties pl)-(p6 Proposition 2.1 relabeled reverse mode applied second order Set fDTOC sparse steepest decent method subP symmetric matrix total number transform Procedure fAD trust region method variables whose indices Wei Yuan XI is positive xn+i Yuying