Effect of inexact adjoint solutions on the discrete-adjoint approach to gradient-based optimization

New algorithms are developed to adapt the convergence tolerances for the constraint and adjoint equations for practical engineering optimization problems solved using the discrete adjoint approach. The algorithms are designed to achieve design order convergence of the optimization algorithm at reduced computational cost. We have found based on analysis and supported by numerical experimentation that adapting both the constraint and adjoint equation tolerances based on the norm of the gradient is sufficient to achieve design order convergence. We have also found that adapting the constraint equation tolerance is necessary, though we were not able to show analytically that adapting the adjoint equation tolerance is necessary. Based on the numerical experimentation, it appears that design order convergence can be achieved without adapting the adjoint equation in some cases but not others. The gain in computational efficiency using the new algorithms over using fixed tolerances is demonstrated through three numerical test problems.