This study proposes an efficient indirect approach for general nonlinear dynamic optimization problems without path constraints.The approach incorporates the virtues both from indirect and direct methods: it solves the optimality conditions like the traditional indirect methods do

but uses a discretization technique inspired from direct methods.Compared with other indirect approaches

the proposed approach has two main advantages: (1) the discretized optimization problem only employs unconstrained nonlinear programming (NLP) algorithms such as BFGS (Broyden-Fletcher-Goldfarb-Shanno)

rather than constrained NLP algorithms

therefore the computational efficiency is increased; (2) the relationship between the number of the discretized time intervals and the integration error of the four-step Adams predictor-corrector algorithm is established

thus the minimal number of time intervals that under desired integration tolerance can be estimated.The classic batch reactor problem is tested and compared in detail with literature reports

and the results reveal the effectiveness of the proposed approach.Dealing with path constraints requires extra techniques

and will be studied in the second paper.