A new method for solving dynamic optimization problems that contain path constraints on the state variables is described. We establish the equivalence between the inequality path-constrained dynamic optimization problem and a hybrid discrete/continuous dynamic optimization problem that contains switching phenomena. The control parameterization method for solving dynamic optimization problems, which transforms the dynamic optimization problem into a finite-dimensional nonlinear program ({NLP}), is combined with an algorithm for constrained dynamic simulation so that any admissible combination of the control parameters produces an initial value problem that is feasible with respect to the path constraints. We show that the dynamic model, which is in general described by a system of differential-algebraic equations ({DAEs}), can become high-index during the state-constrained portions of the trajectory. During these constrained portions of the trajectory, a subset of the control variables are allowed to be determined by the solution of the high-index {DAE}. The algorithm proceeds by detecting activation and deactivation of the constraints during the solution of the initial value problem, and solving the resulting high-index {DAEs} and their related sensitivity systems using the method of dummy derivatives. This method is applicable to a large class of dynamic optimization problems.