It is shown that all stationary and singular points to optimization problems do not necessarily lie in the same valley and are not necessarily smoothly connected. Logarithmic barrier functions are shown to be an effective means of finding smooth connections between distinct valleys, so that the terrain method is guaranteed to explore the entire feasible region. After valleys are connected, different stationary and singular points in separate parts of the feasible region can be calculated and identified and sequentially tracked as the barrier parameter is reduced. The proposed barrier-terrain methodology is used to successfully find all physically meaningful solutions to a small illustrative problem and a collocation model for a spherical catalyst pellet problem with 20 variables. The key contribution of this work is the discovery that barrier methods provide connections between valleys that contain stationary points for intermediate barrier parameter values under mild conditions on the model equations.