Publisher Summary This chapter describes a completely different, novel, and general approach to find physically meaningful solutions and singular points to chemical-process models based on moving up and down the landscape of the least-squares function. The resulting algorithms are called Global Terrain Algorithms and consist of a series of downhill, equation-solving computations, and uphill, predictor–corrector calculations from one stationary point to another. Initial uphill movement from a saddle point, local minimum, or global minimum of the least-squares function always takes place along a direction of smallest positive curvature. Initiating downhill movement from a saddle point or maximum to either a singular point of lower norm or a solution always takes place along an eigendirection, v, of negative curvature that can be calculated from efficient eigendecomposition of the Hessian matrix of the least-squares function using Lanzcos or some other eigenvalue-eigenvector technique. Key concepts and numerical results are illustrated using chemical-process examples.