Locating and Characterizing the Stationary Points of the Extended Rosenbrock Function

Two variants of the extended Rosenbrock function are analyzed in order to find the stationary points. The first variant is shown to possess a single stationary point, the global minimum. The second variant has numerous stationary points for high dimensionality. A previously proposed method is shown to be numerically intractable, requiring arbitrary precision computation in many cases to enumerate candidate solutions. Instead, a standard Newtonian method with multi-start is applied to locate stationary points. The relative magnitude of the negative and positive eigenvalues of the Hessian is also computed, in order to characterize the saddle points. For dimensions up to 100, only two local minimizers are found, but many saddle points exist. Two saddle points with a single negative eigenvalue exist for high dimensionality, which may appear as “near” local minima. The remaining saddle points we found have a predictable form, and a method is proposed to estimate their number. Monte Carlo simulation indicates that it is unlikely to escape these saddle points using uniform random search. A standard particle swarm algorithm also struggles to improve upon a saddle point contained within the initial population.