Very often, the only reliable information available to perform change detection is the description of some unchanged regions. Since sometimes these regions do not contain all the relevant information to identify their counterpart (the changes), we consider the use of unlabeled data to perform Semi-Supervised Novelty detection (SSND). SSND can be seen as an unbalanced classification problem solved using the Cost-Sensitive Support Vector Machine (CS-SVM), but this requires a heavy parameter search. We propose here to use entire solution path algorithms for the CS-SVM in order to facilitate and accelerate the parameter selection for SSND. Two algorithms are considered and evaluated. The first one is an extension of the CS-SVM algorithm that returns the entire solution path in a single optimization. This way, the optimization of a separate model for each hyperparameter set is avoided. The second forces the solution to be coherent through the solution path, thus producing classification boundaries that are nested (included in each other). We also present a low density criterion for selecting the optimal classification boundaries, thus avoiding the recourse to cross-validation that usually requires information about the ``change'' class. Experiments are performed on two multitemporal change detection datasets (flood and fire detection). Both algorithms tracing the solution path provide similar performances than the standard CS-SVM while being significantly faster. The low density criterion proposed achieves results that are close to the ones obtained by cross-validation, but without using information about the changes.