Homotopy parameter bounding in increasing the robustness of homotopy continuation methods in multiplicity studies

Homotopy continuation methods are globally convergent methods, which can also be utilized in multiplicity studies. However, when the starting point and/or solution multiplicities lie on separate homotopy path branches, one or more of the solutions may be missed. This is due to the absence of real space connections between separate homotopy path branches, thus preventing multiple solutions being reached from a single starting point. In this paper, a concept is presented that enables a tracking starting point and solution multiplicities in cases where the standard problem-independent homotopy method fails. The concept is based on homotopy parameter bounding and enables the connection of separate homotopy path branches. The concept performance is examined using distillation column examples. In the examined cases the concept is found to improve robustness by establishing a path in real space such that solutions are approached that would be unattainable using the standard homotopy method.