The bifurcation of constrained optimization optimal solutions and its applications

The appearance and disappearance of the optimal solution for the change of system parameters in optimization theory is a fundamental problem. This paper aims to address this issue by transforming the solutions of a constrained optimization problem into equilibrium points (EPs) of a dynamical system. The bifurcation of EPs is then used to describe the appearance and disappearance of the optimal solution and saddle point through two classes of bifurcation, namely the pseudo bifurcation and saddle-node bifurcation. Moreover, a new class of pseudo-bifurcation phenomena is introduced to describe the transformation of regular and degenerate EPs, which sheds light on the relationship between the optimal solution and a class of infeasible points. This development also promotes the proposal of a tool for predicting optimal solutions based on this phenomenon. The study finds that the bifurcation of the optimal solution is closely related to the bifurcation of the feasible region, as demonstrated by the 5-bus and 9-bus optimal power flow problems.