A systematic search method for obtaining multiple local optimal solutions of nonlinear programming problems

We propose a systematic method to find several local optimal solutions for a general nonlinear optimization problem. Analytical results for quasi-gradient systems and reflected gradient systems are developed and applied to explore the topological and geometric aspects of the critical points of the objective function. A mechanism is devised to escape from a local optimal solution and proceed into another local optimal solution by locating the decomposition point. By properly switching between quasi-gradient systems and reflected gradient systems, our proposed method can obtain a set of local optimal solutions and decomposition points. This algorithm also can find the global optimal solution. It depends on its ability to find all the decomposition points. The main algorithm contains two levels: the lower level is continuous while the upper level is discrete in nature. Further improvements in the algorithm to locate all decomposition points are desirable. The proposed method is applied to one test example with encouraging results