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

We propose, in this paper, a systematic method to find several local optimal solutions for general nonlinear optimization problems. We have developed some analytical results for quasi-gradient systems and reflected gradient systems, applying these results to derive topological and geometric properties of the critical points of the underlying objective function. A mechanism has also been devised to escape from a local optimal solution and proceed into another local optimal solution via decomposition points. By properly switching between quasi-gradient systems and reflected gradient systems, our proposed method can attain a set of local optimal solutions. The proposed method is applied to two test examples with promising results.

[1]  S. Smale.  On Gradient Dynamical Systems , 1961 .

[2]  F. H. Branin.  Widely convergent method for finding multiple solutions of simultaneous nonlinear equations , 1972 .

[3]  H. Th. Jongen,et al.  The index-k-stabilizing differential equation , 1981 .

[4]  L. Chua,et al.  Non-linear optimization with constraints: A cook-book approach , 1983 .

[5]  Leon O. Chua,et al.  Nonlinear programming without computation , 1984 .

[6]  Panos M. Pardalos,et al.  Constrained Global Optimization: Algorithms and Applications , 1987, Lecture Notes in Computer Science.

[7]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[8]  Leon O. Chua,et al.  Neural networks for nonlinear programming , 1988 .

[9]  Felix F. Wu,et al.  Stability regions of nonlinear autonomous dynamical systems , 1988 .

[10]  J. Zaborszky,et al.  On the phase portrait of a class of large nonlinear dynamic systems such as the power system , 1988 .

[11]  Felix F. Wu,et al.  Foundations of the potential energy boundary surface method for power system transient stability analysis , 1988 .

[12]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[13]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[14]  C. DeMarco.  Approximating power system dynamics and energy functions by quasi-gradient models , 1989, IEEE International Symposium on Circuits and Systems,.

[15]  Leon O. Chua,et al.  Global optimization: a naive approach , 1990 .

[16]  Thomas F. Coleman,et al.  Large-Scale Numerical Optimization: Introduction and Overview , 1991 .

[17]  S. Vavasis.  Nonlinear optimization: complexity issues , 1991 .

[18]  Leon O. Chua,et al.  The identification of general pseudoreciprocal vector fields , 1992 .

[19]  N. K. Sancheti,et al.  A new approach to global optimization using ideas from nonlinear stability theory , 1992, [Proceedings] 1992 IEEE International Symposium on Circuits and Systems.