Constrained optimization using a multipoint type chaotic Lagrangian method with a coupling structure

Abstract

This article proposes a new constrained optimization method using a multipoint type chaotic Lagrangian method that utilizes chaotic search trajectories generated by Lagrangian gradient dynamics with a coupling structure. In the proposed method, multiple search points autonomously implement global search using the chaotic search trajectory generated by the coupled Lagrangian gradient dynamics. These points are advected to elite points (which are chosen by considering their objective function values and their feasibility) by the coupling in order to explore promising regions intensively. In this way, the proposed method successfully provides diversification and intensification for constrained optimization problems. The effectiveness of the proposed method is confirmed through application to various types of benchmark problem, including the coil spring design problem, the benchmark problems used in the special session on constrained real parameter optimization in CEC2006, and a high-dimensional and multi-peaked constrained optimization problem.

Keywords:

• constrained optimization
• global optimization
• Lagrangian method
• chaos
• coupled dynamics

Notes

Specifically, . This condition is obtained through a derivation similar to the derivation for unconstrained problems, as in Okamoto and Aiyoshi (2008).

Note that the bounded search space for in Equation (12) is the same as the bounded search space given by Equation (1d).

Regarding inequality conditions, equality conditions with slack variables equivalent to the inequality conditions, appearing in the derivation of the augmented Lagrangian, become (see Luenberger and Ye 2007 for details).

Results for the coil spring design problem are excluded in order to make the comparison with other methods easy. In all experiments, SR for the coil spring design problem is 100.

The steepest descent method and the quasi-Newton method are also applied. The result obtained using the conjugate gradient method is reported, because it gives a better result than results obtained using the first two methods.