A Hybrid Algorithm for the Simple Cell Mapping Method in Multi-objective Optimization

Abstract

Abstract. This paper presents a hybrid gradient free-gradient (GFG) algorithm for the simple cell mapping (SCM) method for multi-objective optimization problems (MOPs). The SCM method is briefly reviewed in the context of the multi-objective optimization problems (MOPs). We present a mixed application of gradient free directed search and gradient search algorithms for the SCM method and discuss its potentials for higher dimensional MOPs. We present several numerical exmaples to demonstrate the effectiveness of the proposed hybrid algorithm. The examples include two simple geometric MOPs, an example with five design parameters, and a proportional-integral-derivative (PID) control design for a second order linear system.

Multi-objective optimal design of active vibration absorber with delayed feedback

“…In this paper, we shall apply the simple cell mapping (SCM) method to obtain the Pareto optimal solutions for the delayed feedback control design [17][18][19][20]. The search of the Pareto optimal solutions is done in the discretized parameter space in a two step manner.…”

“…Recent studies seem to suggest that Pareto front may have fine structures for MOPs of control systems [20,26].…”

“…In particular, we consider a TVA attached to a primary discrete structure such as motors or engines subject to wideband random excitations. The simple cell mapping (SCM) method is applied to solve for the MOP [17][18][19][20]. The continuous time approximation (CTA) method [21,22] is used to obtain the response of the delayed system and to analyze the stability and statistics of the responses.…”

“…max o À ε; k p ; k d A ½À3; 3; τ A ½0:1; 1:5g:(20) Initially, we select the number of divisions of the design parameter domain as N ¼ ½15; 15; 15. The subdivision is ½3; 3; 3.…”

Multi-objective optimal design of sliding mode control with parallel simple cell mapping method

“…Recently, the cell mapping methods originally developed by Hsu in the 1980s for global analysis of nonlinear dynamical systems (Hsu, 1987) are found to be highly effective in discovering the global structure of the Pareto set, the solution of the multi-objective optimization problems (MOPs) (Xiong et al, 2014;Naranjani et al, 2013). The cell mapping methods have been successfully applied to low and moderate dimensional problems and the multi-objective optimal proportional-integralderivative (PID) control design for linear and nonlinear dynamical systems by Hernández et al (2013).…”