On Tchebycheff Decomposition Approaches for Multiobjective Evolutionary Optimization

Tchebycheff decomposition represents one of the most widely used decomposition approaches that can convert a multiobjective optimization problem into a set of scalar optimization subproblems. Nevertheless, the geometric properties of the subproblem objective functions in Tchebycheff decomposition have not been explicitly studied. This paper proposes a Tchebycheff decomposition with lp-norm constraint on direction vectors in which the subproblem objective functions are endowed with clear geometric property. Especially, the Tchebycheff decomposition with l
₂
-norm constraint on direction vectors is taken as an example to illustrate its advantage. A new unary R
₂
indicator is also introduced to approximate the hyper-volume metric and justify the efficiency of the proposed Tchebycheff decomposition. A resultant Tchebycheff decomposition-based multiobjective evolutionary algorithm (MOEA) with l
₂
-norm constraint and a new population update strategy is proposed to solve multiobjective optimization problems. The experimental results on both benchmark and real-world multiobjective optimization problems show that the proposed algorithm is capable of obtaining high quality solutions compared with other state-of-the-art MOEAs.