Evolutionary Computation with Biogeography-based Optimization

Appendix CMulti-objective Benchmark Functions

A multi-objective optimization problem (MOP) involves the minimization of f(x) over all x, where f(x) is a vector, and x is the n-dimensional decision vector. Vector minimization is undefined in the normal sense of the word, and so we define the Pareto set P_s and the Pareto front P_f in Chapter 11. We can then pose an MOP as the problem of finding the “best” possible P_s and P_f.

Detailed information about the multi-objective benchmarks and evaluation metrics for EA competition at the 2009 IEEE Congress on Evolutionary Computation can be found in Zhang et al. [ZHA 08]. The dimension of the independent variable in the benchmarks given below is variable, but the CEC 2009 competition used n = 30.

U01 Unconstrained Problem 1

The two objectives to be minimized:

[C.1]

where J₁ ={j|j is odd and 2 ≤ j ≤ n} and J₂ ={j|j is even and 2 ≤ j ≤ n}.

The search space is [0,1]×[−1,1]ⁿ⁻¹.

Its Pareto front is

[C.2]

Its Pareto set is

[C.3]

U02 Unconstrained Problem 2

The two objectives to be minimized:

[C.4]

where

and

[C.5]

Its search space is [0,1]×[−1,1] ...