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A penalty function-based differential evolution algorithm for constrained global optimization

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Abstract

We propose a differential evolution-based algorithm for constrained global optimization. Although differential evolution has been used as the underlying global solver, central to our approach is the penalty function that we introduce. The adaptive nature of the penalty function makes the results of the algorithm mostly insensitive to low values of the penalty parameter. We have also demonstrated both empirically and theoretically that the high value of the penalty parameter is detrimental to convergence, specially for functions with multiple local minimizers. Hence, the penalty function can dispense with the penalty parameter. We have extensively tested our penalty function-based DE algorithm on a set of 24 benchmark test problems. Results obtained are compared with those of some recent algorithms.

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Notes

  1. Further discussions on the initial choice of U ∗ is presented in the discussion section later in the paper.

  2. If x is the first feasible point generated by CDE then U ∗=f(x). Subsequently, the feasibility of every point x generated by CDE is checked. If the point x is feasible and f(x)<U ∗ holds then U ∗ is updated as U ∗=f(x). The penalty function value L(x) is then calculated.

  3. This is done by calculating ψ(x i,1) which is then used to evaluate L(x i,1) at x i,1 via (8).

  4. The selection rule of CDE is similar to (6). The only difference is that CDE uses the penalty function L(x) in (6) instead of f(x).

  5. We have made this choice based on our observation on results using numerical testing.

  6. Notice that since i=1,2,…,N, we can consider a upper bound \(U_{*}^{i,k}\) corresponding to y i,k . Hence, prior to the creation of T k , the best known upper bound \(U_{*}^{N,k-1} ({=}U_{*})\) is the overall upper bound generated at the end of T k−1. However, all upper bounds, \(U_{*}^{i,k}\), at the k-th iteration are not necessarily distinct. Consider any two consecutive updates at the k-th iteration, say \(U_{*}^{i,k}\) (the update occurred corresponding to y i,k ) and \(U_{*}^{j,k}\) (the update occurred corresponding to y j,k , j>i, j=i+p) then \(U_{*}^{i,k}=U_{*}^{i+1,k}=\cdots=U_{*}^{i+p-1} ({>}U_{*}^{j,k})\). The case when all upper bounds are distinct is explained in the next subsection.

  7. In the example below (17) the two updates \(U_{*}^{1,k}\) and \(U_{*}^{\frac{N}{2},k}\) can be treated as \(\{U_{*}^{1,k},U_{*}^{\frac{N}{2},k}\}=\{U_{*}^{1,k},U_{*}^{2,k}\}\).

  8. The upper bound \(U_{*}^{i,k}\) in the relation corresponds to y i,k ∈T i,k .

  9. The feasible x i,k may also be obtained during the generation of the initial set, i.e., x i,k =x i,1∈S 1. Hence, we can use \(\bar{k}\ge0\) by assuming the members of S 1 as the members of the 0-th trial population where all members are accepted.

  10. Results for CDE are based the decimal points accuracies of f ∗ presented in Table 1.

  11. A slightly high mfe incurs for problems G6, G11 and G24 when we use the population size N=50.

  12. G17 is a non-smooth function, and it is not clear as to how SQP can be implemented.

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Authors and Affiliations

  1. School of Computational and Applied Mathematics, Witwatersrand University, Wits 2050, Johannesburg, South Africa

    M. M. Ali

  2. Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou, 350002, China

    W. X. Zhu

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  1. M. M. Ali
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Correspondence to M. M. Ali.

Additional information

M.M. Ali was supported by the National Research Foundation of South Africa under Grant FA2006042300001.

W.X. Zhu was supported by the National Natural Science Foundation of China under Grant 61170308.

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Ali, M.M., Zhu, W.X. A penalty function-based differential evolution algorithm for constrained global optimization. Comput Optim Appl 54, 707–739 (2013). https://doi.org/10.1007/s10589-012-9498-3

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