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Uniform adaptive scaling of equality and inequality constraints within hybrid evolutionary-cum-classical optimization

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Abstract

The holy grail of constrained optimization is the development of an efficient, scale invariant and generic constraint handling procedure. To address these, the present paper proposes a unified approach of constraint handling, which is capable of handling all inequality, equality and hybrid constraints in a coherent manner. The proposed method also automatically resolves the issue of constraint scaling which is critical in real world and engineering optimization problems. The proposed unified approach converts the single-objective constrained optimization problem into a multi-objective problem. Evolutionary multi-objective optimization is used to solve the modified bi-objective problem and to estimate the penalty parameter automatically. The constrained optimum is further improved using classical optimization. The efficiency of the proposed method is validated on a set of well-studied constrained test problems and compared against without using normalization technique to show the necessity of normalization. The results establish the importance of scaling , especially in constrained optimization and call for further investigation into its use in constrained optimization research.

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Notes

  1. For example, if we have a constraint of the form \(g_i(\mathbf{x}) \ge 0\), then for a candidate \(\mathbf{x}_c\), if \(g_i(\mathbf{x}_c)\) is indeed \(\ge 0\) we take \(v_i = 0\); otherwise, we take \(v_i = - g_i(\mathbf{x}_c)\). Finally, we take \(v = \sum _i v_i\), which is nonnegative because each \(v_i\) is nonnegative. We could also take \(v = \sum _i c_i v_i\), where the \(c_i\) are positive weighting factors, as discussed later in the main text.

  2. It need not have an unconstrained minimum in general. For example, consider \(f(x) = x\) (scalar) with the constraint \(x \ge 0\); if we remove the constraint then the minimum is \(- \infty \) at \(x = -\infty \). However, our design variables are range bound by assumption.

  3. Consider a problem where (say) the 7th largest eigenvalue of a 20 \(\times \) 20 symmetric matrix is constrained to be equal to \(14\). The constraint cannot be expressed as a simple explicit function of the design variables (the matrix elements).

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Acknowledgments

We thank Anindya Chatterjee, Mechanical Engineering, IIT Kanpur for proposing development of this unified approach, and some related suggestions.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India

    Rituparna Datta

  2. Department of Electrical and Computer Engineering, Michigan State University, 428 S. Shaw Lane, 2120 EB, East Lansing, MI, 48824, USA

    Kalyanmoy Deb

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  1. Rituparna Datta
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Correspondence to Rituparna Datta.

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Communicated by V. Loia.

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Datta, R., Deb, K. Uniform adaptive scaling of equality and inequality constraints within hybrid evolutionary-cum-classical optimization. Soft Comput 20, 2367–2382 (2016). https://doi.org/10.1007/s00500-015-1646-0

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