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Learning to Understand the Problem Structure

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Machine Learning Assisted Evolutionary Multi- and Many- Objective Optimization

Part of the book series: Genetic and Evolutionary Computation ((GEVO))

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Abstract

This chapter focuses on an important aspect of learning the preference structure of the objectives, inherent in multi- and many-objective optimization problem formulations. This involves identifying the non-essential (redundant) objectives, and also determining the relative importance of the essential objectives. Such an approach to knowledge discovery is based on the following rationale. Modeling an optimization problem, analytically or through experiments, involves a lot of time and physical resources, possibly from multiple disciplines, in conjunction or isolation from each other. Often, it can be intriguing for analysts or decision makers (DMs) to know if the developed model represents the underlying problem in a minimal form or is marked by redundancy. Any redundancy among objectives, if revealed, could shed insightful light on the physics of the underlying problem, in addition to reducing its complexity and promising greater search efficiency for evolutionary multi- and many-objective optimization algorithms (EMâOAs). Furthermore, the revelation of the relative preferences among the essential objectives that are inherent in the problem models could also be significantly useful, as highlighted below.

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Notes

  1. 1.

    For a given \(0 \le \delta \le 0\), there may be multiple subsets of objectives which ensure that the error associated with the the omission of the remaining objectives does not exceed \(\delta \). Each such subset is referred to as a \(\delta \)-minimal objective subset. However, the \(\delta \)-minimal objective subset having the smallest size is referred to as the \(\delta \)-minimum objective subset.

  2. 2.

    Here, dimensionality refers to the number of objectives that are essential to characterize the complete \(P\!F\).

  3. 3.

    This contribution can be given by \(sc_i = \sum _{j=1}^{N_v} e_jf_{ij}^2\) (Sect. 4.2.2). However, \(e_j\) and \(f_{ij}\) being less than one, may lead to indiscriminately low values for \(sc_i\), hence, adapted in Eq. 4.6.

  4. 4.

    This is also true for DTLZ7 but it is precluded here due to the inconsistency between its formulation and its \(P\!F\), as presented in [14].

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Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Dhish Kumar Saxena

  2. Franklin Templeton, Hyderabad, India

    Sukrit Mittal

  3. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, USA

    Kalyanmoy Deb

  4. Michigan State University, East Lansing, MI, USA

    Erik D. Goodman

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Correspondence to Kalyanmoy Deb .

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Saxena, D.K., Mittal, S., Deb, K., Goodman, E.D. (2024). Learning to Understand the Problem Structure. In: Machine Learning Assisted Evolutionary Multi- and Many- Objective Optimization. Genetic and Evolutionary Computation. Springer, Singapore. https://doi.org/10.1007/978-981-99-2096-9_4

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  • DOI: https://doi.org/10.1007/978-981-99-2096-9_4

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