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Solving dual problems using a coevolutionary optimization algorithm

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Abstract

In solving certain optimization problems, the corresponding Lagrangian dual problem is often solved simply because in these problems the dual problem is easier to solve than the original primal problem. Another reason for their solution is the implication of the weak duality theorem which suggests that under certain conditions the optimal dual function value is smaller than or equal to the optimal primal objective value. The dual problem is a special case of a bilevel programming problem involving Lagrange multipliers as upper-level variables and decision variables as lower-level variables. Another interesting aspect of dual problems is that both lower and upper-level optimization problems involve only box constraints and no other equality of inequality constraints. In this paper, we propose a coevolutionary dual optimization (CEDO) algorithm for co-evolving two populations—one involving Lagrange multipliers and other involving decision variables—to find the dual solution. On 11 test problems taken from the optimization literature, we demonstrate the efficacy of CEDO algorithm by comparing it with a couple of nested smooth and nonsmooth algorithms and a couple of previously suggested coevolutionary algorithms. The performance of CEDO algorithm is also compared with two classical methods involving nonsmooth (bundle) optimization methods. As a by-product, we analyze the test problems to find their associated duality gap and classify them into three categories having zero, finite or infinite duality gaps. The development of a coevolutionary approach, revealing the presence or absence of duality gap in a number of commonly-used test problems, and efficacy of the proposed coevolutionary algorithm compared to usual nested smooth and nonsmooth algorithms and other existing coevolutionary approaches remain as the hallmark of the current study.

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Authors and Affiliations

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

    Kalyanmoy Deb

  2. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Kalyanmoy Deb

  3. Department of Information and Service Economy, Aalto University School of Economics, 00100, Helsinki, Finland

    Kalyanmoy Deb

  4. Naveen Jindal School of Management, University of Texas at Dallas, Richardson, TX, 75080, USA

    Shivam Gupta

  5. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India

    Joydeep Dutta

  6. Simon School of Business, University of Rochester, Rochester, NY, USA

    Bhoomija Ranjan

Authors
  1. Kalyanmoy Deb
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  2. Shivam Gupta
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  3. Joydeep Dutta
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  4. Bhoomija Ranjan
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Correspondence to Kalyanmoy Deb.

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Deb, K., Gupta, S., Dutta, J. et al. Solving dual problems using a coevolutionary optimization algorithm. J Glob Optim 57, 891–933 (2013). https://doi.org/10.1007/s10898-012-9981-5

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  • DOI: https://doi.org/10.1007/s10898-012-9981-5

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