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Augmented Lagrangian cone method for multiobjective optimization problems with an application to an optimal control problem

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Abstract

This paper proposes an augmented Lagrangian method to compute Pareto optimal sets of multiobjective optimization problems. The method neither requires a prior information about the locations of the Pareto surface nor the convexity of the objective and constraint functions. To generate Pareto optimal points, we convert a multiobjective optimization problem into a set of direction-based parametric scalar optimization problems by using the cone method. Subsequently, we apply the augmented Lagrangian method to the direction-based parametric problems to transform them into unconstrained problems. Transformed augmented Lagrangian subproblems are then solved by the steepest descent method with a max-type nonmonotone line search method. A step-wise algorithmic implementation of the proposed method is provided. We discuss the convergence property of the proposed algorithm with regard to a feasibility measure and the global Pareto optimality. Under a few common assumptions, we prove that any subsequential limit of the sequence generated by the proposed algorithm is the global minimizer of an infeasibility measure corresponding to each direction. In addition, the obtained limit is found to be a global minimizer when the feasible region of the given multiobjective optimization problem is nonempty. It is observed that the solution of the proposed method is not affected by variable scaling. The efficiency of the proposed algorithm is shown by solving standard test problems. As a realistic application, we employ the proposed method on a deterministic unemployment optimal control model with the implementation of government policies to create employment and vacancies as their controls.

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Code availability

For Matlab codes of the proposed algorithms, readers can request to the first author (Ashutosh Upadhayay).

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Acknowledgements

The authors would like to thank the editor and the referee for their valuable comments and suggestions, which improved the quality of the paper. Ashutosh Upadhayay thankfully acknowledges financial support from Council of Scientific and Industrial Research, India through a research fellowship (File. No. 09/1217(0047)2018-EMR-I) to carry out this research work. Debdas Ghosh acknowledges the research Grant (MTR/2021/000696) from SERB, India to carry out this research work.

Funding

Ashutosh Upadhayay thankfully acknowledges financial support from Council of Scientific and Industrial Research, India through a research fellowship (File. No. 09/1217(0047)2018-EMR-I) to carry out this research work. Debdas Ghosh acknowledges the research Grant (MTR/2021/000696) from SERB, India to carry out this research work.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi, Uttar Pradesh, 221005, India

    Ashutosh Upadhayay, Debdas Ghosh &  Jauny

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India

    Qamrul Hasan Ansari

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  1. Ashutosh Upadhayay
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  2. Debdas Ghosh
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  3. Qamrul Hasan Ansari
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  4. Jauny
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Upadhayay, A., Ghosh, D., Ansari, Q.H. et al. Augmented Lagrangian cone method for multiobjective optimization problems with an application to an optimal control problem. Optim Eng 24, 1633–1665 (2023). https://doi.org/10.1007/s11081-022-09747-y

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