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Solving the Pareto front for multiobjective Markov chains using the minimum Euclidean distance gradient-based optimization method

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  • Clempner, Julio B.
  • Poznyak, Alexander S.

Abstract

A novel method based on minimizing the Euclidean distance is proposed for generating a well-distributed Pareto set in multi-objective optimization for a class of ergodic controllable Markov chains. The proposed approach is based on the concept of strong Pareto policy. We consider the case where the search space is a non-strictly convex set. For solving the problem we introduce the Tikhonov’s regularization method and implement the Lagrange principle. We formulate the original problem introducing linear constraints over the nonlinear problem employing the c-variable method and constraining the cost-functions allowing points in the Pareto front to have a small distance from one another. As a result, the proposed method generates an even representation of the entire Pareto surface. Then, we propose an algorithm to compute the Pareto front and provide all the details needed to implement the method in an efficient and numerically stable way. As well, we prove the main Theorems for describing the dependence of the saddle point for the regularizing parameter and analyzes its asymptotic behavior. Moreover, we analyze the step size parameter of the Lagrange principle and also its asymptotic behavior. The suggested approach is validated theoretically and verified by a numerical example related to security patrolling that present a technique for visualizing the Pareto front.

Suggested Citation

  • Clempner, Julio B. & Poznyak, Alexander S., 2016. "Solving the Pareto front for multiobjective Markov chains using the minimum Euclidean distance gradient-based optimization method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 119(C), pages 142-160.
    • Handle: RePEc:eee:matcom:v:119:y:2016:i:c:p:142-160
    DOI: 10.1016/j.matcom.2015.08.004
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    References listed on IDEAS

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    1. Pereyra, Victor, 2009. "Fast computation of equispaced Pareto manifolds and Pareto fronts for multiobjective optimization problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(6), pages 1935-1947.
    2. S. Schäffler & R. Schultz & K. Weinzierl, 2002. "Stochastic Method for the Solution of Unconstrained Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 114(1), pages 209-222, July.
    3. M. Dellnitz & O. Schütze & T. Hestermeyer, 2005. "Covering Pareto Sets by Multilevel Subdivision Techniques," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 113-136, January.
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    1. Julio B. Clempner & Alexander S. Poznyak, 2020. "Finding the Strong Nash Equilibrium: Computation, Existence and Characterization for Markov Games," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 1029-1052, September.
    2. Julio B. Clempner, 2018. "Computing multiobjective Markov chains handled by the extraproximal method," Annals of Operations Research, Springer, vol. 271(2), pages 469-486, December.

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