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Solving the cost to go with time penalization using the Lagrange optimization approach

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Abstract

This paper suggests a new method for solving the cost to go with time penalization. We consider the Lagrange approach in order to incorporate the restrictions of the problem and to solve the convex structured minimization problems. The solution is based on an improved version of the proximal method in which the regularization term that asymptotically disappear involves a penalization time parameter. By assuming that the set of equilibria is non-empty, we show that the proximal algorithm involving time penalization generates a sequence that converges to a saddle point of the Lagrange functional. This method improves the traditional approach of proximal algorithms and its applications. Our approach is implemented using continuous-time Markov chains. We analyze a queueing model with applications to waiting time of customers in the branch of a bank, for showing the effectiveness of the method.

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References

  • Antipin AS (2005) An extraproximal method for solving equilibrium programming problems and games. Comput Math Math Phys 45(11):1893–1914

    MathSciNet  Google Scholar 

  • Attouch H, Soubeyran A (2011) Local search proximal algorithms as decision dynamics with costs to move. Set Valued Anal 19:157–177

    Article  MathSciNet  Google Scholar 

  • Bao TQ, Mordukhovich BS, Soubeyran A (2015) Variational analysis in psychological modeling. J Optim Theory Appl 164(1):290–315

    Article  MathSciNet  Google Scholar 

  • Bhat UN (2015) An introduction to queueing theory: modeling and analysis in applications, 2nd edn. Birkhäuser, Basel

    Book  Google Scholar 

  • Carrillo L, Escobar J, Clempner JB, Poznyak AS (2016) Solving optimization problems in chemical reactions using continuous-time Markov chains. J Math Chem 54:1233–1254

    Article  MathSciNet  Google Scholar 

  • Cascone A, Rarita L, Trapel E (2014) Simulation and analysis of a bank’s multi-server queueing system. J Math Sci 196(1):23–29

    Article  Google Scholar 

  • Clempner JB (2020) Penalizing passenger’s transfer time in computing airlines revenue. Omega 97:102099

    Article  Google Scholar 

  • Clempner JB, Poznyak AS (2014) Simple computing of the customer lifetime value: A fixed local-optimal policy approach. J Syst Sci Syst Eng 23(4):439–459

    Article  Google Scholar 

  • Clempner JB, Poznyak AS (2018a) A Tikhonov regularization parameter approach for solving Lagrange constrained optimization problems. Eng Optim. https://doi.org/10.1080/0305215X.2017.1418866, to be published

  • Clempner JB, Poznyak AS (2018b) A Tikhonov regularized penalty function approach for solving polylinear programming problems. J Comput Appl Math 328:267–286

    Article  MathSciNet  Google Scholar 

  • Garcia CB, Zangwill WI (1981) Pathways to solutions, fixed points and equilibria. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Ge Z, Han D, Ni Q, Wang D (2018) An operator splitting method for monotone variational inequalities with a new perturbation strategy. Optim Lett 12(1):103–122

    Article  MathSciNet  Google Scholar 

  • Guo X, Hernández-Lerma O (2009) Continuous-time Markov decision processes: theory and applications. Springer, Berlin

    Book  Google Scholar 

  • Lemaire B (1989) The proximal algorithm. Internat Ser Numer Math 87:73–87

    MathSciNet  MATH  Google Scholar 

  • Martinet B (1970) Breve communication. regularisation d’inequations variationnelles par approximations successives. Esaim Math Model Numer Anal 4(3):154–158

    MATH  Google Scholar 

  • Moreno FG, Oliveira PR, Soubeyran A (2011) A proximal algorithm with quasidistance. application to habit’s formation. Optimization 61:1383–1403

    Article  Google Scholar 

  • Poznyak AS, Najim K, Gomez-Ramirez E (2000) Self-learning control of finite Markov chains. Marcel Dekker Inc, New York

    MATH  Google Scholar 

  • Rockafellar R (1976) Monotone operators and the proximal point algorithm. SIAM J Control Optim 14(5):877–898

    Article  MathSciNet  Google Scholar 

  • Tikhonov AN, Arsenin VY (1977) Solution of Ill-posed problems. Winston & Sons, Washington

    MATH  Google Scholar 

  • Trejo KK, Clempner JB, Poznyak AS (2019) A proximal constrained optimization approach with time penalization. Eng Optim 51(7):1207–1228

    Article  MathSciNet  Google Scholar 

  • Yin G, Zhang Q (1998) Continuous-time Markov chains and applications: a singular perturbations approach. Springer-Verlag, New York

    Book  Google Scholar 

  • Zangwill WI (1969) Nonlinear programming: a unified approach. Prentice-Halt, Englewood Cliffs

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, School of Physics and Mathematics, National Polytechnic Institute, Building 9, Av. Instituto Politécnico Nacional, 07738, San Pedro Zacatenco, Gustavo A. Madero, Mexico City, Mexico

    Julio B. Clempner

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  1. Julio B. Clempner
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Correspondence to Julio B. Clempner.

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Clempner, J.B. Solving the cost to go with time penalization using the Lagrange optimization approach. Soft Comput 25, 4191–4199 (2021). https://doi.org/10.1007/s00500-020-05556-6

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  • DOI: https://doi.org/10.1007/s00500-020-05556-6

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