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Robust continuous linear programs

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Abstract

Continuous linear programs (CLPs) arise in applications such as production/economic planning, continuous-time network flow problems, fluid relaxations of multiclass queueing networks, and control. To the best of the author’s knowledge, this paper proposes the first robust optimization framework for CLPs. The main result of the paper is that the robust counterpart of a CLP is also a CLP. Thus, any computational method for the original problem can be applied to the robust problem. For instance, a recent polynomial-time approximation algorithm applies. Further, if the original problem possesses a so-called separable structure, then the robust problem is also separable. Then existing Simplex-type and other discretization-based solution methods can be applied to the robust problem. The paper also provides a bound on the probability that an optimal solution to the robust counterpart violates a constraint in the original problem. Qualitative properties of this bound are discussed and compared with similar bounds for robust finite-dimensional linear programs.

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References

  1. Anderson, E.J.: A Continuous Model for Job-Shop Scheduling. Ph.D. thesis, University of Cambridge, Cambridge (1978)

  2. Anderson, E.J., Nash, P.: Linear Programming in Infinite-Dimensional Spaces: Theory and Applications. Wiley, Chichester (1987)

    MATH  Google Scholar 

  3. Anderson, E.J., Nash, P., Perold, A.F.: Some properties of a class of continuious linear programs. SIAM J. Control Optim. 21(5), 758–765 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anderson, E.J., Philpott, A.B.: A continuous-time network simplex algorithm. Networks 19(4), 395–425 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bampou, D., Kuhn, D.: Polynomial approximations for continuous linear programs. SIAM J. Optim. 22(2), 628–648 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bellman, R.E.: Bottleneck problem and dynamic programming. Proc. Natl. Acad. Sci. 39(9), 947–951 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben-Tal, A., Nemirovski, A.: Robust solutions of linear programming problems contaminated with uncertain data. Math. Program. 88(3), 411–424 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bertsimas, D., Tsitsiklis, J.: Introduction to Linear Optimization. Athena Scientific, Belmont (1997)

    Google Scholar 

  11. Buie, R.N., Abrham, J.: Numerical solutions to continuous linear programming problems. Math. Methods Oper. Res. 17(3), 107–117 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fleischer, L., Sethuraman, J.: Efficient algorithms for separated continuous linear programs: the multicommodity flow problem with holding costs and extensions. Math. Oper. Res. 30(4), 785–1040 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ghate, A.: Robust optimization in countably infinite linear programs. Optim. Lett. 10(4), 847–863 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grinold, R.C.: Symmetric duality for continuous linear programs. SIAM J. Appl. Math. 18(1), 32–51 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ito, S., Kelley, C.T., Sachs, E.W.: Inexact primal dual interior point iteration for linear program in function spaces. Comput. Optim. Appl. 4(3), 189–201 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover, New York (1975)

    Google Scholar 

  17. Levinson, N.: A class of continuous linear programming problems. J. Math. Anal. Appl. 16(1), 73–83 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  18. Luenberger, D.G.: Optimization by Vector Space Methods. Wiley, New York (1969)

    MATH  Google Scholar 

  19. Luo, X., Bertsimas, D.: A new algorithm for state-constrained separated continuous linear programs. SIAM J. Control Optim. 37(1), 177–210 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Perold, A.F.: Extreme points and basic feasible solutions in continuous time linear programming. SIAM J. Control Optim. 19(1), 52–63 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Philpott, A.B., Craddock, M.: An adaptive discretization algorithm for a class of continuous network programs. Networks 26(1), 1–11 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pullan, M.C.: An algorithm for a class of continuous linear programs. SIAM J. Control Optim. 31(6), 1558–1577 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pullan, M.C.: Forms of optimal solutions for separated continuous linear programs. SIAM J. Control Optim. 33(6), 1952–1977 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pullan, M.C.: A duality theory for separated continuous linear programs. SIAM J. Control Optim. 34(3), 931–965 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pullan, M.C.: Linear optimal control problems with piecewise analytic solutions. J. Math. Anal. Appl. 197(1), 207–226 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pullan, M.C.: Convergence of a general class of algorithms for separated continuous linear programs. SIAM J. Optim. 10(3), 722–731 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pullan, M.C.: An extended algorithm for separated continuous linear programs. Math. Program. 93(3), 415–451 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ross, S.M.: Introduction to Probability Models, 10th edn. Elsevier, Burlington (2010)

    MATH  Google Scholar 

  29. Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)

    MATH  Google Scholar 

  30. Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21(5), 1154–1157 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tyndall, W.F.: A duality theorem for a class of continuous linear programming problems. SIAM J. Appl. Math. 13(3), 644–666 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  32. Tyndall, W.F.: An extended duality theorem for continuous linear programming problems. SIAM J. Appl. Math. 15(5), 1294–1298 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  33. Weiss, G.: A simplex based algorithm to solve separated continuous linear programs. Math. Program. 115(1), 151–198 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This research was funded in part by the National Science Foundation Grant #CMMI 1561918.

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  1. University of Washington, Seattle, WA, USA

    Archis Ghate

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  1. Archis Ghate
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Ghate, A. Robust continuous linear programs. Optim Lett 14, 1627–1642 (2020). https://doi.org/10.1007/s11590-020-01539-6

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  • DOI: https://doi.org/10.1007/s11590-020-01539-6

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