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Some properties of the lower bound of optimal values in interval convex quadratic programming

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Abstract

One of the fundamental problems in interval quadratic programming is to compute the range of optimal values. In this paper, we derive some results on the lower bound of interval convex quadratic programming. We first develop complementary slackness conditions of a quadratic program and its Dorn dual. Then, some interesting and useful characteristics of the lower bound of interval quadratic programming are established based on these conditions. Finally, illustrative examples and remarks are given to get an insight into the problem discussed.

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References

  1. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to interval analysis. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  2. Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  3. Rohn, J.: Strong solvability of interval linear programming problems. Comp. 26, 79–82 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48, 219–225 (1990)

    Article  MATH  Google Scholar 

  5. Chinneck, J.W., Ramadan, K.: Linear programming with interval coefficients. J. Oper. Res. Soc. 51(2), 209–220 (2000)

    Article  MATH  Google Scholar 

  6. Shary, S.P.: A new technique in systems analysis under interval uncertainty and ambiguity. Reliab. Comput. 8(5), 321–418 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wu, X.Y., Huang, G.H., Liu, L., Li, J.B.: An interval nonlinear program for the planning of waste management systems with economies-of-scale effects a case study for the region of Hamilton, Ontario, Canada. Eur. J. Oper. Res. 171 (2), 349–372 (2006)

  8. Inuiguchi, M., Ramik, J., Tanino, T., Vlach, M.: Satisficing solutions and duality in interval and fuzzy linear programming. Fuzzy Sets Syst. 135(1), 151–177 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fiedler, M., Nedoma, J., Ramik, J., Rohn, J., Zimmermann, K.: Linear optimization problems with inexact data. Springer-Verlag, New York (2006)

    MATH  Google Scholar 

  10. Prokopyev, O.A., Butenko, S., Trapp, A.C.: Checking solvability of systems of interval linear equations and inequalities via mixed integer programming. Eur. J. Oper. Res. 199(1), 117–121 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gabrel, V., Murat, C., Remli, N.: Linear programming with interval right hand sides. Int. Trans. Oper. Res. 17(3), 397–408 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, W., Wang, H., Wang, Q.: Localized solutions to interval linear equations. J. Comput. Appl. Math. 238(15), 29–38 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hladík, M.: How to determine basis stability in interval linear programming. Optim. Lett. 8(1), 375–389 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, W., Luo, J., Wang, Q., Li, Y.: Checking weak optimality of the solution to linear programming with interval right-hand side. Optim. Lett. 8(4), 1287–1299 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Luo, J., Li, W.: Strong optimal solutions of interval linear programming. Linear Algebra Appl. 439, 2479–2493 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, W., Liu, X., Li, H.: Generalized solutions to interval linear programmes and related necessary and sufficient optimality conditions. Optim. Methods Softw. 30(3), 516–530 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hladík, M.: Interval linear programming: a survey, In: Zoltan Adam Mann Edits, Linear Programming New Frontiers, pp. 1–46. Nova Science Publishers Inc., New York (2012)

  18. Wang, X., Huang, G.H.: Violation analysis on two-step method for interval linear programming. Inf. Sci. 281, 85–96 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mráz, F.: Calculating the exact bounds of optimal values in LP with interval coefficients. Ann. Oper. Res. 81, 51–62 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, S.T., Wang, R.T.: A numerical solution method to interval quadratic programming. Appl. Math. Comput. 189, 1274–1281 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Li, W., Tian, X.: Numerical solution method for general interval quadratic programming. Appl. Math. Comput. 202, 589–595 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Hladík, M.: Optimal value range in interval linear programming. Fuzzy Optim. Decis. Mak. 8(3), 283–294 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hladík, M.: Optimal value bounds in nonlinear programming with interval data. Top 19, 93–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hladík, M.: Weak and strong solvability of interval linear systems of equations and inequalities. Linear Algebra Appl. 438, 4156–4165 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hladík, M.: On approximation of the best case optimal value in interval linear programming. Optim. Lett. 8, 1985–1997 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, W., Xia, M., Li, H.: New method for computing the upper bound of optimal value in interval quadratic program. J. Comput. Appl. Math. 288, 70–80 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rohn, J.: Interval linear programming, Linear optimization problems with inexact data. In: Fiedler, M. et al. (eds.) Springer, New York (2006)

  28. Li, W., Xia, M., Li, H.: Some results on the upper bound of optimal values in interval convex quadratic programming. J. Comput. Appl. Math. 302, 38–49 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dorn, W.S.: Duality in quadratic programming. Quart. Appl. Math. 18(2), 155–162 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  30. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear programming, Theory and algorithms, 2nd edn. Wiley, New York (1993)

    MATH  Google Scholar 

  31. Farkas, J.: Theorie der einfachen Ungleichungen (Theory of simple inequalities). Zeitschrift für die Reine und Angewandte Mathematik 124, 1–27 (1902)

    Google Scholar 

  32. Ishizaki, T., Koike, M., Ramdani, N., et al.: Interval quadratic programming for day-ahead dispatch of uncertain predicted demand. Automatica 64, 163–173 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hladík, M.: Interval convex quadratic programming problems in a general form. CEJOR (2016). doi:10.1007/s10100-016-0445-8

  34. Hladík, M.: Transformations of interval linear systems of equations and inequalities. Linear Multilinear Algebra (2016). doi:10.1080/03081087.2016.1180339

  35. Li, W.: A note on dependency between interval linear systems. Optim. Lett. 9, 795–797 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank anonymous referees for their comments and suggestions that helped to improve the paper. Especially, Theorem 3.4 is enhanced according to one of the reviewers’ suggestion. The authors were partially supported by the NSF of Zhejiang Province (Grant Nos. LY14A010028, Xinmiao2016R407079) and NNSF of China (Grant Nos. 61673145, 11526184, 71471051, U1509217).

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Authors and Affiliations

  1. Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou, 310018, China

    Wei Li, Jianghong Jin & Mengxue Xia

  2. School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou, 310018, China

    Haohao Li

  3. School of Information and Control, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Qi Luo

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  1. Wei Li
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  2. Jianghong Jin
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  3. Mengxue Xia
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  4. Haohao Li
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  5. Qi Luo
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Correspondence to Haohao Li.

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Li, W., Jin, J., Xia, M. et al. Some properties of the lower bound of optimal values in interval convex quadratic programming. Optim Lett 11, 1443–1458 (2017). https://doi.org/10.1007/s11590-016-1097-2

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