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Journal of Global Optimization

, Volume 46, Issue 3, pp 423–433 | Cite as

Minimal infeasible constraint sets in convex integer programs

  • Wiesława T. Obuchowska
Article

Abstract

In this paper we investigate certain aspects of infeasibility in convex integer programs, where the constraint functions are defined either as a composition of a convex increasing function with a convex integer valued function of n variables or the sum of similar functions. In particular we are concerned with the problem of an upper bound for the minimal cardinality of the irreducible infeasible subset of constraints defining the model. We prove that for the considered class of functions, every infeasible system of inequality constraints in the convex integer program contains an inconsistent subsystem of cardinality not greater than 2 n , this way generalizing the well known theorem of Scarf and Bell for linear systems. The latter result allows us to demonstrate that if the considered convex integer problem is bounded below, then there exists a subset of at most 2 n −1 constraints in the system, such that the minimum of the objective function subject to the inequalities in the reduced subsystem, equals to the minimum of the objective function over the entire system of constraints.

Keywords

Feasibility Infeasibility Convex integer programming 

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References

  1. 1.
    Bank B., Mandel R.: Parametric Integer Optimization. Mathematical Research, vol. 39. Academie-Verlag, Berlin (1988)Google Scholar
  2. 2.
    Bauschke H.H., Borwein J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)CrossRefGoogle Scholar
  3. 3.
    Bell D.E.: A theorem concerning the integer lattice. Stud. Appl. Math. 56, 187–188 (1977)Google Scholar
  4. 4.
    Chinneck J.W., Dravnieks E.W.: Locating minimal infeasible constraint sets in linear programs. ORSA J. Comput. 3, 157–168 (1991)Google Scholar
  5. 5.
    Chinneck J.W.: Feasibility and Infeasibility in Optimization: Algorithms and Computational Methods. Springer, New York (2008)Google Scholar
  6. 6.
    Fiacco A.V., McCormick G.P.: Nonlinear Programming: A Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)Google Scholar
  7. 7.
    Floudas C.A.: Nonlinear and Mixed-Integer Optimization; Fundamentals and Applications. Oxford University Press, New York (1995)Google Scholar
  8. 8.
    Guieu O., Chinneck J.W.: Analyzing infeasible mixed-integer and integer linear programs. INFORMS J. Comput. 11, 63–77 (1999)CrossRefGoogle Scholar
  9. 9.
    Nemhauser G.L., Wolsey L.A.: Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization, New York (1988)Google Scholar
  10. 10.
    Ng C.-K., Li D., Zhang L.-S.: Discrete global method for discrete global optimization and nonlinear linear programming. J. Glob. Optim. 37, 357–379 (2007)CrossRefGoogle Scholar
  11. 11.
    Obuchowska W.T.: Infeasibility analysis for systems of quadratic convex inequalities. Eur. J. Oper. Res. 107, 633–643 (1998)CrossRefGoogle Scholar
  12. 12.
    Obuchowska W.T.: On infeasibility of systems of convex analytic inequalities. J. Math. Anal. Appl. 234, 223–245 (1999)CrossRefGoogle Scholar
  13. 13.
    Obuchowska W.T.: On generalizations of the Frank-Wolfe theorem to convex and quasi-convex programmes. Comput. Optim. Appl. 33(2–3), 349–364 (2006)CrossRefGoogle Scholar
  14. 14.
    Obuchowska W.T.: On boundedness of (quasi-)convex integer optimization problems. Math. Methods Oper. Res. 68, 445–467 (2008)CrossRefGoogle Scholar
  15. 15.
    Patel J., Chinneck J.W.: Active constraint variable ordering for faster feasibility of mixed integer linear programs. Math. Program. 110, 445–474 (2007)CrossRefGoogle Scholar
  16. 16.
    Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)Google Scholar
  17. 17.
    Salkin H.M., Mathur K.: Foundations of Integer Programming. Elsevier, North Holland, New York, Amsterdam, London (1989)Google Scholar
  18. 18.
    Scarf H.E.: An observation on the structure of production sets with indivisibilities. Proc. Natl. Acad. Sci. (Proceedings of the National Academy of Sciences of the United States of America) 74(9), 3637–3641 (1977)CrossRefGoogle Scholar
  19. 19.
    Schrijver A.: Theory of Linear and Integer Programming. Wiley, New York (1986)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of MathematicsEast Carolina UniversityGreenvilleUSA

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