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 2n, 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 2n−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.
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References
Bank B., Mandel R.: Parametric Integer Optimization. Mathematical Research, vol. 39. Academie-Verlag, Berlin (1988)
Bauschke H.H., Borwein J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)
Bell D.E.: A theorem concerning the integer lattice. Stud. Appl. Math. 56, 187–188 (1977)
Chinneck J.W., Dravnieks E.W.: Locating minimal infeasible constraint sets in linear programs. ORSA J. Comput. 3, 157–168 (1991)
Chinneck J.W.: Feasibility and Infeasibility in Optimization: Algorithms and Computational Methods. Springer, New York (2008)
Fiacco A.V., McCormick G.P.: Nonlinear Programming: A Sequential Unconstrained Minimization Techniques. Wiley, New York (1968)
Floudas C.A.: Nonlinear and Mixed-Integer Optimization; Fundamentals and Applications. Oxford University Press, New York (1995)
Guieu O., Chinneck J.W.: Analyzing infeasible mixed-integer and integer linear programs. INFORMS J. Comput. 11, 63–77 (1999)
Nemhauser G.L., Wolsey L.A.: Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization, New York (1988)
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)
Obuchowska W.T.: Infeasibility analysis for systems of quadratic convex inequalities. Eur. J. Oper. Res. 107, 633–643 (1998)
Obuchowska W.T.: On infeasibility of systems of convex analytic inequalities. J. Math. Anal. Appl. 234, 223–245 (1999)
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)
Obuchowska W.T.: On boundedness of (quasi-)convex integer optimization problems. Math. Methods Oper. Res. 68, 445–467 (2008)
Patel J., Chinneck J.W.: Active constraint variable ordering for faster feasibility of mixed integer linear programs. Math. Program. 110, 445–474 (2007)
Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Salkin H.M., Mathur K.: Foundations of Integer Programming. Elsevier, North Holland, New York, Amsterdam, London (1989)
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)
Schrijver A.: Theory of Linear and Integer Programming. Wiley, New York (1986)
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Obuchowska, W.T. Minimal infeasible constraint sets in convex integer programs. J Glob Optim 46, 423–433 (2010). https://doi.org/10.1007/s10898-009-9443-x
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DOI: https://doi.org/10.1007/s10898-009-9443-x