Abstract
In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained integer optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or quasi-convex polynomial function over the feasible set contained in \({\mathbb {Z}^n}\) , and defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities. The conditions for boundedness are provided in the form of an implementable algorithm, terminating after a finite number of iterations, showing that for the considered class of functions, the integer programming problem with nonempty feasible region is unbounded if and only if the associated continuous optimization problem is unbounded. We also prove that for a broad class of objective functions (which in particular includes polynomials with integer coefficients), an optimal solution set of the constrained integer problem is nonempty over any subset of \({\mathbb {Z}^n}\) .
Similar content being viewed by others
References
Auslender A (1997) How to deal with the unbounded in optimization: theory and algorithms. Math Program Ser B 79(1–3): 3–18
Bank B, Mandel R (1988) Parametric integer optimization, Mathematical Research, vol 39. Academie-Verlag, Berlin
Bank B, Guddat J, Klatte B, Kummer B, Tammer K (1983) Nonlinear parametric optimization. Birkhauser Verlag, Basel
Belousov EG, Klatte D (2002) A Frank–Wolfe type theorem for convex polynomial programs. Comput Optim Appl 22(1): 37–48
Floudas CA (1995) Nonlinear and mixed-integer optimization; fundamentals and applications. Oxford University Press, Oxford
Frank M, Wolfe P (1956) An algorithm for quadratic programming. Naval Res Logist Q 3(1–2): 95–110
Hiriart-Urruty J-B, Lemaréchal C (1993) Convex analysis and minimization algorithms I. Springer, Heidelberg
Luo Z-Q, Zhang S (1999) On extensions of the Frank–Wolfe theorems. Comput Optim Appl 13(1–3): 87–110
Nemhauser GL, Wolsey LA (1988) Integer and combinatorial optimization, Interscience Series in Discrete Mathematics and Optimization. Wiley, London
Obuchowska WT (1999) Convex constrained programmes with unattained infima. J Math Anal Appl 234(2): 232–245
Obuchowska WT (2004) Remarks on the analytic centres of convex sets. Comput Optim Appl 29(1): 69–90
Obuchowska WT (2006) On generalizations of the Frank–Wolfe theorem to convex and quasi-convex programmes. Comput Optim Appl 33(2–3): 349–364
Obuchowska WT (2007) Conditions for boundedness in concave programming under reverse convex and convex constraints. Math Methods Oper Res 65(2): 261–279
Rockafellar RT (1970) Some convex programs whose duals are linearly constrained. In: Rosen JB, Mangasarian OL, Ritter K(eds) Nonlinear programming. Academic Press, New York
Rockafellar RT (1979) Convex analysis. Princeton University Press, Princeton
Salkin HM, Mathur K (1989) Foundations of integer programming. North Holland, New York
Schrijver A (1986) Theory of linear and integer programming. Wiley, London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Obuchowska, W.T. On boundedness of (quasi-)convex integer optimization problems. Math Meth Oper Res 68, 445–467 (2008). https://doi.org/10.1007/s00186-007-0196-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-007-0196-3