Abstract
A well-known linearization technique for nonlinear 0–1 maximization problems can be viewed as extending any polynomial in 0–1 variables to a concave function defined on [0, 1]n. Some properties of this “standard” concave extension are investigated. Polynomials for which the standard extension coincides with the concave envelope are characterized in terms of integrality of a certain polyhedron or balancedness of a certain matrix. The standard extension is proved to be identical to another type of concave extension, defined as the lower envelope of a class of affine functions majorizing the given polynomial.
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References
W.P. Adams and P.M. Dearing, “On the equivalence between roof duality and Lagrangian duality for unconstrained 0–1 quadratic programming problems,” Technical Report URI-061, Clemson University (Clemson, SC, 1988).
E. Balas, Notes of the three lectures given at Cornell University in the distinguished lecturer series, GSIA, Carnegie Mellon University, (Pittsburgh, PA, 1987).
E. Balas and J.B. Mazzola. “Nonlinear 0–1 programming: I. Linearization techniques,”Mathematical Programming 30 (1984) 1–22.
M. Conforti, G. Cornuejols and M.R. Rao, “Balanced 0–1 matrices: a recognition algorithm,” Paper presented atthe 14th International Symposium on Mathematical Programming, Amsterdam, The Netherlands (1991).
Y. Crama, “Linearization techniques and concave extensions in nonlinear 0–1 optimization,” RUTCOR Research Report 32-88, Rutgers University, (New Brunswick, NJ, 1988).
Y. Crama, “Recognition problems for special classes of polynomials in 0–1 variables,”Mathematical Programming 44 (1989) 139–155.
Y. Crama, P. Hansen and B. Jaumard, “The basic algorithm for pseudo-Boolean programming revisited,”Discrete Applied Mathematics 29 (1990) 171–185.
J.E. Falk and K.R. Hoffman, “A successive underestimation method for concave minimization problems,”Mathematics of Operations Research 1 (1976) 251–259.
R. Fortet, “Applications de l'algèbre de Boole en recherche opérationelle,”Revue Française de Recherche Opérationelle 4 (1960) 17–26.
F. Glover and E. Woolsey, “Converting the 0–1 polynomial programming problem to a 0–1 linear program,”Operations Research 22 (1974) 180–182.
D. Granot and F. Granot, “Generalized covering relaxation for 0–1 programs,”Operations Research 28 (1980) 1442–1449.
F. Granot and P.L. Hammer, “On the role of generalized covering problems,”Cahiers du Centre d'Etudes de Recherche Opérationnelle 16 (1974) 277–289.
P.L. Hammer, P. Hansen and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,”Mathematical Programming 28 (1984) 121–155.
P.L. Hammer and B. Kalantari, “A bound on the roof duality gap,” in: B. Simeone, ed.,Combinatorial Optimization, Lecture Notes in Mathematics, No. 1403 (Springer, Berlin, 1989) pp. 254–257.
P.L. Hammer and I.G. Rosenberg, “Linear decomposition of a positive group-Boolean function,” in: L. Collatz and W. Wetterling, eds.,Numerische Methoden bei Optimierung, Vol. II (Birkhauser, Basel, 1974) pp. 51–62.
P.L. Hammer and A.A. Rubin, “Some remarks on quadratic programming with 0–1 variables,”Revue Française de Recherche Opérationelle et d'Informatique 4 (1970) 67–79.
P.L. Hammer and S. Rudeanu,Boolean Methods in Operations Research and Related Areas (Springer, Berlin, 1968).
P.L. Hammer and B. Simeone, “Quadratic functions of binary variables,” in: B. Simeone, ed.,Combinatorial Optimization, Lecture Notes in Mathematics, No. 1403, (Springer, Berlin, 1989) pp. 1–56.
P. Hansen, “Methods of nonlinear 0–1 programming,”Annals of Discrete Mathematics 5 (1979) 53–70.
P. Hansen, S.H. Lu and B. Simeone, “On the equivalence of paved-duality and standard linearization in nonlinear 0–1 optimization,”Discrete Applied Mathematics 29 (1990) 187–193.
P. Hansen and B. Simeone, “Unimodular functions,”Discrete Applied Mathematics 14 (1986) 269–281.
R.M. Karp, “Reducibility among combinatorial problems,” in: R.E. Miller and J.W. Thatcher, eds.,Complexity of Computer Computations, (Plenum Press, New York, 1972) pp. 85–104.
L. Lovász,An Algorithmic Theory of Numbers, Graphs and Convexity (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1986).
S.H. Lu and A.C. Williams, “Roof duality for polynomial 0–1 optimization,”Mathematical Programming 37 (1987) 357–360.
S.H. Lu and A.C. Williams, “Roof duality and linear relaxation for quadratic and polynomial 0–1 optimization,” RUTCOR Research Report 8-87, Rutgers University, (New Brunswick, NJ, 1987).
R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
A. Schrijver,Theory of Linear and Integer Programming (Wiley, Chichester, 1986).
I. Singer, “Extensions of functions of 0–1 variables and applications to combinatorial optimization,”Numerical Functional Analysis and Optimization 7 (1984–85) 23–62.
K. Truemper, “Alpha-balanced graphs and matrices and GF(3)-representability of matroids,”Journal of Combinatorial Theory Series B 32 (1982) 112–139.
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Crama, Y. Concave extensions for nonlinear 0–1 maximization problems. Mathematical Programming 61, 53–60 (1993). https://doi.org/10.1007/BF01582138
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DOI: https://doi.org/10.1007/BF01582138