Concave extensions for nonlinear 0–1 maximization problems

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.

[1]  Peter L. Hammer,et al.  Boolean Methods in Operations Research and Related Areas , 1968 .

[2]  Yves Crama,et al.  Recognition problems for special classes of polynomials in 0–1 variables , 1989, Math. Program..

[3]  Daniel Granot,et al.  Technical Note - Generalized Covering Relaxation for 0-1 Programs , 1980, Oper. Res..

[4]  Warren P. Adams,et al.  On the Equivalence Between Roof Duality and Lagrangian Duality for Unconstrained 0-1 Quadratic Programming Problems , 1994, Discret. Appl. Math..

[5]  S. H. Lu,et al.  Roof duality for polynomial 0–1 optimization , 1987, Math. Program..

[6]  P. Hammer,et al.  Linear Decomposition of a Positive Group-Boolean Function , 1974 .

[7]  Pierre Hansen,et al.  The basic algorithm for pseudo-Boolean programming revisited , 1988, Discret. Appl. Math..

[8]  Klaus Truemper,et al.  Alpha-balanced graphs and matrices and GF(3)-representability of matroids , 1982, J. Comb. Theory, Ser. B.

[9]  Peter L. Hammer,et al.  Some remarks on quadratic programming with 0-1 variables , 1970 .

[10]  P. Hammer,et al.  Quadratic functions of binary variables , 1989 .

[11]  Pierre Hansen,et al.  Roof duality, complementation and persistency in quadratic 0–1 optimization , 1984, Math. Program..

[12]  P. Hansen.  Methods of Nonlinear 0-1 Programming , 1979 .

[13]  Fred W. Glover,et al.  Technical Note - Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program , 1974, Oper. Res..

[14]  P. Hammer,et al.  A bound on the roof-duality gap , 1989 .

[15]  Pierre Hansen,et al.  On the equivalence of paved-duality and standard linearization in nonlinear 0-1 optimization , 1990, Discret. Appl. Math..

[16]  László Lovász,et al.  Algorithmic theory of numbers, graphs and convexity , 1986, CBMS-NSF regional conference series in applied mathematics.

[17]  Pierre Hansen,et al.  Unimodular functions , 1986, Discrete Applied Mathematics.

[18]  James E. Falk,et al.  A Successive Underestimation Method for Concave Minimization Problems , 1976, Math. Oper. Res..

[19]  Daniel Granot,et al.  Generalized Covering Relaxation for 0-1 Programs. , 1978 .

[20]  I. Singer.  Extensions of functions of 0-1 variables and applications to combinatorial optimization , 1985 .

[21]  Egon Balas,et al.  Nonlinear 0–1 programming: I. Linearization techniques , 1984, Math. Program..