Analysis of Bounds for Multilinear Functions

AbstractWe analyze four bounding schemes for multilinear functions and theoretically compare their tightness. We prove that one of the four schemes provides the convex envelope and that two schemes provide the concave envelope for the product of p variables over R+p .

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

[2]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

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

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

[5]  Hanif D. Sherali,et al.  CONVEX ENVELOPES OF MULTILINEAR FUNCTIONS OVER A UNIT HYPERCUBE AND OVER SPECIAL DISCRETE SETS , 1997 .

[6]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[7]  Egon Balas,et al.  Nonlinear 0–1 programming: II. Dominance relations and algorithms , 1983, Math. Program..

[8]  Yves Crama Concave extensions for nonlinear 0–1 maximization problems , 1993, Math. Program..

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

[10]  Fred W. Glover,et al.  Further Reduction of Zero-One Polynomial Programming Problems to Zero-One linear Programming Problems , 1973, Oper. Res..

[11]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

[12]  Hanif D. Sherali,et al.  Linearization Strategies for a Class of Zero-One Mixed Integer Programming Problems , 1990, Oper. Res..

[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]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[15]  Pierre Hansen,et al.  Constrained Nonlinear 0-1 Programming , 1989 .

[16]  Anatoliy D. Rikun,et al.  A Convex Envelope Formula for Multilinear Functions , 1997, J. Glob. Optim..