RLT Hierarchy for Mixed-Integer Zero-One Problems

  • Hanif D. Sherali
  • Warren P. Adams
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 31)


Consider a linear mixed-integer zero-one programming problem whose (nonempty) feasible region is given as follows:
$$\begin{gathered} X = \{ (x,y):\sum\limits_{j = 1}^n {{a_{rj}}{x_j}} + \sum\limits_{k = 1}^m {{r_{rk}}{y_k}} \geqslant {\beta _r}for = 1, \ldots ,R, \hfill \\ 0 \leqslant x \leqslant {e_n},x\operatorname{int} eger,0 \leqslant y \leqslant {e_m}\} , \hfill \\ \end{gathered} $$
where e n and e m are, respectively, column vectors of n and m entries of 1, and where the continuous variables y k are assumed to be bounded and appropriately scaled to lie in the interval [0, 1] for k = 1,...,m. (Upper bounds on the continuous variables are imposed here only for convenience in exposition, as we comment on later in the discussion.) Note that any equality constraints present in the formulation can be accommodated in a similar manner as are the inequalities in the following derivation, and we omit writing them explicitly in (2.1) only to simplify the presentation. However, we will show later that the equality constraints can be treated in a special manner which, in fact, might sometimes encourage the writing of the R inequalities in (2.1) as equalities by using slack variables.


Feasible Solution Extreme Point Equality Constraint Valid Inequality Linear Programming Relaxation 
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Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Hanif D. Sherali
    • 1
  • Warren P. Adams
    • 2
  1. 1.Department of Industrial and Systems EngineeringVirginia Polytechnic Institute and State UniversityBlacksburgUSA
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA

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