Convex hull relaxation

Lee and Grossmann (2000) have derived the convex hull relaxation of problem (GDP). The basic idea is as follows. Consider a disjunction keK that has convex constraints. 

To strengthen the lower bounds one can derive cutting planes using the convex hull relaxation (CRP). To generate a cutting plane, the following 2-nonn separation problem (SP), a convex QP, is solved ... 

Grossmann I.E. and Lee S. 2003. Generalized disjunctive progranuning nonlinear convex hull relaxation and algorithms, Comput. Optimization AppL, 26, 83-100. 

Convex hull formulations of MILPs and MINLPs lead to relaxed problems that have much tighter lower bounds. This leads to the examination of far fewer nodes in the branch and bound tree. See Grossmann and Lee, Comput. Optim. Applic. 26 83 (2003) for more details. 

The convex hull of a set of points is the volume surrounding these points such that any segment between any two of these points stay inside the volume. For proteins, we might relax this definition to include inside the protein volume, the closed protein cavities that are not open to bulk solvent. A more adequate... 

Another application of the cutting plane is to determine if the convex hull formulation yields a good relaxation of a disjunction. If the value of x is large, then it... 

Then at the solution point of the NLP subproblem, the nonlinear constraints are linearized and the disjunction is relaxed by convex hull to build a master MILP subproblem which will yield a new discrete choice of (y, T) for the next iteration. 

Sherali H.D. and Adams W.P. 1990. A hierarchy of relaxations between the continnous and convex hull representations for zero-one programming problems, SIAM 1. Discrete Math., 3(3), 411 30. 

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