2,060 research outputs found
Valid Inequalities and Reformulation Techniques for Mixed Integer Nonlinear Programming
One of the most important breakthroughs in the area of Mixed Integer Linear Programming (MILP) is the characterization of the convex hull of specially structured non-convex polyhedral sets in order to develop valid inequalities or cutting planes. Development of strong valid inequalities such as Split cuts, Gomory Mixed Integer (GMI) cuts, and Mixed Integer Rounding (MIR) cuts has resulted in highly effective branch-and-cut algorithms. While such cuts are known to be equivalent, each of their characterizations provides different advantages and insights.
The study of cutting planes for Mixed Integer Nonlinear Programming (MINLP) is still much more limited than that for MILP, since characterizing cuts for MINLP requires the study of the convex hull of a non-convex and non-polyhedral set, which has proven to be significantly harder than the polyhedral case. However, there has been significant work on the computational use of cuts in MINLP. Furthermore, there has recently been a significant interest in extending the associated theoretical results from MILP to the realm of MINLP.
This dissertation is focused on the development of new cuts and extended formulations for Mixed Integer Nonlinear Programs. We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. We also study the relation between the introduced cuts and some known classes of cutting planes from MILP. Furthermore, we show how an aggregation technique can be easily extended to characterize the convex hull of sets defined by two quadratic or by a conic quadratic and a quadratic inequality. We also computationally evaluate the performance of the introduced cuts and extended formulations on two classes of MINLP problems
Inner and Outer Rounding of Boolean Operations on Lattice Polygonal Regions
Robustness problems due to the substitution of the exact computation on real
numbers by the rounded floating point arithmetic are often an obstacle to
obtain practical implementation of geometric algorithms. If the adoption of the
--exact computation paradigm--[Yap et Dube] gives a satisfactory solution to
this kind of problems for purely combinatorial algorithms, this solution does
not allow to solve in practice the case of algorithms that cascade the
construction of new geometric objects. In this report, we consider the problem
of rounding the intersection of two polygonal regions onto the integer lattice
with inclusion properties. Namely, given two polygonal regions A and B having
their vertices on the integer lattice, the inner and outer rounding modes
construct two polygonal regions with integer vertices which respectively is
included and contains the true intersection. We also prove interesting results
on the Hausdorff distance, the size and the convexity of these polygonal
regions
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