186 research outputs found
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
On the composition of convex envelopes for quadrilinear terms
International audienceWithin the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting
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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
Topics in exact precision mathematical programming
The focus of this dissertation is the advancement of theory and computation related to exact precision mathematical programming. Optimization software based on floating-point arithmetic can return suboptimal or incorrect resulting because of round-off errors or the use of numerical tolerances. Exact or correct results are necessary for some applications. Implementing software entirely in rational arithmetic can be prohibitively slow. A viable alternative is the use of hybrid methods that use fast numerical computation to obtain approximate results that are then verified or corrected with safe or exact computation. We study fast methods for sparse exact rational linear algebra, which arises as a bottleneck when solving linear programming problems exactly. Output sensitive methods for exact linear algebra are studied. Finally, a new method for computing valid linear programming bounds is introduced and proven effective as a subroutine for solving mixed-integer linear programming problems exactly. Extensive computational results are presented for each topic.Ph.D.Committee Chair: Dr. William J. Cook; Committee Member: Dr. George Nemhauser; Committee Member: Dr. Robin Thomas; Committee Member: Dr. Santanu Dey; Committee Member: Dr. Shabbir Ahmed; Committee Member: Dr. Zonghao G
A branch, price, and cut approach to solving the maximum weighted independent set problem
The maximum weight-independent set problem (MWISP) is one of the most
well-known and well-studied NP-hard problems in the field of combinatorial
optimization.
In the first part of the dissertation, I explore efficient branch-and-price (B&P)
approaches to solve MWISP exactly. B&P is a useful integer-programming tool for
solving NP-hard optimization problems. Specifically, I look at vertex- and edge-disjoint
decompositions of the underlying graph. MWISPĂÂąĂĂs on the resulting subgraphs are less
challenging, on average, to solve. I use the B&P framework to solve MWISP on the
original graph G using these specially constructed subproblems to generate columns. I
demonstrate that vertex-disjoint partitioning scheme gives an effective approach for
relatively sparse graphs. I also show that the edge-disjoint approach is less effective than
the vertex-disjoint scheme because the associated DWD reformulation of the latter
entails a slow rate of convergence.
In the second part of the dissertation, I address convergence properties associated
with Dantzig-Wolfe Decomposition (DWD). I discuss prevalent methods for improving the rate of convergence of DWD. I also implement specific methods in application to the
edge-disjoint B&P scheme and show that these methods improve the rate of
convergence.
In the third part of the dissertation, I focus on identifying new cut-generation
methods within the B&P framework. Such methods have not been explored in the
literature. I present two new methodologies for generating generic cutting planes within
the B&P framework. These techniques are not limited to MWISP and can be used in
general applications of B&P. The first methodology generates cuts by identifying faces
(facets) of subproblem polytopes and lifting associated inequalities; the second
methodology computes Lift-and-Project (L&P) cuts within B&P. I successfully
demonstrate the feasibility of both approaches and present preliminary computational
tests of each
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