Convex Hulls of Algebraic Sets

This article describes a method to compute successive convex approximations of the convex hull of a set of points in R^n that are the solutions to a system of polynomial equations over the reals. The method relies on sums of squares of polynomials and the dual theory of moment matrices. The main feature of the technique is that all computations are done modulo the ideal generated by the polynomials defining the set to the convexified. This work was motivated by questions raised by Lov\'asz concerning extensions of the theta body of a graph to arbitrary real algebraic varieties, and hence the relaxations described here are called theta bodies. The convexification process can be seen as an incarnation of Lasserre's hierarchy of convex relaxations of a semialgebraic set in R^n. When the defining ideal is real radical the results become especially nice. We provide several examples of the method and discuss convergence issues. Finite convergence, especially after the first step of the method, can be described explicitly for finite point sets.Comment: This article was written for the "Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications

Playing Billiard in Version Space

A ray-tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a set of linear separable examples. While the Bayes-optimum requires a majority decision over all Perceptrons separating the example set, the problem considered here corresponds to finding the single Perceptron with best average generalization probability. For randomly distributed examples the billiard estimate agrees with known analytic results. In real-life classification problems the generalization error is consistently reduced compared to the maximal stability Perceptron.Comment: uuencoded, gzipped PostScript file, 127576 bytes To recover 1) save file as bayes.uue. Then 2) uudecode bayes.uue and 3) gunzip bayes.ps.g

Convex hulls of curves of genus one

Let C be a real nonsingular affine curve of genus one, embedded in affine n-space, whose set of real points is compact. For any polynomial f which is nonnegative on C(R), we prove that there exist polynomials f_i with f \equiv \sum_i f_i^2 (modulo I_C) and such that the degrees deg(f_i) are bounded in terms of deg(f) only. Using Lasserre's relaxation method, we deduce an explicit representation of the convex hull of C(R) in R^n by a lifted linear matrix inequality. This is the first instance in the literature where such a representation is given for the convex hull of a nonrational variety. The same works for convex hulls of (singular) curves whose normalization is C. We then make a detailed study of the associated degree bounds. These bounds are directly related to size and dimension of the projected matrix pencils. In particular, we prove that these bounds tend to infinity when the curve C degenerates suitably into a singular curve, and we provide explicit lower bounds as well.Comment: 1 figur

Convex Hulls, Oracles, and Homology

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem POLYTOPE-COMPLETENESS-COMBINATORIAL, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the ``no''-case of POLYTOPE-COMPLETENESS-COMBINATORIAL has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).Comment: 11 pages, 2 figure

Multidimensional indexing structure for use with linear optimization queries

Linear optimization queries, which usually arise in various decision support and resource planning applications, are queries that retrieve top N data records (where N is an integer greater than zero) which satisfy a specific optimization criterion. The optimization criterion is to either maximize or minimize a linear equation. The coefficients of the linear equation are given at query time. Methods and apparatus are disclosed for constructing, maintaining and utilizing a multidimensional indexing structure of database records to improve the execution speed of linear optimization queries. Database records with numerical attributes are organized into a number of layers and each layer represents a geometric structure called convex hull. Such linear optimization queries are processed by searching from the outer-most layer of this multi-layer indexing structure inwards. At least one record per layer will satisfy the query criterion and the number of layers needed to be searched depends on the spatial distribution of records, the query-issued linear coefficients, and N, the number of records to be returned. When N is small compared to the total size of the database, answering the query typically requires searching only a small fraction of all relevant records, resulting in a tremendous speedup as compared to linearly scanning the entire dataset