A critical index algorithm for nearest point problems on simplicial cones

We consider the linear complementarity problem ( q, M ) in which M is a positive definite symmetric matrix of order n. This problem is equivalent to a nearest point problem [ Γ; b ] in which Γ = { A . 1 , ⋯, A. n } is a basis for R n , b is a given point in R n ; and it is required to find the nearest point in the simplicial cone Pos( Γ ) to b. We develop an algorithm for solving the linear complementarity problem ( q, M ) or the equivalent nearest point problem [ Γ; b ]. Computational experience in comparison with an existing algorithm is presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47910/1/10107_2005_Article_BF01583789.pd

Decomposition Methods for Nonlinear Optimization and Data Mining

We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope into special polyhedra. We use these decompositions and develop methods for computing a special class of integrals exactly. Namely, we are interested in computing the exact value of integrals of polynomial functions over convex polyhedra. We present prior work and new extensions of the integration algorithms. Every integration method we present requires that the polynomial has a special form. We explore two special polynomial decomposition algorithms that are useful for integrating polynomial functions. Both polynomial decompositions have strengths and weaknesses, and we experiment with how to practically use them. After developing practical algorithms and efficient software tools for integrating a polynomial over a polytope, we focus on the problem of maximizing a polynomial function over the continuous domain of a polytope. This maximization problem is NP-hard, but we develop approximation methods that run in polynomial time when the dimension is fixed. Moreover, our algorithm for approximating the maximum of a polynomial over a polytope is related to integrating the polynomial over the polytope. We show how the integration methods can be used for optimization. The second central topic in this dissertation is on problems in data science. We first consider a heuristic for mixed-integer linear optimization. We show how many practical mixed-integer linear have a special substructure containing set partition constraints. We then describe a nice data structure for finding feasible zero-one integer solutions to systems of set partition constraints. Finally, we end with an applied project using data science methods in medical research.Comment: PHD Thesis of Brandon Dutr

CP-rays in simplicial cones

An interior point of a triangle is called CP-point if its orthogonal projection on the line containing each side lies in the relative interior of that side. In classical mathematics, interest in the concept of regularity of a triangle is mainly centered on the property of every interior point of the triangle being a CP-point. We generalize the concept of regularity using this property, and extend this work to simplicial cones in ℝ n , and derive necessary and sufficient conditions for this property to hold in them. These conditions highlight the geometric properties of Z-matrices. We show that these concepts have important ramifications in algorithmic studies of the linear complementarity problem. We relate our results to other well known properties of square matrices.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47921/1/10107_2005_Article_BF01582265.pd

Towards Stratification Learning through Homology Inference

A topological approach to stratification learning is developed for point cloud data drawn from a stratified space. Given such data, our objective is to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving a stratification for each radius level. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions; some geometric intuition for these topological conditions is also provided. Our correctness result is then given a probabilistic flavor: we give bounds on the minimum number of sample points required to infer, with probability, which points belong to the same strata. Finally, we give an explicit algorithm for the clustering, prove its correctness, and apply it to some simulated data.Comment: 48 page

Nearest Points on Toric Varieties

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.Comment: 20 page