Current mixed-integer linear programming solvers are based on linear programming routines that use floating point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. It is shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size.

Three extended real interval systems are defined and distinguished by their implementation complexity and result sharpness. The three systems are closed with respect to interval arithmetic and the enclosure of functions and relations, notwithstanding domain restrictions or the presence of singularities. 1 Overview Section 2 introduces the problem of defining closed interval systems. In Section 3, real and extended points and intervals are defined. In Section 4, the empty and entire intervals are used to close the extended interval system. Section ?? shows how incorrect conclusions have been reached about the result of certain interval arithmetic operator-operand combinations. The author is grateful to Professor Arnold Neumaier for originally raising this issue. Section 9 describes how to legitimately use IEEE floating-point arithmetic to obtain the sharp results described in Section ??. Section 6 generalizes extended interval arithmetic to define interval enclosures of functions, with...

We present efficient algorithms based on a combination of numeric and symbolic techniques for evaluating one-dimensional algebraic sets in a subset of the real domain. Given a description of a one-dimensional algebraic set, we compute its projection using resultants. We represent the resulting plane curve as a singular set of a matrix polynomial as opposed to roots of a bivariate polynomial. Given the matrix formulation, we make use of algorithms from numerical linear algebra to compute start points on all the components, partition the domain such that each resulting region contains only one component and evaluate it accurately using marching methods. We also present techniques to handle singularities for well-conditioned inputs. The resulting algorithm is iterative and its complexity is output sensitive. It has been implemented in oating-point arithmetic and we highlight its performance in the context of computing intersection of high-degree algebraic surfaces.

Deformable models are a useful modeling paradigm in computer vision. A deformable model is a curve, a surface, or a volume, whose shape, position, and orientation are controlled through a set of parameters. They can represent manufactured objects, human faces and skeletons, and even bodies of fluid.

. We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twice-continuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the "midpoint test," but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of boxes around each minimizer. In a previous paper, we analyzed this problem for univariate objective functions. In this paper, we generalize that analysis to multidimensional objective functions. As in the univariate case, the results show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension. 1. Introduction and Basic Concepts Our underlying problem is: (1) find all global minimizers to f(x) subject to x 2 X; where X ae R m is a compact right parallelepiped with face...

. GLOPT is a Fortran77 program for global minimization of a blockseparable objective function subject to bound constraints and block-separable constraints. It finds a nearly globally optimal point that is near a true local minimizer. Unless there are several local minimizers that are nearly global, we thus find a good approximation to the global minimizer. GLOPT uses a branch and bound technique to split the problem recursively into subproblems that are either eliminated or reduced in their size. This is done by an extensive use of the block separable structure of the optimization problem. In this paper we discuss a new reduction technique for boxes and new ways for generating feasible points of constrained nonlinear programs. These are implemented as the first stage of our GLOPT project. The current implementation of GLOPT uses neither derivatives nor simultaneous information about several constraints. Numerical results are already encouraging. Work on an extension using curvature inf...

iii This document and the work it represents was impossible without the support of my wife Ginger. Often one needs non-technical advice to make clear what one is contemplating. Also one always needs a financial supporter. My thesis advisor Erik Antonsson helped focus many of my thoughts. In addition to providing me with technical assistance, he as well provided instruction on the process of conducting academic research, the communication of ideas both orally and written, and the approach to a developing field. I also owe much to my colleagues in the Engineering and Applied Science Division at Caltech. Their comments and advice maintained my comprehension and rigor. Andrew Lewis in particular provided me with invaluable support. Many of the technical proofs were impossible without him. This material and the work it represented were made possible, in part, by a fellowship from the AT&T-Bell Laboratories Ph.D. scholar program, sponsored by the AT&T foundation. Also, the National Science Foundation provided funding under a Presidential Young

The classical isoperimetric inequality in R 3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an angle of 120 ◦. 1.

We present efficient and accurate algorithms to compute solutions of zero-dimensional multivariate polynomial equations in a given domain. The total number of solutions correspond to the Bezout bound for dense polynomial systems or the Bernstein bound for sparse systems. In most applications the actual number of solutions in the domain of interest is much lower than the Bezout or Bernstein bound. Our approach is based on global symbolic formulation of the problem using resultants and matrix computations and localizing it to find selected solutions based on numerical computations. The problem of finding roots is reduced to computing eigenvalues of a generalized companion matrix and we use the structure of the matrix to compute the solutions in the domain of interest only. The resulting algorithm combines symbolic preprocessing with numerical iterations and works well in practice. We discuss its performance on a number of applications. 1 Introduction Finding roots of polynomial system...

One of the main problems of interval computations is, given a function f(x 1 ; :::; x n ) and n intervals x 1 ; :::; x n , to compute the range y = f(x 1 ; :::; x n ). This problem is feasible for linear functions f , but for generic polynomials, it is known to be computationally intractable. Because of that, traditional interval techniques usually compute the enclosure of y, i.e., an interval that contains y. The closer this enclosure to y, the better. It is desirable to describe cases in which we can compute the optimal enclosure, i.e., the range itself.