The Chebyshev center as an alternative to the analytic center in the feasibility pump

© The Author(s) 2023As a heuristic for obtaining feasible points of mixed integer linear problems, the feasibility pump (FP) generates two sequences of points: one of feasible solutions for the relaxed linear problem; and another of integer points obtained by rounding the linear solutions. In a previous work, the present authors proposed a variant of FP, named analytic center FP, which obtains integer solutions by rounding points in the segment between the linear solution and the analytic center of the polyhedron of the relaxed problem. This work introduces a new FP variant that replaces the analytic center with the Chebyshev center. Two of the benefts of using the Chebyshev center are: (i) it requires the solution of a linear optimization problem (unlike the analytic center, which involves a convex nonlinear optimization problem for its exact solution); and (ii) it is invariant to redundant constraints (unlike the analytic center, which may not be well centered within the polyhedron for problems with highly rank-defcient matrices). The computational results obtained with a set of more than 200 MIPLIB2003 and MIPLIB2010 instances show that the Chebyshev center FP is competitive and can serve as an alternative to other FP variants.This research has been supported by the MCIN/AEI/FEDER project RTI2018-097580-B-I00. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer NaturePeer ReviewedPostprint (published version

An algorithm for the rapid numerical evaluation of Bessel functions of real orders and arguments

We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel functions themselves are expensive to represent via piecewise polynomial expansions, the logarithms of certain solutions of Bessel's equation are not. We exploit this observation by numerically precomputing the logarithms of carefully chosen Bessel functions and representing them with piecewise bivariate Chebyshev expansions. Our scheme is able to evaluate Bessel functions of orders between $0$ and 1\sep,000\sep,000\sep,000 at essentially any positive real argument. In that regime, it is competitive with existing methods for the rapid evaluation of Bessel functions and has several advantages over them. First, our approach is quite general and can be readily applied to many other special functions which satisfy second order ordinary differential equations. Second, by calculating the logarithms of the Bessel functions rather than the Bessel functions themselves, we avoid many issues which arise from numerical overflow and underflow. Third, in the oscillatory regime, our algorithm calculates the values of a nonoscillatory phase function for Bessel's differential equation and its derivative. These quantities are useful for computing the zeros of Bessel functions, as well as for rapidly applying the Fourier-Bessel transform. The results of extensive numerical experiments demonstrating the efficacy of our algorithm are presented. A Fortran package which includes our code for evaluating the Bessel functions as well as our code for all of the numerical experiments described here is publically available

The monic integer transfinite diameter

We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval I. The monic integer transfinite diameter t_M(I) is defined as the infimum of all such supremums. We show that if I has length 1 then t_M(I) = 1/2. We make three general conjectures relating to the value of t_M(I) for intervals I of length less that 4. We also conjecture a value for t_M([0, b]) where 0 < b < 1. We give some partial results, as well as computational evidence, to support these conjectures. We define two functions that measure properties of the lengths of intervals I with t_M(I) on either side of t. Upper and lower bounds are given for these functions. We also consider the problem of determining t_M(I) when I is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.Comment: 32 pages, 5 figure