Hans D. Mittelmann

An independent benchmarking of SDP and SOCP solvers

Abstract. This work reports the results of evaluating all computer codes submitted to the Seventh DIMACS Implementation Challenge on Semidefinite and Related Optimization Problems. The codes were run on a standard platform and on all the benchmark problems provided by the organizers of the challenge. A total of ten codes were tested on fifty problems in twelve categories. For each code the most important information is summarized. Together with the tabulated and commented benchmarking results this provides an overview of the state of the art in this field.

MIPLIB 2010 - Mixed Integer Programming Library version 5

This paper reports on the fifth version of the Mixed Integer Programming Library. The miplib 2010 is the first miplib release that has been assembled by a large group from academia and from industry, all of whom work in integer programming. There was mutual consent that the concept of the library had to be expanded in order to fulfill the needs of the community. The new version comprises 361 instances sorted into several groups. This includes the main benchmark test set of 87 instances, which are all solvable by today’s codes, and also the challenge test set with 164 instances, many of which are currently unsolved. For the first time, we include scripts to run automated tests in a predefined way. Further, there is a solution checker to test the accuracy of provided solutions using exact arithmetic.

Energy stability of thermocapillary convection in a model of the float-zone crystal-growth process

Energy stability theory has been applied to a basic state of thermocapillary convection occurring in a cylindrical half-zone of finite length to determine conditions under which the flow will be stable. Because of the finite length of the zone, the basic state must be determined numerically. Instead of obtaining stability criteria by solving the related Euler–Lagrange equations, the variational problem is attacked directly by discretization of the integrals in the energy identity using finite differences. Results of the analysis are values of the Marangoni number, Ma E , below which axisymmetric disturbances to the basic state will decay, for various values of the other parameters governing the problem.

On multi-grid methods for variational inequalities

SummaryWe consider here a general class of algorithms for the numerical solution of variational inequalities. A convergence proof is given and in particular a multi-grid method is described. Numerical results are presented for the finite-difference discretization of an obstacle problem for minimal surfaces

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.

Recent Developments in Barycentric Rational Interpolation

In 1945, W. Taylor discovered the barycentric formula for evaluating the interpolating polynomial. In 1984, W. Werner has given first consequences of the fact that the formula usually is a rational interpolant. We review some advances since the latter paper in the use of the formula for rational interpolation.

Linear‐stability theory of thermocapillary convection in a model of the float‐zone crystal‐growth process

Linear‐stability theory has been applied to a basic state of thermocapillary convection in a model half‐zone to determine values of the Marangoni number above which instability is guaranteed. The basic state must be determined numerically since the half‐zone is of finite, O(1) aspect ratio with two‐dimensional flow and temperature fields. This, in turn, means that the governing equations for disturbance quantities are nonseparable partial differential equations. The disturbance equations are treated by a staggered‐grid discretization scheme. Results are presented for a variety of parameters of interest in the problem, including both terrestrial and microgravity cases; they complement recent calculations of the corresponding energy‐stability limits.

Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval

Abstract Polynomial interpolation between large numbers of arbitrary nodes does notoriously not, in general, yield useful approximations of continuous functions. Following [1], we suggest to determine for each set of n + 1 nodes another denominator of degree n , and then to interpolate every continuous function by a rational function with that same denominator, so that the resulting interpolation process remains a linear projection. The optimal denominator is chosen so as to minimize the Lebesgue constant for the given nodes. It has to be computed numerically. For that purpose, the barycentric representation of rational interpolants, which displays the linearity of the interpolation and reduces the determination of the denominator to that of the barycentric weights, is used. The optimal weights can then be computed by solving an optimization problem with simple bounds which could not be solved accurately by the first author in [1]. We show here how to do so, and we present numerical results.

Interior Point Methods for Second-Order Cone Programming and OR Applications

Interior point methods (IPM) have been developed for all types of constrained optimization problems. In this work the extension of IPM to second order cone programming (SOCP) is studied based on the work of Andersen, Roos, and Terlaky. SOCP minimizes a linear objective function over the direct product of quadratic cones, rotated quadratic cones, and an affine set. It is described in detail how to convert several application problems to SOCP. Moreover, a proof is given of the existence of the step for the infeasible long-step path-following method. Furthermore, variants are developed of both long-step path-following and of predictor-corrector algorithms. Numerical results are presented and analyzed for those variants using test cases obtained from a number of application problems.

Numerical Methods for Bifurcation Problems — A Survey and Classification

The purpose of this paper is to give an account of recent developments in numerical methods for the solution of bifurcation problems. For readers not too familiar with our subject we shall first summarize important applications of bifurcation and dicuss some of the basic ideas, problems and tools of bifurcation theory.