Rigorous results in electronic structure calculations

Electronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced … Read more

VSDP: A Matlab toolbox for verified semidefinite-quadratic-linear programming

VSDP is a software package that is designed for the computation of verified results in conic programming. The current version of VSDP supports the constraint cone consisting of the product of semidefinite cones, second-order cones and the nonnegative orthant. It provides functions for computing rigorous error bounds of the true optimal value, verified enclosures of … Read more

A Comparison of Software Packages for Verified Linear Programming

Linear programming is arguably one of the most basic forms of optimization. Its theory and algorithms can not only be applied to linear optimization problems but also to relaxations of nonlinear problems and branch-and-bound methods for mixed-integer and global optimization problems. Recent research shows that against intuition bad condition numbers frequently occur in linear programming. … Read more

VSDP: Verified SemiDefinite Programming

VSDP is a MATLAB software package for rigorously solving semidefinite programming problems. It expresses these problems in a notation closely related to the form given in textbooks and scientific papers. Functions for computing verified forward error bounds of the true optimal value and verified certificates of feasibility and infeasibility are provided. All rounding errors due … Read more

Rigorous Error Bounds for the Optimal Value in Semidefinite Programming

A wide variety of problems in global optimization, combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how … Read more

Computational Experience with Rigorous Error Bounds for the Netlib Linear Programming Library

The Netlib library of linear programming problems is a well known suite containing many real world applications. Recently it was shown by Ordonez and Freund that 71% of these problems are ill-conditioned. Hence, numerical difficulties may occur. Here, we present rigorous results for this library that are computed by a verification method using interval arithmetic. … Read more