Efficient Calculation of the Genomic Relationship Matrix

Since the calculation of a genomic relationship matrix needs a large number of arithmetic operations, fast implementations are of interest. Our fastest algorithm is more accurate and 25× faster than a AVX double precision floating-point implementation.

Convex Optimization and Numerical Issues

This chapter aims at providing the intuition behind convex optimization algorithms and addresses their effective use with floating-point implementation. It first briefly presents the algorithms, assuming a real semantics. As outlined in Chapter 4, convex conic programming is supported by different methods depending on the cone considered. The most known approach for linear constraints is the simplex method by Dantzig. While having an exponential-time complexity with respect to the number of constraints, the simplex method performs well in general. Another method is the set of interior point methods, initially proposed by Karmarkar and made popular by Nesterov and Nemirovski. They can be characterized as path-following methods in which a sequence of local linear problems are solved, typically by Newton's method. After these algorithms are considered, the chapter discusses approaches to obtain sound results.

Experiments on the feasibility of using a floating-point simplex in an SMT solver

SMT solvers use simplex-based decision procedures to solve decision problems whose formulas are quantifier-free and atoms are linear constraints over the rationals. State-of-art SMT solvers use rational (exact) simplex implementations, which have shown good performance for typical software, hardware or protocol verification problems over the years.Yet, most other scientific and technical fields use (inexact) floating-point computations, which are deemed far more efficient than exact ones.It is therefore tempting to use a floating-point simplex implementation inside an SMT solver, though special precautions must be taken to avoid unsoundness.In this work, we describe experimental results, over common benchmarks (SMT-LIB) of the integration of a mature floating-point implementation of the simplex algorithm (GLPK) into an existing SMT solver (OpenSMT).We investigate whether commonly cited reasons for and against the use of floating-point truly apply to real cases from verification problems.