In this paper the interval global optimisation approach is recast, by reformulation of the global optimisation problem, as a bound constrained linear relaxation, Two new classes of linear under-estimator, derived from the natural extension and mean value forms of interval analysis, respectively, and applicable to any once differentiable function, are introduced. These under-estimators are combined with the interval bounded linear program to create a rigorous global optimisation algorithm for constrained global optimisation. The value of the approach is validated by application to selected test problems from the process engineering and global optimisation literature. The results indicate that the interval LP is more efficient than other interval methods for constrained problems whilst retaining a wide applicability.