This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

Many marine ecosystems have the remarkable property that the abundance of organisms of a given body size is approximately proportional to the inverse square of that size. Size-structured models have been developed for which this “invariance-of-biomass” state is an equilibrium solution. These models are built on the coupling of predator growth to prey abundance, where prey suitability is determined by a size-based function referred to as a feeding kernel. In this paper, the local stability of the equilibrium state is investigated in a limiting case where predators only consume prey of a preferred size. In this special case, it is shown analytically that the equilibrium state is always unstable. It is concluded that some degree of diet breadth, in terms of the range of prey sizes consumed by a predator, is an essential prerequisite for the invariance-of-biomass state to be stable, as widely observed in the field.

We introduce a full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular function to provide the feasibility step and to measure proximity to the central path. The result of polynomial complexity coincides with the best known iteration bound for infeasible interior-point methods.

In this paper we study optimal control computation based on the control parameterization method for a class of optimal control problems involving nonlinear systems with multiple time delays subject to continuous state inequality constraints. Both the state and the control are allowed to have different time delays, and they are uncorrelated in this system. The control of the dynamical system is approximated by a piecewise constant function whose heights are taken as decision vectors. The formulae for computing the gradients of the cost and constraint functions are then derived. Based on this, a computational method for finding the optimal control is developed by utilizing the Sequential Quadratic Programming (SQP) algorithm with an active set strategy. The computational method is applied to an industrial problem arising in the purification process of zinc hydrometallurgy. Numerical simulation shows that the amount of zinc powder that is needed can be decreased significantly, thus avoiding wastage of resources.