Wind farm layout optimisation has become a very challenging and widespread problem in recent years. In many publications, the main goal is to achieve the maximum power output and minimum wind farm cost. This may be accomplished by applying single or multi-objective optimisation techniques. In this paper, we apply a single objective hill-climbing algorithm (HCA) and three multi-objective evolutionary algorithms (NSGA-II, SPEA2 and PESA-II) to a well-known benchmark optimisation problem proposed by Mosetti et al., which includes three different wind scenarios. We achieved better results by applying single- and multi-objective algorithms. Furthermore, we showed that the best performing multi-objective algorithm was NSGA-II. Finally, an extensive comparison of the results of past publications is made.

Real-world portfolio optimisation problems are often NP-hard, their efficient frontiers (EFs) in practice being calculated by randomised algorithms. In this work, a deterministic method of decomposition of EFs into a short sequence of sub-EFs is presented. These sub-EFs may be calculated by a quadratic programming algorithm, the collection of such sub-EFs then being subjected to a sifting process to produce the full EF. Full EFs of portfolio optimisation problems with small-cardinality constraints are computed to a high resolution, providing a fast and practical alternative to randomised algorithms. The method may also be used with other practical classes of portfolio problems, complete with differing measures of risk. Finally, it is shown that the identified sub-EFs correspond closely to local optima of the objective function of a case study evolutionary algorithm.

This work assesses the efficacy of evolutionary algorithms (EAs) using an intuitive Multi-Dimensional Scaling (MDS) visualisation of the evolution of a population. We propose the use of Landmark Multi-Dimensional Scaling (LMDS) to overcome computational challenges inherent to visualising many-objective and complex problems with MDS. For the benchmark problems we tested, LMDS is akin to MDS visually, whilst requiring less than 1% of the time and memory necessary to produce an MDS visualisation of the same objective space solutions, leading to the possibility of online visualisations for multi-and many-objective optimisation evaluation. Using multi-and many-objective problems from the DTLZ and WFG benchmark test suites, we analyse how Landmark MDS visualisations can offer far greater insight into algorithm performance than using traditional algorithm performance metrics such as hypervolume alone, and can be used to complement explicit performance metrics. Ultimately, this visualisation allows visual identification of problem features and assists the decision maker in making intuitive recommendations for algorithm parameters/operators for creating and testing better EAs to solve multi-and many-objective problems.