Some applications of continuous variable neighbourhood search metaheuristic (mathematical modelling)

In the real world, many problems are continuous in nature. In some cases, finding the global solutions for these problems is difficult. The reason is that the problem’s objective function is non convex, nor concave and even not differentiable. Tackling these problems is often computationally too expensive. Although the development in computer technologies are increasing the speed of computations, this often is not adequate, particularly if the size of the problem’s instance are large. Applying exact methods on some problems may necessitate their linearisation. Several new ideas using heuristic approaches have been considered particularly since they tackle the problems within reasonable computational time and give an approximate solution. In this thesis, the variable neighbourhood search (VNS) metaheuristic (the framework for building heuristic) has been considered. Two variants of variable neighbourhood search metaheuristic have been developed, continuous variable neighbourhood search and reformulation descent variable neighbourhood search. The GLOB-VNS software (Drazić et al., 2006) hybridises the Microsoft Visual Studio C++ solver with variable neighbourhood search metaheuristics. It has been used as a starting point for this research and then adapted and modified for problems studied in this thesis. In fact, two problems have been considered, censored quantile regression and the circle packing problem. The results of this approach for censored quantile regression outperforms other methods described in the literature, and the near-optimal solutions are obtained in short running computational time. In addition, the reformulation descent variable neighbourhood search variant in solving circle packing problems is developed and the computational results are provided.