Variable space transformation techniques and new algorithms for global optimization

Solving a global optimization problem is a hard task. In the chapters of this thesis variable space transformation techniques and new algorithmic approaches are proposed to deal with such hard problems. In the first research investigation some variable space transformation techniques are defined as a tool that can be helpfully integrated in (almost) all algorithm frameworks. In particular the focus will be on piecewise linear and non-linear transformations that allow to tackle the problem advantageously. After introducing the theory, preliminary numerical experiments are reported exploiting the transformations in a simple multi-start framework. The idea is to gather the information obtained during a multi-start approach and to apply a sequence of transformations in the variable space that makes the exploration easier. The aim is to expand the attraction basins of global minimizers shrinking those of the local minima already found. Preliminary considerations are made about the possibility to use these transformations as derivative-free preconditioners. The second research investigation concerns the definition of an efficient algorithm on a wide spectrum of global optimization problems. In particular, will be discussed how to do an accurate exploratory geometry and a space search reduction strategy, recently renamed in literature as zoom-in strategy, in a probabilistic algorithm that can speed up significantly the convergence towards better solutions. After introducing the algorithm framework named GABRLS, presented as the winner of the generalization-based Contest in Global Optimization (GENOPT 2017, [61]), the approach is extended to handle also non-continuous variables. The resulting algorithm has been tested in a real case study of design optimization of electric motor. The case study provides evidences about the promising exploratory geometry of the approach in quickly finding feasible and optimal solutions to a mixed-integer constrained problem. In the last research investigation, a new black-box approach is proposed to tackle a real case study of the spare part management problem of a fleet of aircraft. In particular, for this specific type of inventory problem, a black-box model and a tailored global optimization algorithm is defined. The aim is to address the non-linearity of the problem as is, without any decomposition in sub-problem and without any approximation or necessity to check ex post the feasibility of the solution. The main contribution consists of advancing the existing literature for multi-item inventory systems through an enhanced time-effective optimization algorithm tested in a single-echelon system.