Safe global optimization

This chapter addresses the recent emerging topic known as safe global optimization, with \textit{safe} meaning that no \textit{safety constraints} are violated while sequentially performing evaluations of the noisy black-box objective function. Safety constraints can also be black-box. Thus, the research is focused on designing and developing algorithms which could -- more or less accurately -- estimate safety of a new point \textit{before} evaluating the function at this point. The main algorithms proposed during last years are revised here, with most of them designed to respond to real-life applications-specific needs, in domains such as robotics, engineering, systems control, and many others. The most widely proposed and extended approaches use a model -- typically a Gaussian Process Regression -- to approximate both the objective function and the safety constraints, and then include Lipschitz continuity to make the safety estimation more accurate. One can refer to these algorithms with the term \textit{model-based}: due to modelling errors they can only offer probabilistic guarantees of safety. A recent \textit{model-free} algorithm is also revised, which overcomes limitations of the model-based ones by using Lipschitz continuity and directly dealing with the noise. Moreover, it offers sure guarantees of safety, under reasonable assumptions about the noise. The chapter is concluded by summarizing benefits as well as limitations of different algorithms; challenges and perspectives for future research are also indicated.