Balancing risk and expected gain in kriging-based global optimization

Kriging-based Global optimization has been proposed and extensively used for solving black-box optimization problems with expensive function evaluations. The performance of such algorithm relies heavily on the effectiveness of the infill criterion that is used to decide which point to evaluate next. Two common infill criteria are, the probability of improvement (PI) and the expected improvement (EI). The PI results in solutions that have a low risk of failure, that is the solution is not improved in the next round. However the expected gain can be small. EI has a higher risk of failure but maximizes expected gain. This paper views the maximization of PI and EI as a bi-objective optimization problem and suggest a fast and precise gradient-based hypervolume ascent algorithm to compute the Pareto front. The computed Pareto front can be beneficial in different scenarios: Firstly, it can be used for decision making in experimental optimization, when a human decision maker has to decide between a `low risk, low gain' or a `high risk, high gain' strategy. Another application is to use different points on the Pareto front in a multi-point efficient global optimization strategy to balance between exploitation and exploration. The algorithm is validated on two objective functions and compared to other well-known multi-objective optimization algorithms. As a side result we also analyze the landscape of the PI and EI infill criterion and provide closed-form gradient expressions of them.