A Framework for Optimization under Limited Information

In many real world problems, optimization decisions have to be made with limited information. The decision maker may have no a priori or posteriori data about the often nonconvex objective function except from on a limited number of points that are obtained over time through costly observations. This paper presents an optimization framework that takes into account the information collection (observation), estimation (regression), and optimization (maximization) aspects in a holistic and structured manner. Explicitly quantifying the information acquired at each optimization step using the entropy measure from information theory, the (nonconvex) objective function to be optimized (maximized) is modeled and estimated by adopting a Bayesian approach and using Gaussian processes as a state-of-the-art regression method. The resulting iterative scheme allows the decision maker to solve the problem by expressing preferences for each aspect quantitatively and concurrently.