Monotonic grey box direct search optimization

We are interested in blackbox optimization for which the user is aware of monotonic behaviour of some constraints defining the problem. That is, when increasing a variable, the user is able to predict if a function increases or decreases, but is unable to quantify the amount by which it varies. We refer to this type of problems as “monotonic grey box” optimization problems. Our objective is to develop an algorithmic mechanism that exploits this monotonic information to find a feasible solution as quickly as possible. With this goal in mind, we have built a theoretical foundation through a thorough study of monotonicity on cones of multivariate functions. We introduce a trend matrix and a trend direction to guide the Mesh Adaptive Direct Search (Mads) algorithm when optimizing a monotonic grey box optimization problem. Different strategies are tested on a some analytical test problems, and on a real hydroelectric dam optimization problem.