Multiobjective Optimization of Expensive Black-Box Functions via Expected Maximin Improvement

Many engineering design optimization problems contain multiple objective functions all of which it is desired to minimize, say. One approach to solving this problem is to identify those inputs to the objective functions that produce an output (vector) on the Pareto Front; the inputs that produce outputs on the Pareto Front form the Pareto Set. This paper proposes a method for identifying the Pareto Front and the Pareto Set when the objective functions are expensive to compute. The method replaces the objective function evaluations by a rapidly computable approximator based on an interpolating Gaussian process (GP) model. It sequentially selects new input sites guided by an improvement function; the next input to evaluate each output is that vector which maximizes the conditional expected value of this improvement function given the current data. The method introduced in this paper provides two advances within this framework. First, it proposes an improvement function based on the modified maximin fitness function. Second, it uses a family of GP models that allow for dependent output functions but which permits zero covariance should the data be consistent with a model of no association. GP models with dependent component functions have the potential to provide more precise predictions of competing objectives than independent GPs. A closed-form expression is derived for the conditional expectation of the proposed improvement function when there are two objective functions; simulation is used to evaluate this expectation when there are three or more objectives. Examples from the multiobjective optimization literature are presented to show that the proposed procedure can improve substantially previously proposed statistical improvement criteria for the computationally intensive multiobjective optimization setting.