Efficient Global Optimization with Adaptive Target Setting

Abstract

S URROGATE-BASED optimization is becoming increasingly popular due to savings in computational time [1–12]. Surrogatebased optimization proceeds through cycles, selecting new sampling points that contribute toward global optimization in each cycle. Algorithms like the popular efficient global optimization (EGO) of Jones et al. [11,12] use both the surrogate prediction and its error estimates. The most common EGO variant uses prediction and prediction variance to seek the point of maximum expected improvement (EI) [12]. Jones [12] also discusses a version of EGO that uses the probability of improvement (PI) beyond a given target as the selection criterion. The use of PI was first introduced by Kushner in 1964 [13] for a one-dimensional algorithm. It was extended heuristically to higher dimensions by Stuckman [14], Elder [15], and Mockus [16]. Maximizing PI can balance local and global searches, but its performance can be sensitive to the target value [12]. If the target is too ambitious, the search is excessively global and slow to focus on promising areas. If the target is too modest, there is exhaustive search around the present best solution (PBS) before moving to global search. This issuemay account for the lack of popularity of EGOwith PI. Jones [12] proposed to address the issue of target setting by considering several target values to add multiple points per cycle, calling the method “a highly promising approach”. Queipo et al. [17] proposed away to estimate the optimum that could be used as a target. In this work, we propose an adaptive target method that adapts the target for each EGO cycle according to the success of meeting the target in the previous cycle. We dub this variant EGO-AT (for “adaptive target”). EGO-AT learns from the history of progress to predict what to expect in the next cycle. Traditionally, EGO-like algorithms add one point per cycle. When simulations take long to complete, it is attractive to run in parallel multiple simulations per cycle. Consequently, there has beenwork on including multiple points [18–21]. Selecting multiple points to maximize EI is computationally expensive [20]. However, there are methods such as kriging believer [20] and constant liar [20] to moderate the expense of multipoint EI. Using PI for finding multiple points using a single surrogate has been shown to be quite cheap [12,22]. The joint probability of all the points being added can also be easily calculated for a single target [22]. The major objective of this work is to bring PI on equal footing with EI and remove issues with using EGO-PI. In addition, EGO-AT provides two ingredients that may be useful for decisions on stopping: amount of improvement to target and the probability of targeted improvement [23,24].