Many practical applications require optimization of multiple, computationally expensive, and possibly competing objectives that are well-suited for multi-objective Bayesian optimization (MOBO) procedures. However, for many types of biomedical data, measures of data analysis workflow success are often heuristic and therefore it is not known a priori which objectives are useful. Thus, MOBO methods that return the full Pareto front may be suboptimal in these cases. Here we propose a novel MOBO method that adaptively updates the scalarization function using properties of the posterior of a multi-output Gaussian process surrogate function. This approach selects useful objectives based on a flexible set of desirable criteria, allowing the functional form of each objective to guide optimization. We demonstrate the qualitative behaviour of our method on toy data and perform proof-of-concept analyses of single-cell RNA sequencing and highly multiplexed imaging datasets.