In the active-learning phase, at each iteration, we query a new sample and refine the surrogate function by re-partitioning and re-fitting the surrogates using PARC, incorporating both existing and newly acquired samples. As merely minimizing the surrogate may overlook the global optimum, PWAS sums the surrogate function, f (X), with a distance-based exploration function, E(X), i.e., a(X) = f (X)θ E E(X), where a(X) represents an acquisition function that balances between the exploitation of the PWA surrogates, and the exploration of the design space. Here, E(X) is a pure exploration term that quantifies the distance between X and existing samples X 1 , . . . , X N , which is independent of function evaluations. A user-defined trade-off parameter, denoted as θ E , is introduced with a default setting of 0.5. This default value has been validated across a variety of benchmarks to ensure optimal performance . In PWAS, two types of exploration methods, which can be directly reformulated as mixed-integer problems, are considered: a max-box approach for continuous and integer variables, and the hamming distance method for binary encoded categorical variables. These exploration functions are selected as they can be reformulated as MILPs. Therefore, the acquisition function can be minimized with a standard MILP solver to determine the next sample for testing, with the option to directly incorporate linear constraints, if present. Subsequently, a new experiment or simulation is selected and evaluated to assess the objective function value. This iterative process continues until reaching the maximum number of iterations (N max ).