  Counterfactual explanation is one branch of interpretable machine learning
that produces a perturbation sample to change the model's original decision.
The generated samples can act as a recommendation for end-users to achieve
their desired outputs. Most of the current counterfactual explanation
approaches are the gradient-based method, which can only optimize the
differentiable loss functions with continuous variables. Accordingly, the
gradient-free methods are proposed to handle the categorical variables, which
however have several major limitations: 1) causal relationships among features
are typically ignored when generating the counterfactuals, possibly resulting
in impractical guidelines for decision-makers; 2) the counterfactual
explanation algorithm requires a great deal of effort into parameter tuning for
dertermining the optimal weight for each loss functions which must be conducted
repeatedly for different datasets and settings. In this work, to address the
above limitations, we propose a prototype-based counterfactual explanation
framework (ProCE). ProCE is capable of preserving the causal relationship
underlying the features of the counterfactual data. In addition, we design a
novel gradient-free optimization based on the multi-objective genetic algorithm
that generates the counterfactual explanations for the mixed-type of continuous
and categorical features. Numerical experiments demonstrate that our method
compares favorably with state-of-the-art methods and therefore is applicable to
existing prediction models. All the source codes and data are available at
\url{https://github.com/tridungduong16/multiobj-scm-cf}.
