The SVDD algorithm can be extended to handle multi-objective optimization by incorporating a weighted sum or Pareto front approach to combine multiple reward functions into a single composite reward signal. In the context of SVDD, this could involve defining a new reward function ( r_{\ ext{multi}}(x) ) that aggregates the individual reward functions ( r_1(x), r_2(x), \\ldots, r_n(x) ) as follows:
[
r_{\ ext{multi}}(x) = \\sum_{i=1}^{n} w_i r_i(x)
]
where ( w_i ) are the weights assigned to each reward function, reflecting their relative importance in the optimization process. This composite reward can then be used in the SVDD framework to guide the sampling process.
Alternatively, a Pareto optimization approach could be employed, where the algorithm seeks to find a set of solutions that represent the best trade-offs among the multiple objectives. In this case, the soft value functions could be adapted to evaluate the trade-offs between different reward functions, allowing the SVDD algorithm to explore a diverse set of solutions that are optimal with respect to multiple criteria. This would enhance the algorithm's ability to generate designs that are not only high-performing in one aspect but also balanced across various objectives, thus broadening the applicability of SVDD in complex design spaces.