On the Convergence of Adaptive Stochastic Search Methods for Constrained and Multi-objective Black-Box Optimization

Stochastic search methods for global optimization and multi-objective optimization are widely used in practice, especially on problems with black-box objective and constraint functions. Although there are many theoretical results on the convergence of stochastic search methods, relatively few deal with black-box constraints and multiple black-box objectives and previous convergence analyses require feasible iterates. Moreover, some of the convergence conditions are difficult to verify for practical stochastic algorithms, and some of the theoretical results only apply to specific algorithms. First, this article presents some technical conditions that guarantee the convergence of a general class of adaptive stochastic algorithms for constrained black-box global optimization that do not require iterates to be always feasible and applies them to practical algorithms, including an evolutionary algorithm. The conditions are only required for a subsequence of the iterations and provide a recipe for making any algorithm converge to the global minimum in a probabilistic sense. Second, it uses the results for constrained optimization to derive convergence results for stochastic search methods for constrained multi-objective optimization.