Black-Box Min-Max Continuous Optimization Using CMA-ES withWorst-case Ranking Approximation

In this study, we investigate the problem of min-max continuous optimization in a black-box setting min.. max(y) (x,y). A popular approach updates.. and.. simultaneously or alternatingly. However, two major limitations have been reported in existing approaches. (I) As the influence of the interaction term between x and y (e.g x(T)By) on the Lipschitz smooth and strongly convex-concave function y increases, the approaches converge to an optimal solution at a slower rate. (II) The approaches fail to converge if.. is not Lipschitz smooth and strongly convex-concave around the optimal solution. To address these difficulties, we propose minimizing the worst-case objective function f (x) = max(y) f(x,y) directly using the covariance matrix adaptation evolution strategy, in which the rankings of solution candidates are approximated by our proposed worst-case ranking approximation (WRA) mechanism. Compared with existing approaches, numerical experiments show two important findings regarding our proposed method. (1) The proposed approach is efficient in terms of f-calls on a Lipschitz smooth and strongly convex-concave function with a large interaction term. (2) The proposed approach can converge on functions that are not Lipschitz smooth and strongly convex-concave around the optimal solution, whereas existing approaches fail.