  We study the ridge method for min-max problems, and investigate its
convergence without any convexity, differentiability or qualification
assumption. The central issue is to determine whether the "parametric
optimality formula" provides a conservative field, a notion of generalized
derivative well suited for optimization. The answer to this question is
positive in a semi-algebraic, and more generally definable, context. The proof
involves a new characterization of definable conservative fields which is of
independent interest. As a consequence, the ridge method applied to definable
objectives is proved to have a minimizing behavior and to converge to a set of
equilibria which satisfy an optimality condition. Definability is key to our
proof: we show that for a more general class of nonsmooth functions,
conservativity of the parametric optimality formula may fail, resulting in an
absurd behavior of the ridge method.
