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The ridge method for tame min-max problems


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.
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Dates and versions

hal-03186676 , version 1 (31-03-2021)
hal-03186676 , version 2 (04-02-2022)
hal-03186676 , version 3 (26-12-2022)
hal-03186676 , version 4 (23-06-2023)


  • HAL Id : hal-03186676 , version 1


Edouard Pauwels. The ridge method for tame min-max problems. 2021. ⟨hal-03186676v1⟩


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