Abstract: Many interesting problems in applied mathematics can be written in terms of infinite-dimensional optimisation problems with dense conic constraints. While linear constraints can be effectively tackled using existing numerical schemes, such as the collocation method, leading to adaptive algorithms, this adaptivity does not extend to problems involving dense conic constraints. Non-convex optimization and optimal transport between densities are two examples where current methods fail to overcome the curse of dimensionality: despite lower bounds suggesting the potential for adaptive numerical algorithms, existing approaches suffer from worst-case complexities that prevent scalability to large-dimensional problems, even for regular instances.

We propose an extension of the collocation method that introduces adaptivity to infinite-dimensional optimization with dense conic constraints. The approach is based on a novel class of non-negative functions, of independent interest, that enjoys desirable properties from both analytical and computational perspectives. Using the proposed method we derive numerical algorithms for non-convex optimization and optimal transport between densities which exhibit computational complexities close to the lower bounds and that effectively address the curse of dimensionality for regular instances. Our research, conducted  over the past three years within the framework of my ERC, opens the path to adaptive scaling algorithms in large-dimensional instances across various classes of applied mathematics problems to address contemporary applications.

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