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Density of convex intersections in Sobolev spaces with applications

A significant number of problems in optimization involve, either directly or through Fenchel dualization, constraint sets determined by pointwise bounds (possibly spatially variant) on function values, their gradient or divergence. It is shown within a general framework that through regularization, discretization, and dualization of such problems, a density question arises: Are smooth functions satisfying the initial constraint dense, in some norm, in the constraint set consisting of less regular functions? A number of positive and negative answer scenarios to this question are given. In particular, it is shown that a continuous embedding is not enough to guarantee the density property even if the upper bound in the constraint set is constant, and counterexamples in the Sobolev setting are provided. In addition, we identify diverse classes of upper bound functions for which the answer to the density question is positive. The talk is finalized with many applications in image reconstruction, and elastoplasticity.

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