On Convex Envelopes and Regularization of Non-convex Functionals Without Moving Global Minima

Marcus Carlsson ()
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Marcus Carlsson: University of Lund

Journal of Optimization Theory and Applications, 2019, vol. 183, issue 1, No 4, 66-84

Abstract: Abstract We provide theory for the computation of convex envelopes of non-convex functionals including an $$\ell ^2$$ ℓ 2 -term and use these to suggest a method for regularizing a more general set of problems. The applications are particularly aimed at compressed sensing and low-rank recovery problems, but the theory relies on results which potentially could be useful also for other types of non-convex problems. For optimization problems where the $$\ell ^2$$ ℓ 2 -term contains a singular matrix, we prove that the regularizations never move the global minima. This result in turn relies on a theorem concerning the structure of convex envelopes, which is interesting in its own right. It says that at any point where the convex envelope does not touch the non-convex functional, we necessarily have a direction in which the convex envelope is affine.

Keywords: Fenchel conjugate; Convex envelope; Regularization; Non-convex/non-smooth optimization; Proximal hull; 49M20; 65K10; 90C26 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10957-019-01541-8

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