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Moreau’s decomposition theorem revisited. (English) Zbl 0675.90093

Summary: Given two convex functions g and h on a Hilbert space, verifying \(g+h=\| \cdot \|^ 2\), we show there necessarily exists a lower- semicontinuous convex function F such that \(g=F\square \| \cdot \|^ 2\) and \(h=F^*\square \| \cdot \|^ 2\). An explicit formulation of F is given as a deconvolution of a convex function by another one. The approach taken here as well as the way of factorizing g and h shed a new light on what is known as Moreau’s theorem in the literature on Convex Analysis.

MSC:

90C48 Programming in abstract spaces
49J52 Nonsmooth analysis
49J50 Fréchet and Gateaux differentiability in optimization
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References:

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