## 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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