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

Hiriart-Urruty, J. B.; Plazanet, Ph.

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, Suppl., 325-338 (1989).

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

Keywords:

Hilbert space; lower-semicontinuous convex function; deconvolution

References:

[1]	Asplund, E., Differentiability of the metric projection in finite-dimensional Euclidean space, Proceedings of the Amer. Math. Society, 38, 218-219, (1973) · Zbl 0269.52002
[2]	D. Azé and J-P. Penot, Uniformly convex functions and uniformly smooth convex functions, preprint (1987).
[3]	Hiriart-Urruty, J.-B., Generalized differentiability, duality and optimization for problems dealing with differences of convex functions, Convexity and Duality in Optimization, Lecture Notes in Economics and Mathematical Systems, 256, 37-70, (1986) · Zbl 0591.90073
[4]	Hiriart‐Urruty, J.-B.; Mazure, M.-L., Formulations variationnelles de l’addition parallele et de la soustraction parallele d’opératours semi-définis positits, C.R. Acad. Sc. Paris, 1. 302, n° 15, Série I, 527-530, (1986) · Zbl 0597.15008
[5]	Hiriart−Urruty, J.-B., A general formula on the conjugate of the difference of functions, Canad. Math. Bull, Vol 29, 4, 482-485, (1986) · Zbl 0608.90087
[6]	Lasry, J.-M.; Lions, P. L., A remark on regularization in Hilbert spaces, Israel J. of Mathematics, 55, 257-266, (1986) · Zbl 0631.49018
[7]	Moreau, J.-J., Décomposition orthogonale d’un espace hilbertion selon deux cones mutuellement polaires, C. R. Acad. Sc., t.255, 238-240, (1962) · Zbl 0109.08105
[8]	Moreau, J.-J., Proximité et dualité dans un espace hilberation, Bull. Soc. Math. France, 93, 273-299, (1965) · Zbl 0136.12101
[9]	Moreau, J.-J., Weak and strong solutions of dual problems, (Zarantonello, E., Contributions to Nonlinear Functional Analysis, (1971), Academic Press) · Zbl 0274.49007
[10]	M. Volle, private communication (June 1987).
[11]	Wierzbicki, A. P.; Kurcyuz, S., Projection on a cone, penalty functionals and duality theory for problems with inequality constraints in Hilbert space, SIAM J. Control and optimization, 15, 25-26, (1977) · Zbl 0355.90045