zbMATH — the first resource for mathematics

On the convergence of the continuous gradient projection method. (English) Zbl 1430.90462
Summary: We investigate the long time behaviour of the solutions \(x(t)\) to the first order differential inclusion \[ x'(t)+x(t) \in P_Q (x(t)-\lambda (t)\partial\Phi (x(t))), \] where \(\partial\Phi\) is the subgradient of a given convex and continuous function defined on a real Hilbert space \(\mathcal{H}\), the operator \(P_Q:\mathcal{H}\to Q\) is the orthogonal projection onto a closed, nonempty and convex subset \(Q\) of \(\mathcal{H}\), and \(\lambda :[0,+\infty [\to ]0,+\infty [\) is an absolutely continuous function. We establish that if the objective function \(\Phi\) has at least one minimizer over \(Q\) and \(\lambda (t)\) behaviours, for \(t\) large enough, like \(t^\theta\) for some constant \(\theta >-1\) then any solution \(x(t)\) to (the above equation) converges weakly to a minimizer of \(\Phi\) over \(Q\) and satisfies the following fast decay property: \[ \Phi (x(t))-\Phi^* =o \left(\frac{1}{t^{\theta+1}}\right)\text{ as }\to +\infty, \] where \(\Phi^* =\min_{x\in Q} \Phi (x)\). Moreover, we prove the strong convergence of the solutions \(x(t)\) under some simple geometrical assumptions on the function \(\Phi\).
90C25 Convex programming
90C30 Nonlinear programming
Full Text: DOI arXiv
[1] Peypouquet, J., Convex optimization in normed spaces: theory, methods, and examples. With a foreword by Hedy Attouch. Springer Briefs in Optimization, xiv-124, (2015), Cham: Springer, Cham
[2] Antipin, As., Minimization of convex functions on convex sets by means of differential equations, Differ Equ, 30, 9, 1365-1376, (1994)
[3] Rockafellar, Rt., Convex analysis, (1970), Princeton, NJ: Princeton University Press, Princeton, NJ
[4] Nestrov, Y., Introductory lectures on convex optimization: a basic course, (2004), Dordrech: Kluwer, Dordrech
[5] Bauschke, Hh; Combettes, Pl., Convex analysis and monotone operator theory in Hilbert spaces. With a foreword by Hedy Attouch. CMS Books in Mathematics, xvi-468, (2011), New York: Springer, New York
[6] Bolte, J., Continuous gradient projection method in Hilbert spaces, J Optim Theory Appl, 119, 2, 235-259, (2003) · Zbl 1055.90069
[7] The continuous gradient projection method. arXiv:1808.07705 (October 2018)
[8] Baillon, Jb., Un exemple concernant le comportement asymptotique de la solution du probleme \(##?##\), J Funct Anal, 28, 369-376, (1978) · Zbl 0386.47041
[9] Bruck, Re., Asymptotic convergence of nonlinear contraction semigroups in Hilbert space, J Funct Anal, 18, 1, 15-26, (1975) · Zbl 0319.47041
[10] Brezis, H., Operateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert, vi-183, (1973), Amesterdam: North Holland Publishing CO, Amesterdam
[11] Polyak, Bp., Minimization of unsmooth functionals, USSR Comput Math Math Phys, 9, 3, 14-29, (1969) · Zbl 0229.65056
[12] Cheney, W.; Goldestein, Aa., Proximity maps for convex sets, Proc Amer Math Soc, 10, 448-450, (1959) · Zbl 0092.11403
[13] Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull Amer Math Soc, 73, 591-598, (1967) · Zbl 0179.19902
[14] Semi groupes de contraction. École normale supérieure Paris. 1978
[15] Attouch, H.; Goudou, X.; Redont, P., The heavy ball with friction method, I: the continuous dynamical system: global exploration of the local minima of a real valued function by asymptotic analysis of a dissipative dynamical system, Commun Contemp Math, 2, 1-34, (2000) · Zbl 0983.37016
[16] Alvarez, F., On the Minimizing Property of a Second Order Dissipative System in Hilbert Spaces, SIAM J Cont Optim, 38, 1102-1119, (2000) · Zbl 0954.34053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.