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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 \(\Phi\).
MSC:
90C25 Convex programming
90C30 Nonlinear programming
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[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
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