Abstract
We deal with monotone inclusion problems of the form 0 ∈ A x + D x + N C (x) in real Hilbert spaces, where A is a maximally monotone operator, D a cocoercive operator and C the nonempty set of zeros of another cocoercive operator. We propose a forward-backward penalty algorithm for solving this problem which extends the one proposed by Attouch et al. (SIAM J. Optim. 21(4): 1251-1274, 2011). The condition which guarantees the weak ergodic convergence of the sequence of iterates generated by the proposed scheme is formulated by means of the Fitzpatrick function associated to the maximally monotone operator that describes the set C. In the second part we introduce a forward-backward-forward algorithm for monotone inclusion problems having the same structure, but this time by replacing the cocoercivity hypotheses with Lipschitz continuity conditions. The latter penalty type algorithm opens the gate to handle monotone inclusion problems with more complicated structures, for instance, involving compositions of maximally monotone operators with linear continuous ones.
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Attouch, H., Czarnecki, M.-O.: Asymptotic behavior of coupled dynamical systems with multiscale aspects. J. Differ. Equ. 248(6), 1315–1344 (2010)
Attouch, H., Czarnecki, M.-O., Peypouquet, J.: Prox-penalization and splitting methods for constrained variational problems. SIAM J. Optim. 21(1), 149–173 (2011)
Attouch, H., Czarnecki, M.-O., Peypouquet, J.: Coupling forward-backward with penalty schemes and parallel splitting for constrained variational inequalities. SIAM J. Optim. 21(4), 1251–1274 (2011)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics. Springer, New York (2011)
Bauschke, H.H., McLaren, D.A., Sendov, H.S.: Fitzpatrick functions: inequalities, examples and remarks on a problem by S. Fitzpatrick. J. Convex Anal. 13(3–4), 499–523 (2006)
Borwein, J.M.: Maximal monotonicity via convex analysis. J. Convex Anal. 13(3–4), 561–586 (2006)
Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010)
Boţ, R.I.: Conjugate duality in convex optimization: Lecture Notes in Economics and Mathematical Systems, vol. 637. Springer, Berlin Heidelberg (2010)
Boţ, R.I., Csetnek, E.R.: An application of the bivariate inf-convolution formula to enlargements of monotone operators. Set-Valued Anal. 16(7–8), 983–997 (2008)
Boţ, R.I., Csetnek, E.R., Heinrich, A.: A primal-dual splitting algorithm for finding zeros of sums of maximal monotone operators. SIAM J. Optim. 23(4), 2011–2036 (2013)
Boţ, R.I., Hendrich, C.: Convergence analysis for a primal-dual monotone + skew splitting algorithm with applications to total variation minimization. J. Math. Imaging Vision. (2014) doi:10.1007/s10851-013-0486-8
Briceño-Arias, L.M., Combettes, P.L.: A monotone + skew splitting model for composite monotone inclusions in duality. SIAM J. Optim. 21(4), 1230-1250 (2011)
Burachik, R.S., Svaiter, B.F.: Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal. 10(4), 297–316 (2002)
Combettes, P.L.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optim. 53(5–6), 475–504 (2004)
Combettes, P.L., Pesquet, J.-C.: Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators. Set-Valued Var. Anal. 20(2), 307–330 (2012)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland Publishing Company, Amsterdam (1976)
Fitzpatrick, S.: Representing monotone operators by convex functions. In: Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proceedings of the Centre for Mathematical Analysis 20, Australian National University, Canberra, pp. 59–65 (1988)
Noun, N., Peypouquet, J.: Forward-backward penalty scheme for constrained convex minimization without inf-compactness. J. Optim. Theory Appl. 158(3), 787–795 (2013)
Peypouquet, J.: Coupling the gradient method with a general exterior penalization scheme for convex minimization. J. Optim. Theory Appl. 153(1), 123-138 (2012)
Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33(1), 209–216 (1970)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J Control. Optim. 14(5), 877–898 (1976)
Simons, S.: From Hahn-Banach to Monotonicity. Springer, Berlin (2008)
Tseng, P.: A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control. Optim. 38(2), 431–446 (2000)
Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38(3), 667–681 (2013)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)
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Boţ, R.I., Csetnek, E.R. Forward-Backward and Tseng’s Type Penalty Schemes for Monotone Inclusion Problems. Set-Valued Var. Anal 22, 313–331 (2014). https://doi.org/10.1007/s11228-014-0274-7
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DOI: https://doi.org/10.1007/s11228-014-0274-7
Keywords
- Maximally monotone operator
- Fitzpatrick function
- Resolvent
- Cocoercive operator
- Lipschitz continuous operator
- Forward-backward algorithm
- Forward-backward-forward algorithm
- Subdifferential
- Fenchel conjugate