Computing the resolvent of the sum of operators with application to best approximation problems
- Authors: Dao, Minh , Phan, Hung
- Date: 2019
- Type: Text , Journal article
- Relation: Optimization letters Vol. 14, no. 5 (2019), p. 1193-1205
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- Description: We propose a flexible approach for computing the resolvent of the sum of weakly monotone operators in real Hilbert spaces. This relies on splitting methods where strong convergence is guaranteed. We also prove linear convergence under Lipschitz continuity assumption. The approach is then applied to computing the proximity operator of the sum of weakly convex functions, and particularly to finding the best approximation to the intersection of convex sets.
Nonconvex bundle method with application to a delamination problem
- Authors: Dao, Minh , Gwinner, Joachim , Noll, Dominikus , Ovcharova, Nina
- Date: 2016
- Type: Text , Journal article
- Relation: Computational optimization and applications Vol. 65, no. 1 (2016), p. 173-203
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- Description: Delamination is a typical failure mode of composite materials caused by weak bonding. It arises when a crack initiates and propagates under a destructive loading. Given the physical law characterizing the properties of the interlayer adhesive between the bonded bodies, we consider the problem of computing the propagation of the crack front and the stress field along the contact boundary. This leads to a hemivariational inequality, which after discretization by finite elements we solve by a nonconvex bundle method, where upper- C 1 criteria have to be minimized. As this is in contrast with other classes of mechanical problems with non-monotone friction laws and in other applied fields, where criteria are typically lower- C 1 , we propose a bundle method suited for both types of nonsmoothness. We prove its global convergence in the sense of subsequences and test it on a typical delamination problem of material sciences.
On Slater’s condition and finite convergence of the Douglas–Rachford algorithm for solving convex feasibility problems in Euclidean spaces
- Authors: Bauschke, Heinz , Dao, Minh , Noll, Dominikus , Phan, Hung
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of global optimization Vol. 65, no. 2 (2016), p. 329-349
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- Description: The Douglas–Rachford algorithm is a classical and very successful method for solving optimization and feasibility problems. In this paper, we provide novel conditions sufficient for finite convergence in the context of convex feasibility problems. Our analysis builds upon, and considerably extends, pioneering work by Spingarn. Specifically, we obtain finite convergence in the presence of Slater’s condition in the affine-polyhedral and in a hyperplanar-epigraphical case. Various examples illustrate our results. Numerical experiments demonstrate the competitiveness of the Douglas–Rachford algorithm for solving linear equations with a positivity constraint when compared to the method of alternating projections and the method of reflection–projection.
The Douglas–Rachford algorithm for a hyperplane and a doubleton
- Authors: Bauschke, Heinz , Dao, Minh , Lindstrom, Scott
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of global optimization Vol. 74, no. 1 (2019), p. 79-93
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- Description: The Douglas–Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which—somewhat surprisingly—depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.
Union averaged operators with applications to proximal algorithms for min-convex functions
- Authors: Dao, Minh , Tam, Matthew
- Date: 2018
- Type: Text , Journal article
- Relation: Journal of optimization theory and applications Vol. 181, no. 1 (2018), p. 61-94
- Full Text: false
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- Description: In this paper, we introduce and study a class of structured set-valued operators, which we call union averaged nonexpansive . At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations, where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions, where the first is convex and the second can be expressed as the minimum of finitely many convex functions.