@article{bui_2022, title={A decomposition method for solving multicommodity network equilibria}, volume={50}, ISSN={["1872-7468"]}, DOI={10.1016/j.orl.2021.12.002}, abstractNote={We consider the numerical aspect of the multicommodity network equilibrium problem proposed by Rockafellar in 1995. Our method relies on the flexible monotone operator splitting framework recently proposed by Combettes and Eckstein.}, number={1}, journal={OPERATIONS RESEARCH LETTERS}, author={Bui, Minh N.}, year={2022}, month={Jan}, pages={40–44} }
 @article{bui_combettes_2021, title={Multivariate Monotone Inclusions in Saddle Form}, volume={12}, ISSN={["1526-5471"]}, DOI={10.1287/moor.2021.1161}, abstractNote={ We propose a novel approach to monotone operator splitting based on the notion of a saddle operator. Under investigation is a highly structured multivariate monotone inclusion problem involving a mix of set-valued, cocoercive, and Lipschitzian monotone operators, as well as various monotonicity-preserving operations among them. This model encompasses most formulations found in the literature. A limitation of existing primal-dual algorithms is that they operate in a product space that is too small to achieve full splitting of our problem in the sense that each operator is used individually. To circumvent this difficulty, we recast the problem as that of finding a zero of a saddle operator that acts on a bigger space. This leads to an algorithm of unprecedented flexibility, which achieves full splitting, exploits the specific attributes of each operator, is asynchronous, and requires to activate only blocks of operators at each iteration, as opposed to activating all of them. The latter feature is of critical importance in large-scale problems. The weak convergence of the main algorithm is established, as well as the strong convergence of a variant. Various applications are discussed, and instantiations of the proposed framework in the context of variational inequalities and minimization problems are presented. }, journal={MATHEMATICS OF OPERATIONS RESEARCH}, author={Bui, Minh N. and Combettes, Patrick L.}, year={2021}, month={Dec} }
 @article{bui_2022, title={Projective Splitting as a Warped Proximal Algorithm}, volume={85}, ISSN={["1432-0606"]}, DOI={10.1007/s00245-022-09868-x}, abstractNote={We show that the asynchronous block-iterative primal-dual projective splitting framework introduced by P. L. Combettes and J. Eckstein in their 2018 Math. Program. paper can be viewed as an instantiation of the recently proposed warped proximal algorithm.}, number={2}, journal={APPLIED MATHEMATICS AND OPTIMIZATION}, author={Bui, Minh N.}, year={2022}, month={Apr} }
 @article{bui_combettes_2021, title={Bregman Forward-Backward Operator Splitting}, volume={29}, ISBN={1877-0541}, url={https://doi.org/10.1007/s11228-020-00563-z}, DOI={10.1007/s11228-020-00563-z}, abstractNote={We establish the convergence of the forward-backward splitting algorithm based on Bregman distances for the sum of two monotone operators in reflexive Banach spaces. Even in Euclidean spaces, the convergence of this algorithm has so far been proved only in the case of minimization problems. The proposed framework features Bregman distances that vary over the iterations and a novel assumption on the single-valued operator that captures various properties scattered in the literature. In the minimization setting, we obtain rates that are sharper than existing ones.}, number={3}, journal={Set-Valued and Variational Analysis}, author={Bui, Minh N. and Combettes, Patrick L.}, year={2021}, month={Sep}, pages={583–603} }
 @article{bùi_combettes_2020, title={The Douglas--Rachford Algorithm Converges Only Weakly}, volume={58}, ISSN={0363-0129 1095-7138}, url={http://dx.doi.org/10.1137/19m1308451}, DOI={10.1137/19M1308451}, abstractNote={We show that the weak convergence of the Douglas--Rachford algorithm for finding a zero of the sum of two maximally monotone operators cannot be improved to strong convergence. Likewise, we show that strong convergence can fail for the method of partial inverses.}, number={2}, journal={SIAM Journal on Control and Optimization}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Bùi, Minh N. and Combettes, Patrick L.}, year={2020}, month={Jan}, pages={1118–1120} }
 @article{bùi_combettes_2020, title={Warped proximal iterations for monotone inclusions}, volume={491}, ISSN={0022-247X}, url={http://dx.doi.org/10.1016/j.jmaa.2020.124315}, DOI={10.1016/j.jmaa.2020.124315}, abstractNote={Resolvents of set-valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions. We propose a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. The properties of warped resolvents are investigated and connections are made with existing notions. Abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems.}, number={1}, journal={Journal of Mathematical Analysis and Applications}, publisher={Elsevier BV}, author={Bùi, Minh N. and Combettes, Patrick L.}, year={2020}, month={Nov}, pages={124315} }
 @article{bauschke_bui_wang_2020, title={Applying FISTA to optimization problems (with or) without minimizers}, volume={184}, ISSN={["1436-4646"]}, DOI={10.1007/s10107-019-01415-x}, abstractNote={Beck and Teboulle's FISTA method for finding a minimizer of the sum of two convex functions, one of which has a Lipschitz continuous gradient whereas the other may be nonsmooth, is arguably the most important optimization algorithm of the past decade. While research activity on FISTA has exploded ever since, the mathematically challenging case when the original optimization problem has no minimizer has found only limited attention. In this work, we systematically study FISTA and its variants. We present general results that are applicable, regardless of the existence of minimizers.}, number={1-2}, journal={MATHEMATICAL PROGRAMMING}, author={Bauschke, Heinz H. and Bui, Minh N. and Wang, Xianfu}, year={2020}, month={Nov}, pages={349–381} }
 @article{bauschke_bui_wang_2019, title={On sums and convex combinations of projectors onto convex sets}, volume={242}, ISSN={["1096-0430"]}, DOI={10.1016/j.jat.2019.02.001}, abstractNote={The projector onto the Minkowski sum of closed convex sets is generally not equal to the sum of individual projectors. In this work, we provide a complete answer to the question of characterizing the instances where such an equality holds. Our results unify and extend the case of linear subspaces and Zarantonello's results for projectors onto cones. A detailed analysis in the case of convex combinations is carried out, and we also establish the partial sum property for projectors onto convex cones.}, journal={JOURNAL OF APPROXIMATION THEORY}, author={Bauschke, Heinz H. and Bui, Minh N. and Wang, Xianfu}, year={2019}, month={Jun}, pages={31–57} }