Abstract
The recent explosion in size and complexity of datasets and the increased availability of computational resources has led us to what is sometimes called the big data era. In many big data fields, mathematical optimization has over the last decade emerged as a vital tool in extracting information from the data sets and creating predictors for unseen data. The large dimension of these data sets and the often parallel, distributed, or decentralized computational structures used for storing and handling the data, set new requirements on the optimization algorithms that solve these problems. This has led to a dramatic shift in focus in the optimization community over this period. Much effort has gone into developing algorithms that scale favorably with problem dimension and that can exploit structure in the problem as well as the computational environment. This is also the main focus of this book, which is comprised of individual chapters that further contribute to this development in different ways. In this introductory chapter, we describe the individual contributions, relate them to each other, and put them into a wider context.
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References
H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. (Springer, New York, 2017)
A. Beck, M. Teboulle, Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper. Res. Lett. 31(3), 167–175 (2003). ISSN: 0167-6377
A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imag. Sci. 2(1), 183–202 (2009)
D.P. Bertsekas, Incremental Aggregated Proximal and Augmented Lagrangian Algorithms, Sept. 2015, arXiv:1509.09257
D.P. Bertsekas, J.N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods (Prentice-Hall, Upper Saddle River, NJ, 1989)
S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)
A. Chambolle, T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vision 40(1), 120–145 (2011)
P.L. Combettes, J.-C. Pesquet, Proximal splitting methods in signal processing, in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, ed. by H.H. Bauschke, R. S. Burachik, P.L. Combettes, V. Elser, D.R. Luke, H. Wolkowicz (Springer, New York, 2011), pp. 185–212
D. Davis, W. Yin, A three-operator splitting scheme and its optimization applications. Set-Valued Var. Anal. 25(4), 829–858 (2017)
J. Douglas, H.H. Rachford, On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)
J. Eckstein, Splitting methods for monotone operators with applications to parallel optimization. PhD thesis, MIT, 1989
M. Frank, P. Wolfe, An algorithm for quadratic programming. Naval Res. Log. Q. 3, 95–110 (1956)
D. Gabay, B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)
R. Glowinski, A. Marroco, Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité d’une classe de problémes de dirichlet non linéaires. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 9, 41–76 (1975)
M. Jaggi, Revisiting Frank-Wolfe: projection-free sparse convex optimization, in Proceedings of the 30th International Conference on Machine Learning, Atlanta, ed. by S. Dasgupta, D. McAllester. Proceedings of Machine Learning Research, vol. 28 of number 1, pp. 427–435 (2013)
D.P. Kingma, J. Ba, Adam: a method for stochastic optimization, Dec. 2014. arXiv:1412.6980
P. Latafat, P. Patrinos, Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators. Comput. Optim. Appl. 68(1), 57–93 (2017)
P.L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)
Y. Nesterov, A method of solving a convex programming problem with convergence rate O (1/k2). Sov. Math. Dokl. 27(2), 372–376 (1983)
Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, 1st edn. (Springer, Boston, 2003)
Y. Nesterov, Efficiency of coordinate descent methods on huge-scale optimization problems. SIAM J. Optim. 22(2), 341–362 (2012)
Y. Nesterov, A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994)
N. Parikh, S. Boyd, Proximal algorithms. Found. Trends Optim. 1(3), 123–231 (2014)
G. Pierra, Decomposition through formalization in a product space. Math. Program. 28(1), 96–115 (1984)
E.K. Ryu, S. Boyd, A primer on monotone operator methods. Appl. Comput. Math. 15(1), 3–43 (2016)
P. Tseng, A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431–446 (2000)
L. Vandenberghe, M.S. Andersen, Chordal graphs and semidefinite optimization. Found. Trends Optim. 1(4), 241–433 (2015)
Y. Ye, M.J. Todd, S. Mizuno, An o(\(\sqrt {n}L\))-iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19(1), 53–67 (1994)
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Giselsson, P., Rantzer, A. (2018). Large-Scale and Distributed Optimization: An Introduction. In: Giselsson, P., Rantzer, A. (eds) Large-Scale and Distributed Optimization. Lecture Notes in Mathematics, vol 2227. Springer, Cham. https://doi.org/10.1007/978-3-319-97478-1_1
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