Abstract
In this paper, we consider a class of optimization problems with a strongly convex objective function and the feasible set given by an intersection of a simple convex set with a set given by a number of linear equality and inequality constraints. Quite a number of optimization problems in applications can be stated in this form, examples being entropy-linear programming, ridge regression, elastic net, regularized optimal transport, etc. We extend the Fast Gradient Method applied to the dual problem in order to make it primal-dual, so that it allows not only to solve the dual problem, but also to construct nearly optimal and nearly feasible solution of the primal problem. We also prove a theorem about the convergence rate for the proposed algorithm in terms of the objective function residual and the linear constraints infeasibility.
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Notes
- 1.
The absolute value here is crucial since \(x_k\) may not satisfy linear constraints and, hence, \(f(x_k)-Opt[P_1]\) could be negative.
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Acknowledgements
The research by A. Gasnikov and P. Dvurechensky presented in Sect. 3 was conducted in IITP RAS and supported by the Russian Science Foundation grant (project 14-50-00150), the research by A. Gasnikov and P. Dvurechensky presented in Sect. 4 was partially supported by RFBR, research project No. 15-31-20571 mol_a_ved. The research by A. Chernov presented in Sect. 4 was partially supported by RFBR, research project No.14-01-00722-a.
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Chernov, A., Dvurechensky, P., Gasnikov, A. (2016). Fast Primal-Dual Gradient Method for Strongly Convex Minimization Problems with Linear Constraints. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds) Discrete Optimization and Operations Research. DOOR 2016. Lecture Notes in Computer Science(), vol 9869. Springer, Cham. https://doi.org/10.1007/978-3-319-44914-2_31
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