Abstract
We consider the classical optimization problem of minimizing a strongly convex, non-smooth, Lipschitz-continuous function with one Lipschitz-continuous constraint. We develop the approach in [10] and propose two methods for the considered problem with adaptive stopping rules. The main idea of the methods is using the dichotomy method and solving an auxiliary one-dimensional problem at each iteration. Theoretical estimates for the proposed methods are obtained. Partially, for smooth functions, we prove the linear rate of convergence of the methods. We also consider theoretical estimates in the case of non-smooth functions. The results for some examples of numerical experiments illustrating the advantages of the proposed methods and the comparison with some adaptive optimal method for non-smooth strongly convex functions are also given.
The authors are very grateful to Alexander V. Gasnikov and Anastasiya S. Ivanova for fruitful discussions. The research of F. Stonyakin in Algorithm 1, Theorem 2 and Lemma 2 was supported by Russian Foundation for Basic Research according to the project 18-29-03071 mk. The research of F. Stonyakin in Subsects. 3.4 and 3.5 was supported by Russian Science Foundation grant 18-71-10044.
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Stonyakin, F.S., Alkousa, M.S., Titov, A.A., Piskunova, V.V. (2019). On Some Methods for Strongly Convex Optimization Problems with One Functional Constraint. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Lecture Notes in Computer Science(), vol 11548. Springer, Cham. https://doi.org/10.1007/978-3-030-22629-9_7
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