arXiv subjects

Oleg Burdakov

Publications and source records attributed to Oleg Burdakov.

4 recordsLinked to original sources

arXiv2021

Algorithm XXX: SC-SR1: Matlab software for solving shape-changing L-SR1 trust-region subproblems

We present a MATLAB implementation of the symmetric rank-one (SC-SR1) method that solves trust-region. subproblems when a limited-memory symmetric rank-one (L-SR1) matrix is used in place of the true Hessian matrix, which can be used for large-scale optimization. The method takes advantage of two shape-changing norms[Burdakov and Yuan 2002; Burdakov et al. 2017] to decompose the trust-region subproblem into two separate problems. Using one of the proposed norms, the resulting subproblems have closed-form solutions. Meanwhile, using the other proposed norm, one of the resulting subproblems has a closed-form solution while the other is easily solvable using techniques that exploit the structure of L-SR1 matrices. Numerical results suggest that the SC-SR1 method is able to solve trust-region subproblems to high accuracy even in the so-called "hard case". When integrated into a trust-region algorithm, extensive numerical experiments suggest that the proposed algorithms perform well, when compared with widely used solvers, such as truncated CG.

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arXiv2019

Stabilized Barzilai-Borwein method

The Barzilai-Borwein (BB) method is a popular and efficient tool for solving large-scale unconstrained optimization problems. Its search direction is the same as for the steepest descent (Cauchy) method, but its stepsize rule is different. Owing to this, it converges much faster than the Cauchy method. A feature of the BB method is that it may generate too long steps, which throw the iterates too far away from the solution. Moreover, it may not converge, even when the objective function is strongly convex. In this paper, a stabilization technique is introduced. It consists in bounding the distance between each pair of successive iterates, which often allows for decreasing the number of BB iterations. When the BB method does not converge, our simple modification of this method makes it convergent. Under suitable assumptions, we prove its global convergence, despite the fact that no line search is involved, and only gradient values are used. Since the number of stabilization steps is proved to be finite, the stabilized version inherits the fast local convergence of the BB method. The presented results of extensive numerical experiments show that our stabilization technique often allows the BB method to solve problems in a fewer iterations, or even to solve problems where the latter fails.

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arXiv2019

A Dense Initialization for Limited-Memory Quasi-Newton Methods

We consider a family of dense initializations for limited-memory quasi-Newton methods. The proposed initialization exploits an eigendecomposition-based separation of the full space into two complementary subspaces, assigning a different initialization parameter to each subspace. This family of dense initializations is proposed in the context of a limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) trust-region method that makes use of a shape-changing norm to define each subproblem. As with L-BFGS methods that traditionally use diagonal initialization, the dense initialization and the sequence of generated quasi-Newton matrices are never explicitly formed. Numerical experiments on the CUTEst test set suggest that this initialization together with the shape-changing trust-region method outperforms other L-BFGS methods for solving general nonconvex unconstrained optimization problems. While this dense initialization is proposed in the context of a special trust-region method, it has broad applications for more general quasi-Newton trust-region and line search methods. In fact, this initialization is suitable for use with any quasi-Newton update that admits a compact representation and, in particular, any member of the Broyden class of updates.

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arXiv2017

Multipoint secant and interpolation methods with nonmonotone line search for solving systems of nonlinear equations

Multipoint secant and interpolation methods are effective tools for solving systems of nonlinear equations. They use quasi-Newton updates for approximating the Jacobian matrix. Owing to their ability to more completely utilize the information about the Jacobian matrix gathered at the previous iterations, these methods are especially efficient in the case of expensive functions. They are known to be local and superlinearly convergent. We combine these methods with the nonmonotone line search proposed by Li and Fukushima (2000), and study global and superlinear convergence of this combination. Results of numerical experiments are presented. They indicate that the multipoint secant and interpolation methods tend to be more robust and efficient than Broyden's method globalized in the same way.

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