2017 , Volume 22, ¹ 5, p.47-57
Applications of quasi-Newton algorithms for solving large scale problems
In this paper, numerical stability of quasi-Newton algorithms is studied, and their efficiency is estimated. Two types of quasi-Newton algorithms are considered. In the BFGS algorithm, iterative approximations to the Hessian matrix 𝐵𝑘, are constructed. The matrix 𝐵𝑘 is maintained in a factored form, 𝐵 = 𝐿𝑘𝐷𝑘𝐿𝑇 , by a procedure based 𝑘 on the reflection method.
In the limited - memory quasi-Newton algorithm, L-BFGS, approximations to the inverse Hessian matrix are constructed. Instead of the inverse Hessian 𝐻𝑘, ○ IAO SB RAS, 2017 L-BFGS stores a few vectors that represent the quasi-Newton updates. The accuracy and efficiency of the BFGS algorithms are compared by solving some test problems. A parallel L-BFGS algorithm based on OpenMP programming interface is developed for solving large scale problems.
The algorithm is tested on problems with a large number of variables. The use of the parallel algorithm makes it possible to significantly reduce the execution time in a wide range of the problem dimensions.
Keywords: quasi-Newton algorithms, quasi-Newton algorithms with limited memory, unconstrained minimization, OpenMP technology
Zabinyako Gerard Idelfonovich
PhD. , Senior Scientist
Position: Head of Laboratory
Office: ICMMG SB RAS
Address: Russia, Novosibirsk
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Zabinyako G.I. Applications of quasi-Newton algorithms for solving large scale problems // Computational technologies. 2017. V. 22. ¹ 5. P. 47-57