A framework around limited-memory partitioned quasi-Newton methods

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We present a Julia framework dedicated to partially-separable problems whose element function are detected automatically. This framework takes advantage of the JuliaSmoothOptimizer ecosystem for minimizing unconstrained smooth partially-separable problems with several partitioned quasi-Newton trust-region methods. We provide three new limited-memory partitioned quasi-Newton operators: PLBFGS, PLSR1 and PLSE. These new limited-memory partitioned quasi-Newton operators rely on LBFGS and LSR1 operators to approximate element Hessians instead of dense matrices. Therefore, the memory footprint and the linear application complexity drop from \(\Theta(\sum_{i=1}^N \frac{n_i(n_i+2)}{2})\) to \(\Theta(\sum_{i=1}^N 2m n_i)\). Under the assumption that element function gradients are Lipchitz continuous, we establish a global convergence proof for these new methods. The numerical results compare the methods presented to the state of the art of limited-memory or partitioned quasi-Newton methods through performance profiles. To legitimate partitioned limited-memory operators, we study limits of classic partitioned quasi-Newton methods where large element functions exist.

, 27 pages

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