This paper by Dong C. Liu and Jorge Nocedal focuses on the numerical performance of the Limited Memory BFGS (L-BFGS) method for large-scale optimization. The authors compare the L-BFGS method with the method developed by Buckley and LeNir (1985), which combines BFGS steps and conjugate direction steps. Numerical tests show that the L-BFGS method is faster and more efficient in using additional storage to accelerate convergence. The paper also explores ways to further enhance the L-BFGS method through diagonal scalings and examines its performance on very large problems. Additionally, the L-BFGS method is compared with the partitioned quasi-Newton method of Griewank and Toint (1982a), revealing that while the partitioned quasi-Newton method is superior for some problems, the L-BFGS method remains competitive due to its low iteration cost. The authors also provide a convergence analysis of the L-BFGS method, proving global convergence for uniformly convex problems.This paper by Dong C. Liu and Jorge Nocedal focuses on the numerical performance of the Limited Memory BFGS (L-BFGS) method for large-scale optimization. The authors compare the L-BFGS method with the method developed by Buckley and LeNir (1985), which combines BFGS steps and conjugate direction steps. Numerical tests show that the L-BFGS method is faster and more efficient in using additional storage to accelerate convergence. The paper also explores ways to further enhance the L-BFGS method through diagonal scalings and examines its performance on very large problems. Additionally, the L-BFGS method is compared with the partitioned quasi-Newton method of Griewank and Toint (1982a), revealing that while the partitioned quasi-Newton method is superior for some problems, the L-BFGS method remains competitive due to its low iteration cost. The authors also provide a convergence analysis of the L-BFGS method, proving global convergence for uniformly convex problems.