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Table 3 Numerical results of the benchmark problems

From: A modified nonmonotone BFGS algorithm for unconstrained optimization

Problem/ \(\boldsymbol{x_{0}}\) Dim BFGS-WP NI/NFG/Time BFGS-WP-Zhang NI/NFG/Time BFGS-Non NI/NFG/Time BFGS-M-Non NI/NFG/Time
Sphere/\(x_{Sph10}\) 30 2/19/1.562500e−001 2/19/1.562500e−002 2/19/4.687500e−002 2/19/4.687500e−002
500 2/19/2.031250e−001 2/19/3.125000e−001 2/19/2.656250e−001 2/19/2.187500e−001
1,000 2/19/1.015625e+000 2/19/1.093750e+000 2/19/1.062500e+000 2/19/1.046875e+000
Sphere/\(x_{Sph20}\) 30 2/19/0 2/19/0 2/19/0 2/19/0
500 2/19/1.875000e−001 2/19/2.500000e−001 2/19/2.187500e−001 2/19/1.875000e−001
1,000 2/19/9.531250e−001 2/19/1.046875e+000 2/19/1.031250e+000 2/19/1.218750e+000
Sphere/\(x_{Sph30}\) 30 2/19/0 2/19/0 2/19/0 2/19/0
500 2/19/2.031250e−001 2/19/2.812500e−001 2/19/2.343750e−001 2/19/1.718750e−001
1,000 2/19/1.015625e+000 2/19/9.687500e−001 2/19/9.531250e−001 2/19/9.843750e−001
Sphere/\(x_{Sph40}\) 30 2/19/0 2/19/0 2/19/0 2/19/0
500 2/19/1.718750e−001 2/19/2.343750e−001 2/19/2.187500e−001 2/19/1.250000e−001
1,000 2/19/9.218750e−001 2/19/1 2/19/1 2/19/1.015625e+000
Schwefel’s/\(x_{SchDs10}\) 30 3/32/0 3/32/6.250000e−002 3/32/6.250000e−002 3/32/0
50 3/32/0 3/32/0 3/32/6.250000e−002 3/32/6.250000e−002
100 4/45/1.562500e−001 4/45/2.500000e−001 6/70/3.750000e−001 6/70/4.062500e−001
Schwefel’s/\(x_{SchDs20}\) 30 2/19/6.250000e−002 2/19/0 2/19/0 2/19/0
50 2/19/0 2/19/6.250000e−002 2/19/0 2/19/0
100 3/32/1.875000e−001 3/32/1.250000e−001 3/32/1.875000e−001 3/32/1.718750e−001
Schwefel’s/\(x_{SchDs30}\) 30 3/32/0 3/32/6.250000e−002 3/32/0 3/32/0
50 3/32/6.250000e−002 3/32/0 3/32/0 3/32/6.250000e−002
100 3/32/1.875000e−001 3/32/1.250000e−001 3/32/1.875000e−001 3/32/1.250000e−001
Schwefel’s/\(x_{SchDs40}\) 30 2/19/0 2/19/0 2/19/0 2/19/0
50 2/19/0 2/19/6.250000e−002 2/19/0 2/19/0
100 2/19/6.250000e−002 2/19/6.250000e−002 2/19/1.250000e−001 2/19/6.250000e−002
Griewank/\(x_{Gri10}\) 30 3/37/0 3/37/0 11/258/6.250000e−002 9/130/6.250000e−002
500 2/24/5.781250e−001 2/24/5.312500e−001 2/24/5.781250e−001 2/24/6.406250e−001
1,000 2/24/1.984375e+000 2/24/1.656250e+000 2/24/1.671875e+000 2/24/1.625000e+000
Griewank/\(x_{Gri20}\) 30 4/75/0 4/75/4.687500e−002 4/59/0 4/58/0
500 2/24/6.718750e−001 2/24/3.437500e−001 2/24/4.062500e−001 2/24/6.562500e−001
1,000 2/24/1.765625e+000 2/24/1.796875e+000 2/24/1.859375e+000 2/24/1.640625e+000
Griewank/\(x_{Gri30}\) 30 3/38/0 3/37/4.687500e−002 11/394/1.250000e−001 9/178/0
500 2/24/5.625000e−001 2/24/5.468750e−001 2/24/5.625000e−001 2/24/5.781250e−001
1,000 2/24/2.046875e+000 2/24/1.531250e+000 2/24/1.468750e+000 2/24/1.421875e+000
Griewank/\(x_{Gri40}\) 30 15/200/6.250000e−002 19/249/6.250000e−002 9/502/6.250000e−002 18/446/1.250000e−001
500 2/24/6.093750e−001 2/24/2.968750e−001 2/24/5.468750e−001 2/24/5.468750e−001
1,000 2/24/1.843750e+000 2/24/1.468750e+000 2/24/1.828125e+000 2/24/1.781250e+000
Rosenbrock/\(x_{Ros10}\) 30 34/483/1.406250e−001 5/116/0 2/19/0 2/19/0
500 30/419/3.431250e+001 5/116/2.031250e+000 2/19/2.187500e−001 2/19/1.875000e−001
1,000 28/393/2.136875e+002 6/152/2.207813e+001 2/19/1.078125e+000 2/19/9.375000e−001
Rosenbrock/\(x_{Ros20}\) 30 30/467/9.375000e−002 5/121/0 2/19/0 2/19/0
500 16/268/1.650000e+001 3/38/6.250000e−001 2/19/1.875000e−001 2/19/2.187500e−001
1,000 17/286/1.181094e+002 3/38/3.453125e+000 2/19/1.062500e+000 2/19/9.062500e−001
Rosenbrock/\(x_{Ros30}\) 30 8/134/0 7/141/0 2/19/0 2/19/0
500 9/154/6.828125e+000 6/110/3.546875e+000 2/19/2.031250e−001 2/19/2.187500e−001
1,000 7/115/3.090625e+001 5/92/1.373438e+001 2/19/1.125000e+000 2/19/1.156250e+000
Rosenbrock/\(x_{Ros40}\) 30 8/140/0 5/102/0 2/19/6.250000e−002 2/19/0
500 12/186/1.185938e+001 6/105/5.203125e+000 2/19/2.343750e−001 2/19/2.031250e−001
1,000 15/226/101 6/105/2.275000e+001 2/19/1.062500e+000 2/19/1.015625e+000
Ackley/\(x_{Ack10}\) 30 5/68/6.250000e−002 6/80/0 6/83/0 6/80/0
500 5/67/2.343750e+000 5/64/1.937500e+000 5/67/2.046875e+000 5/68/2.171875e+000
1,000 5/66/1.407813e+001 6/79/2.229688e+001 5/66/1.410938e+001 6/79/2.278125e+001
Ackley/\(x_{Ack20}\) 30 2/42/0 2/42/0 7/99/6.250000e−002 7/97/6.250000e−002
500 6/79/3.250000e+000 6/77/3.640625e+000 6/79/3.671875e+000 6/77/3.593750e+000
1,000 5/66/1.354688e+001 5/63/1.443750e+001 5/65/1.423438e+001 5/66/1.429688e+001
Ackley/\(x_{Ack30}\) 30 9/126/0 5/67/0 9/126/6.250000e−002 6/83/0
500 6/88/3.500000e+000 4/50/1.187500e+000 6/88/3.437500e+000 6/78/2.828125e+000
1,000 4/53/7.531250e+000 4/51/7.671875e+000 7/95/3.085938e+001 6/77/2.229688e+001
Ackley/\(x_{Ack40}\) 30 4/56/6.250000e−002 4/57/6.250000e−002 8/108/0 7/92/4.687500e−002
500 4/55/1.343750e+000 4/54/1.015625e+000 7/98/4.062500e+000 7/92/4.562500e+000
1,000 6/84/2.232813e+001 6/79/2.256250e+001 6/84/2.310938e+001 6/77/2.254688e+001
Total CPU Time 516.1562 161.5781 115.0938 115.0156