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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