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 |