from pyscipopt import*
R=input()
m=Model()
V,C=m.addVar,m.addCons
a,b,c=V(),V(),V()
m.setObjective(c)
C(a*b<=c)
P=[]
for r in R:
x,y=V(),V();C(r<=x);C(x<=a-r);C(r<=y);C(y<=b-r)
for u,v,s in P:C((x-u)**2+(y-v)**2>=(r+s)**2)
P+=(x,y,r),
m.optimize()
m.printBestSol()
怎么运行的
我们将问题写成如下：在变量a，b，c，x 1，y 1，…，x n，y n上最小化c，其中
- AB ≤ Ç ;
- ř 我 ≤ X 我 ≤ 一 - - [R 我和- [R 我 ≤ ÿ 我 ≤ b - ÿ 我，1≤ 我 ≤ Ñ ;
- （x i − x j）2 +（y我 - Ÿ Ĵ） 2 ≥（ - [R 我 + ř Ĵ） 2，1≤ Ĵ <我 ≤ Ñ。
显然，我们在这些约束条件上使用了外部优化库，但您不能仅将它们提供给任何旧的优化程序，甚至Mathematica NMinimize对于这些微小的测试用例都会陷入局部最小值。如果密切注视约束，您会发现它们构成了一个二次约束二次程序，并且发现非凸QCQP的全局最优是NP难的。因此，我们需要一些难以置信的强大魔力。我选择了工业强度求解器SCIP，这是我可以找到的唯一的全球QCQP求解器，它具有免费的学术使用许可。幸运的是，它具有一些非常不错的Python绑定。
输入输出
在stdin上传递半径列表，例如[5,3,1.5]。输出显示objective value:的矩形区域，x1，x2矩形尺寸，x3再次矩形区域，x4，x5第一圆心坐标，x6，x7第二圆心坐标等
测试用例
[5,3,1.5] ↦ 157.459666673757
SCIP Status : problem is solved [optimal solution found]
Solving Time (sec) : 0.04
Solving Nodes : 187
Primal Bound : +1.57459666673757e+02 (9 solutions)
Dual Bound : +1.57459666673757e+02
Gap : 0.00 %
objective value: 157.459666673757
x1 10 (obj:0)
x2 15.7459666673757 (obj:0)
x3 157.459666673757 (obj:1)
x4 5 (obj:0)
x5 5 (obj:0)
x6 7 (obj:0)
x7 12.7459666673757 (obj:0)
x8 1.5 (obj:0)
x9 10.4972522849871 (obj:0)
[9,4,8,2] ↦ 709.061485909243
这比OP的解决方案更好。确切尺寸为18 x 29 +6√3。
SCIP Status : problem is solved [optimal solution found]
Solving Time (sec) : 1.07
Solving Nodes : 4650
Primal Bound : +7.09061485909243e+02 (6 solutions)
Dual Bound : +7.09061485909243e+02
Gap : 0.00 %
objective value: 709.061485909243
x1 18 (obj:0)
x2 39.3923047727357 (obj:0)
x3 709.061485909243 (obj:1)
x4 9 (obj:0)
x5 30.3923047727357 (obj:0)
x6 14 (obj:0)
x7 18.3923048064677 (obj:0)
x8 8 (obj:0)
x9 8 (obj:0)
x10 2 (obj:0)
x11 19.6154311552252 (obj:0)
[18,3,1] ↦ 1295.999999999
SCIP Status : problem is solved [optimal solution found]
Solving Time (sec) : 0.00
Solving Nodes : 13
Primal Bound : +1.29599999999900e+03 (4 solutions)
Dual Bound : +1.29599999999900e+03
Gap : 0.00 %
objective value: 1295.999999999
x1 35.9999999999722 (obj:0)
x2 36 (obj:0)
x3 1295.999999999 (obj:1)
x4 17.9999999999722 (obj:0)
x5 18 (obj:0)
x6 32.8552571627738 (obj:0)
x7 3 (obj:0)
x8 1 (obj:0)
x9 1 (obj:0)
奖金案
[1,2,3,4,5] ↦ 230.244214912998
SCIP Status : problem is solved [optimal solution found]
Solving Time (sec) : 401.31
Solving Nodes : 1400341
Primal Bound : +2.30244214912998e+02 (16 solutions)
Dual Bound : +2.30244214912998e+02
Gap : 0.00 %
objective value: 230.244214912998
x1 13.9282031800476 (obj:0)
x2 16.530790960676 (obj:0)
x3 230.244214912998 (obj:1)
x4 1 (obj:0)
x5 9.60188492354373 (obj:0)
x6 11.757778088743 (obj:0)
x7 3.17450418828415 (obj:0)
x8 3 (obj:0)
x9 13.530790960676 (obj:0)
x10 9.92820318004764 (obj:0)
x11 12.530790960676 (obj:0)
x12 5 (obj:0)
x13 5 (obj:0)
[3,4,5,6,7] ↦ 553.918025310597
SCIP Status : problem is solved [optimal solution found]
Solving Time (sec) : 90.28
Solving Nodes : 248281
Primal Bound : +5.53918025310597e+02 (18 solutions)
Dual Bound : +5.53918025310597e+02
Gap : 0.00 %
objective value: 553.918025310597
x1 21.9544511351279 (obj:0)
x2 25.2303290086403 (obj:0)
x3 553.918025310597 (obj:1)
x4 3 (obj:0)
x5 14.4852813557912 (obj:0)
x6 4.87198593295855 (obj:0)
x7 21.2303290086403 (obj:0)
x8 16.9544511351279 (obj:0)
x9 5 (obj:0)
x10 6 (obj:0)
x11 6 (obj:0)
x12 14.9544511351279 (obj:0)
x13 16.8321595389753 (obj:0)
[3,4,5,6,7,8] ↦ 777.87455544487
SCIP Status : problem is solved [optimal solution found]
Solving Time (sec) : 218.29
Solving Nodes : 551316
Primal Bound : +7.77874555444870e+02 (29 solutions)
Dual Bound : +7.77874555444870e+02
Gap : 0.00 %
objective value: 777.87455544487
x1 29.9626413867546 (obj:0)
x2 25.9614813640722 (obj:0)
x3 777.87455544487 (obj:1)
x4 13.7325948669477 (obj:0)
x5 15.3563780595534 (obj:0)
x6 16.0504838821134 (obj:0)
x7 21.9614813640722 (obj:0)
x8 24.9626413867546 (obj:0)
x9 20.7071098175984 (obj:0)
x10 6 (obj:0)
x11 19.9614813640722 (obj:0)
x12 7 (obj:0)
x13 7 (obj:0)
x14 21.9626413867546 (obj:0)
x15 8.05799919177801 (obj:0)
