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Hands Tied
Hard optimization Portfolio Optimization Time Series
Time limit: 8 s per test  ·  Memory: 256 MB

You are given n daily log prices of one asset and, for each of N trading strategies, the full history of the position it held on each day. Every strategy is a fixed deterministic rule mapping the price history to a position in [-1, 1]. The rules are not disclosed.

You cannot trade the asset yourself. All you may do is allocate to the strategies: submit one weight w_i per strategy, positive or negative. The judge normalizes the weights to gross exposure 1, i.e. divides them by sum |w_i|.

The prices come from a hidden regime-switching process whose parameters vary between test cases. The judge continues the process beyond the end of the history, from its true hidden state, along many future paths, runs every strategy on each path, and computes your book's daily return

R_t = sum_i w_i * pos_{i,t-1} * (p_t - p_{t-1}) - c * sum_i |w_i| * |pos_{i,t} - pos_{i,t-1}|

where p_t is the log price, pos_{i,t} the position of strategy i on day t and c the transaction cost per unit traded. Your objective is the annualized Sharpe ratio of R_t over each future path, averaged over the paths.

Input

The first line contains a case identifier, n, N and c. The second line contains the N strategy names. Each of the next n lines contains one log price, in time order, with 6 digits after the decimal point. Each of the next N lines contains the n positions of one strategy, in the order of the names.

Output

N real numbers: your weights, in the order of the names.

Scoring

An output is rejected (score 0) unless it has exactly N finite values that are not all zero. Otherwise the case score is

score = clip(S / S_ref, 0, 1)

where S is your averaged Sharpe ratio on the hidden future paths and S_ref is the averaged Sharpe ratio of the best weights the judge can find on those same paths. The total score is the mean over the cases.

Constraints

n = 2000

N = 10

c = 0.0005

The future paths have 500 days each.

Examples

input
3001 2000 10 0.0005
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10
-0.007627
-0.015770
-0.035771
-0.040003
-0.014992
0.009170
0.015852
0.025760
0.025564
0.031090
0.018299
0.011684
0.014060
0.032036
0.027320
0.011677
-0.001332
-0.007369
-0.015473
-0.061310
-0.111185
-0.135814
-0.136086
-0.125222
-0.093669
-0.106860
-0.109894
-0.097840
-0.108371
-0.135473
-0.151868
-0.175053
-0.206327
-0.235299
-0.258314
-0.254835
-0.259011
-0.268984
-0.271324
-0.275159
-0.274011
-0.257362
-0.234286
-0.234942
-0.239432
-0.241666
-0.234807
-0.252185
-0.255608
-0.255686
-0.271483
-0.286390
-0.305510
-0.303792
-0.287073
-0.272617
-0.262916
-0.265571
-0.278265
-0.271165
-0.266502
-0.281186
-0.301619
-0.329533
-0.363572
-0.397452
-0.420343
-0.430365
-0.446048
-0.459780
-0.482459
-0.494341
-0.500867
-0.484817
-0.480419
-0.496581
-0.506346
-0.508387
-0.527380
-0.550284
-0.558141
-0.575335
-0.577226
-0.599849
-0.600880
-0.602156
-0.624593
-0.651899
-0.671382
-0.674585
-0.661420
-0.663620
-0.653955
-0.669332
-0.694280
-0.710270
-0.705722
-0.689254
-0.677361
-0.670332
-0.660894
-0.646460
-0.634529
-0.617044
-0.625390
-0.646137
-0.653632
-0.647643
-0.644402
-0.627924
-0.599161
-0.598195
-0.600538
-0.588121
-0.584991
-0.573275
-0.576071
-0.562000
-0.545634
-0.555058
-0.571911
-0.606247
-0.627111
-0.648850
-0.673548
-0.682875
-0.668253
-0.666265
-0.656230
-0.646732
-0.633494
-0.617189
-0.614011
-0.596316
-0.587635
-0.577392
-0.583186
-0.591835
-0.603631
-0.623466
-0.653446
-0.699309
-0.713658
-0.691012
-0.664772
-0.636677
-0.609350
-0.556550
-0.504093
-0.466545
-0.441415
-0.410857
-0.403829
-0.407999
-0.398041
-0.389807
-0.396897
-0.396405
-0.398189
-0.414710
-0.439961
-0.444268
-0.439556
-0.423757
-0.422817
-0.430576
-0.448582
-0.462743
-0.490069
-0.498285
-0.506087
-0.512023
-0.514367
-0.525235
-0.523177
-0.525929
-0.517848
-0.505757
-0.495929
-0.505714
-0.511712
-0.518972
-0.509333
-0.504722
-0.479047
-0.483196
-0.487812
-0.490300
-0.487591
-0.485831
-0.483045
-0.487108
-0.506967
-0.519728
-0.513245
-0.511671
-0.493548
-0.463310
-0.443251
-0.456310
-0.449074
-0.401823
-0.377373
-0.364522
-0.349283
-0.341123
-0.350079
-0.360312
-0.358170
-0.372133
-0.374584
-0.359335
-0.350496
-0.337065
-0.304712
-0.309776
-0.312800
-0.308165
-0.297678
-0.307424
-0.304931
-0.305253
-0.317052
-0.338672
-0.346170
-0.348380
-0.364147
-0.376186
-0.362359
-0.353015
-0.343077
-0.337017
-0.322801
-0.338857
-0.333083
-0.327271
-0.321721
-0.291078
-0.258924
-0.225384
-0.192697
-0.192579
-0.189148
-0.172892
-0.152348
-0.127322
-0.125909
-0.112755
-0.103270
-0.082753
-0.051357
-0.036330
-0.037776
-0.028884
-0.000185
0.027428
0.052677
0.066923
0.072523
0.081202
0.096172
0.119806
0.107950
0.101356
0.097112
0.112890
0.132754
0.136700
0.140090
0.150703
0.145115
0.124264
0.117426
0.115336
0.135148
0.133563
0.112247
0.098605
0.106903
0.108011
0.088297
0.061499
0.044882
0.049723
0.038507
0.038119
0.033242
0.029955
0.045769
0.066225
0.058467
0.047759
0.066157
0.093177
0.110704
0.116211
0.119653
0.108535
0.107256
0.116412
0.136641
0.156467
0.165672
0.160120
0.162903
0.158356
0.150729
0.152022
0.148208
0.154659
0.174308
0.190789
0.210681
0.234809
0.254479
0.264105
0.275144
0.279690
0.275336
0.263373
0.264563
0.258843
0.265820
0.276413
0.268294
0.296482
0.301522
0.329531
0.353863
0.367061
0.388689
0.393942
0.390384
0.392437
0.368122
0.356393
0.345081
0.357859
0.376810
0.393017
0.419176
0.407755
0.394022
0.393004
0.409315
0.430719
0.408598
0.416337
0.415888
0.403670
0.415383
0.402938
0.408794
0.445368
0.467748
0.489083
0.488259
0.488240
0.487101
0.476250
0.455662
0.457917
0.455146
0.483822
0.515171
0.519213
0.520034
0.512293
0.520207
0.512654
0.495118
0.491041
0.474421
0.454241
0.445719
0.438191
0.470783
0.484969
0.505272
0.519148
0.550854
0.568183
0.543429
0.519741
0.504161
0.485686
0.474110
0.464894
0.458606
0.457946
0.454928
0.446558
0.434285
0.414686
0.400094
0.363867
0.324942
0.287823
0.283056
0.276847
0.294883
0.302408
0.306664
0.314452
0.307855
0.283689
0.275626
0.274585
0.285585
0.308408
0.330261
0.339869
0.368748
0.382841
0.381571
0.399126
0.404663
0.422424
0.426261
0.436950
0.445113
0.446767
0.454196
0.462710
0.456065
0.477190
0.492168
0.507650
0.525869
0.516143
0.514980
0.525698
0.572586
0.612206
0.648446
0.675750
0.693384
0.713310
0.738370
0.776581
0.817936
0.841366
0.859792
0.877079
0.894821
0.880783
0.864513
0.867847
0.858835
0.847793
0.836321
0.859044
0.869999
0.869836
0.874160
0.884523
0.879398
0.857365
0.858915
0.869542
0.877032
0.890304
0.898352
0.893988
0.919205
0.935554
0.952314
0.961351
0.957821
0.946108
0.920030
0.887170
0.884295
0.889506
0.908776
0.919229
0.934547
0.947370
0.931070
0.905831
0.907215
0.910787
0.894039
0.869364
0.873070
0.873071
0.876566
0.879939
0.875718
0.861952
0.852388
0.816188
0.777841
0.757276
0.749671
0.741255
0.737045
0.741597
0.732953
0.714545
0.703062
0.698789
0.693140
0.692475
0.700360
0.693715
0.684981
0.678727
0.666021
0.626905
0.598980
0.579265
0.557944
0.548067
0.548003
0.549232
0.561113
0.555263
0.546826
0.537797
0.510796
0.488096
0.488113
0.490125
0.454566
0.459605
0.451237
0.448438
0.415978
0.389080
0.374681
0.360928
0.335114
0.329589
0.335080
0.363451
0.369081
0.387557
0.397764
0.402831
0.393221
0.413508
0.415025
0.431893
0.444688
0.455035
0.461204
0.476227
0.492319
0.521071
0.551488
0.548612
0.540492
0.547519
0.556937
0.585358
0.600815
0.629290
0.632751
0.639920
0.643374
0.635018
0.636622
0.638769
0.628578
0.619864
0.611021
0.602378
0.603891
0.583418
0.586459
0.585029
0.589601
0.607576
0.629568
0.638824
0.642111
0.661428
0.664830
0.661494
0.664264
0.646215
0.649642
0.655468
0.644991
0.633229
0.623277
0.607398
0.566929
0.540396
0.524627
0.544146
0.556775
0.557388
0.571495
0.585638
0.596993
0.600868
0.609801
0.634152
0.668284
0.698395
0.721174
0.727923
0.734646
0.768328
0.774151
0.775988
0.768294
0.753454
0.747471
0.751110
0.732111
0.705695
0.681128
0.658168
0.667202
0.671095
0.672809
0.682920
0.690443
0.700872
0.681037
0.662258
0.639998
0.632157
0.610833
0.602906
0.621536
0.631663
0.643146
0.652378
0.684767
0.706776
0.717443
0.727779
0.734673
0.783665
0.811157
0.838866
0.853581
0.859981
0.855515
0.837583
0.824595
0.802678
0.786391
0.775764
0.760074
0.753729
0.744555
0.736275
0.724881
0.716789
0.695902
0.690125
0.679681
0.664021
0.638485
0.637527
0.634958
0.651583
0.683483
0.710715
0.705802
0.692701
0.696898
0.698775
0.708900
0.696558
0.697198
0.705566
0.697045
0.708604
0.733103
0.755394
0.769557
0.765846
0.752437
0.738356
0.737718
0.735417
0.758955
0.789544
0.789356
0.798653
0.804462
0.818346
0.838952
0.858564
0.872657
0.879766
0.901243
0.914299
0.907521
0.906616
0.920783
0.925938
0.918091
0.925638
0.924747
0.939984
0.944975
0.958633
0.951586
0.955002
0.972244
0.991787
0.987759
1.007573
1.035952
1.051314
1.043379
1.020026
1.008603
0.972873
0.939050
0.935839
0.929789
0.919110
0.914210
0.914467
0.907806
0.894645
0.884027
0.905481
0.934940
0.950289
0.930493
0.907708
0.904933
0.926944
0.942759
0.954095
0.959155
0.978386
0.983594
0.994506
1.029737
1.054586
1.067283
1.072566
1.073670
1.081247
1.061507
1.047838
1.052128
1.060934
1.077755
1.097058
1.102914
1.107303
1.115755
1.116644
1.124615
1.112215
1.084979
1.057874
1.039517
1.038167
1.047848
1.054890
1.043758
1.038424
1.062896
1.066700
1.035159
1.026769
1.044873
1.040580
1.046786
1.026719
1.030503
1.021510
1.021911
1.021007
1.023384
1.036567
1.012451
1.038819
1.028586
1.055602
1.052268
1.059006
1.059649
1.072421
1.086746
1.077061
1.084265
1.083936
1.070585
1.057554
1.056448
1.046903
1.045700
1.051122
1.076099
1.066577
1.063053
1.061956
1.045187
1.049993
1.069597
1.074840
1.078189
1.089488
1.097910
1.118898
1.151061
1.174876
1.194966
1.190298
1.190284
1.212417
1.235043
1.246494
1.267834
1.284620
1.304371
1.327688
1.346834
1.354919
1.353342
1.351530
1.347267
1.356864
1.371669
1.386658
1.383707
1.393804
1.412938
1.442628
1.450210
1.446606
1.458407
1.468390
1.485732
1.489832
1.497189
1.503883
1.517156
1.504587
1.501757
1.527604
1.544683
1.543440
1.527303
1.539647
1.541909
1.552511
1.575226
1.567401
1.548941
1.547077
1.549982
1.528139
1.525078
1.530982
1.523901
1.534105
1.525914
1.529813
1.558359
1.574136
1.573400
1.581477
1.573906
1.591016
1.599635
1.599165
1.592517
1.602438
1.588013
1.590413
1.591840
1.578375
1.579580
1.580907
1.584087
1.590224
1.591778
1.591148
1.596704
1.601382
1.592881
1.597504
1.619168
1.621936
1.632865
1.625546
1.610594
1.596521
1.593927
1.598797
1.590111
1.593313
1.605575
1.616072
1.606173
1.620079
1.614410
1.622402
1.599849
1.590228
1.574591
1.571518
1.588324
1.570982
1.566301
1.565346
1.579654
1.567294
1.568127
1.578675
1.587543
1.582625
1.593648
1.598277
1.596773
1.568918
1.537945
1.527550
1.549066
1.552579
1.564529
1.578821
1.603490
1.618796
1.638293
1.650534
1.632106
1.625547
1.614601
1.596511
1.597681
1.602723
1.606153
1.603183
1.601235
1.613518
1.627314
1.633426
1.623654
1.625661
1.629431
1.639865
1.638096
1.634444
1.638370
1.659788
1.680260
1.699734
1.705128
1.725862
1.744387
1.761219
1.785525
1.799368
1.804747
1.799313
1.782466
1.813551
1.809702
1.802629
1.809668
1.825593
1.855872
1.835968
1.846233
1.828277
1.832199
1.827637
1.808133
1.805686
1.804665
1.819712
1.821443
1.797178
1.788092
1.778169
1.781096
1.800330
1.814235
1.833865
1.830445
1.845633
1.856214
1.842202
1.848110
1.834680
1.805009
1.796529
1.785425
1.806422
1.824117
1.841817
1.857168
1.856125
1.881854
1.879921
1.871604
1.867893
1.862231
1.853946
1.845721
1.857884
1.843244
1.831630
1.836202
1.826677
1.845800
1.847395
1.849979
1.856093
1.873211
1.879436
1.883870
1.867802
1.859807
1.852774
1.850905
1.845635
1.841629
1.823639
1.821892
1.830593
1.833491
1.829193
1.829163
1.798243
1.787372
1.767753
1.749325
1.740032
1.749227
1.773001
1.774119
1.751716
1.724499
1.702447
1.658895
1.640839
1.647520
1.644319
1.640390
1.636164
1.624031
1.616012
1.613611
1.592471
1.594921
1.596485
1.583783
1.576326
1.576864
1.557568
1.545159
1.536893
1.534279
1.539673
1.526417
1.509273
1.506943
1.491696
1.484476
1.477662
1.469101
1.473701
1.469151
1.468652
1.469745
1.471901
1.487887
1.485828
1.497843
1.496125
1.496806
1.505433
1.510049
1.503519
1.470480
1.466948
1.458397
1.437070
1.422870
1.389291
1.364628
1.342417
1.336835
1.324233
1.324056
1.327540
1.340708
1.327455
1.311207
1.302089
1.277889
1.270330
1.247941
1.226840
1.189568
1.125360
1.079396
1.058771
1.037474
1.008209
0.981045
0.944358
0.905819
0.881819
0.846468
0.821445
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output
0.1687
-0.3136
-0.5200
0.2691
-0.4864
0.2657
2.1489
1.3341
-0.2218
-0.3663
Python 3.13 i Execution environment Isolated microVM · 1 vCPU · no internet access Time and memory limits are set per problem Available packages numpy 2.5.0scipy 1.18.0pandas 3.0.0scikit-learn 1.9.0statsmodels 0.15.0cvxpy 1.9.2
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