SlowGuess/NeurIPS_2023_OCR huggingface dataset:
# of features	method	upper bound	time(s)	gap(%)
100	big-M50 + warm start	-9.11026 · 107	0.04	0.00
100	big-M20 + warm start	-9.11026 · 107	0.03	0.00
100	big-M5 + warm start	-9.11026 · 107	0.03	0.00
100	ours	-9.11026 · 107	0.31	0.00
500	big-M50 + warm start	-9.12047 · 107	14.34	0.00
500	big-M20 + warm start	-9.12047 · 107	28.63	0.00
500	big-M5 + warm start	-9.12047 · 107	18.60	0.00
500	ours	-9.12047 · 107	0.38	0.00
1000	big-M50 + warm start	-9.04075 · 107	3600.16	0.01
1000	big-M20 + warm start	-9.04075 · 107	3600.11	0.20
1000	big-M5 + warm start	-9.04075 · 107	1687.23	0.00
1000	ours	-9.04075 · 107	0.52	0.00
3000	big-M50 + warm start	-9.12761 · 107	3611.02	0.59
3000	big-M20 + warm start	-9.12761 · 107	3601.67	0.59
3000	big-M5 + warm start	-9.12761 · 107	3602.09	222997914.92
3000	ours	-9.12761 · 107	20.88	0.00
5000	big-M50 + warm start	-9.12916 · 107	3606.64	1.00
5000	big-M20 + warm start	-9.12916 · 107	3610.82	1.00
5000	big-M5 + warm start	-9.12916 · 107	3632.24	618133893.79
5000	ours	-9.12916 · 107	58.07	0.00
Table 80: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.9, λ2 = 0.1). All baselines use our beam search solution as a warm start. The method big-M5 has some large optimality gaps because Gurobi couldn't finish solving the root relaxation within 1h.
G.3.5 Synthetic 1 Benchmark with  $\lambda_{2} = 10.0$  with No Warmstart
# of features	method	upper bound	time(s)	gap(%)
100	big-M50	-2.10868 · 10^6	0.28	0.00
100	big-M20	-2.10868 · 10^6	0.44	0.00
100	big-M5	-2.10868 · 10^6	0.32	0.00
100	ours	-2.10868 · 10^6	0.30	0.00
500	big-M50	-2.12819 · 10^6	42.85	0.00
500	big-M20	-2.12823 · 10^6	28.73	0.00
500	big-M5	-2.12823 · 10^6	57.10	0.00
500	ours	-2.12823 · 10^6	0.38	0.00
1000	big-M50	-2.10995 · 10^6	1072.15	0.00
1000	big-M20	-2.10995 · 10^6	451.79	0.00
1000	big-M5	-2.10995 · 10^6	904.01	0.00
1000	ours	-2.10995 · 10^6	0.57	0.00
3000	big-M50	-7.92671 · 10^5	3605.01	166.77
3000	big-M20	-9.76712 · 10^5	3602.25	116.51
3000	big-M5	-	3623.24	-
3000	ours	-2.10220 · 10^6	1.68	0.00
5000	big-M50	-7.89596 · 10^5	3617.52	168.16
5000	big-M20	-9.77684 · 10^5	3612.02	116.57
5000	big-M5	-1.87959 · 10^6	3600.23	12.65
5000	ours	-2.09625 · 10^6	5.09	0.00
Table 81: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.1, λ2 = 10.0).
# of features	method	upper bound	time(s)	gap(%)
100	big-M50	-5.28713 · 10^6	0.40	0.00
100	big-M20	-5.28716 · 10^6	0.39	0.00
100	big-M5	-5.28716 · 10^6	0.35	0.00
100	ours	-5.28716 · 10^6	0.30	0.00
500	big-M50	-5.31819 · 10^6	89.34	0.00
500	big-M20	-5.31835 · 10^6	73.89	0.00
500	big-M5	-5.31842 · 10^6	52.55	0.00
500	ours	-5.31842 · 10^6	0.41	0.00
1000	big-M50	-5.26964 · 10^6	1869.66	0.00
1000	big-M20	-5.26964 · 10^6	1241.04	0.00
1000	big-M5	-5.26964 · 10^6	711.84	0.00
1000	ours	-5.26964 · 10^6	0.51	0.00
3000	big-M50	-3.96610 · 10^6	3600.32	33.83
3000	big-M20	-3.98698 · 10^6	3601.87	33.13
3000	big-M5	-	3604.15	-
3000	ours	-5.27658 · 10^6	1.76	0.00
5000	big-M50	-3.96757 · 10^6	3607.78	34.19
5000	big-M20	-4.37277 · 10^6	3610.74	21.76
5000	big-M5	-5.27116 · 10^6	3600.14	1.00
5000	ours	-5.27116 · 10^6	4.32	0.00
Table 82: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.3, λ2 = 10.0).
# of features	method	upper bound	time(s)	gap(%)
100	big-M50	-1.10083 · 107	0.44	0.00
100	big-M20	-1.10083 · 107	0.37	0.00
100	big-M5	-1.10083 · 107	0.33	0.00
100	ours	-1.10083 · 107	0.32	0.00
500	big-M50	-1.10515 · 107	91.37	0.00
500	big-M20	-1.10517 · 107	74.71	0.00
500	big-M5	-1.10517 · 107	99.37	0.00
500	ours	-1.10517 · 107	0.41	0.00
1000	big-M50	-1.09505 · 107	1534.37	0.00
1000	big-M20	-1.09507 · 107	1102.78	0.00
1000	big-M5	-1.09507 · 107	814.92	0.00
1000	ours	-1.09507 · 107	0.55	0.00
3000	big-M50	-	3603.26	-
3000	big-M20	-	3606.10	-
3000	big-M5	-	3606.02	-
3000	ours	-1.10011 · 107	1.73	0.00
5000	big-M50	-9.62941 · 106	3608.37	15.34
5000	big-M20	-1.00932 · 107	3602.48	10.04
5000	big-M5	-	3633.80	-
5000	ours	-1.09965 · 107	4.24	0.00
Table 83: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.5, λ2 = 10.0).
# of features	method	upper bound	time(s)	gap(%)
100	big-M50	-2.43574 · 10^7	0.42	0.00
100	big-M20	-2.43574 · 10^7	0.36	0.00
100	big-M5	-2.43574 · 10^7	0.37	0.00
100	ours	-2.43574 · 10^7	0.30	0.00
500	big-M50	-2.44178 · 10^7	85.12	0.00
500	big-M20	-2.44184 · 10^7	70.81	0.00
500	big-M5	-2.44184 · 10^7	53.85	0.00
500	ours	-2.44184 · 10^7	0.44	0.00
1000	big-M50	-2.41986 · 10^7	1415.19	0.00
1000	big-M20	-2.41987 · 10^7	878.61	0.00
1000	big-M5	-2.41988 · 10^7	613.98	0.00
1000	ours	-2.41988 · 10^7	0.52	0.00
3000	big-M50	-	3615.34	-
3000	big-M20	-	3632.97	-
3000	big-M5	-2.43712 · 10^7	3600.25	0.56
3000	ours	-2.43712 · 10^7	1.81	0.00
5000	big-M50	-2.43693 · 10^7	3607.06	1.00
5000	big-M20	-2.43693 · 10^7	3603.18	1.00
5000	big-M5	-	3626.02	-
5000	ours	-2.43693 · 10^7	4.08	0.00
Table 84: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.7, λ2 = 10.0).
# of features	method	upper bound	time(s)	gap(%)
100	big-M50	-9.11025 · 107	0.17	0.00
100	big-M20	-9.11025 · 107	0.12	0.00
100	big-M5	-9.11025 · 107	0.10	0.00
100	ours	-9.11025 · 107	0.32	0.00
500	big-M50	-9.12046 · 107	117.86	0.00
500	big-M20	-9.12046 · 107	128.18	0.00
500	big-M5	-9.12046 · 107	184.18	0.00
500	ours	-9.12046 · 107	0.38	0.00
1000	big-M50	-9.04074 · 107	3600.04	0.01
1000	big-M20	-9.04074 · 107	3600.17	0.20
1000	big-M5	-9.04074 · 107	2000.72	0.00
1000	ours	-9.04074 · 107	0.52	0.00
3000	big-M50	-	3611.90	-
3000	big-M20	-	3625.57	-
3000	big-M5	-	3607.35	-
3000	ours	-9.12760 · 107	20.84	0.00
5000	big-M50	-	3615.45	-
5000	big-M20	-	3613.78	-
5000	big-M5	-	3613.28	-
5000	ours	-9.12915 · 107	57.41	0.00
Table 85: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.9, λ2 = 10.0).
G.3.6 Synthetic 1 Benchmark with  $\lambda_{2} = 10.0$  with Warmstart
# of features	method	upper bound	time(s)	gap(%)
100	big-M50 + warm start	-2.10868 · 10^6	0.25	0.00
100	big-M20 + warm start	-2.10868 · 10^6	0.17	0.00
100	big-M5 + warm start	-2.10868 · 10^6	0.05	0.00
100	ours	-2.10868 · 10^6	0.30	0.00
500	big-M50 + warm start	-2.12823 · 10^6	3.76	0.00
500	big-M20 + warm start	-2.12823 · 10^6	2.68	0.00
500	big-M5 + warm start	-2.12823 · 10^6	2.17	0.00
500	ours	-2.12823 · 10^6	0.38	0.00
1000	big-M50 + warm start	-2.10995 · 10^6	23.00	0.00
1000	big-M20 + warm start	-2.10995 · 10^6	18.30	0.00
1000	big-M5 + warm start	-2.10995 · 10^6	26.24	0.00
1000	ours	-2.10995 · 10^6	0.57	0.00
3000	big-M50 + warm start	-2.10220 · 10^6	241.10	0.00
3000	big-M20 + warm start	-2.10220 · 10^6	302.29	0.00
3000	big-M5 + warm start	-2.10220 · 10^6	273.21	0.00
3000	ours	-2.10220 · 10^6	1.68	0.00
5000	big-M50 + warm start	-2.09625 · 10^6	1309.75	0.00
5000	big-M20 + warm start	-2.09625 · 10^6	1142.75	0.00
5000	big-M5 + warm start	-2.09625 · 10^6	1636.89	0.00
5000	ours	-2.09625 · 10^6	5.09	0.00
Table 86: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.1, λ2 = 10.0). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	big-M50 + warm start	-5.28716 · 10^6	0.25	0.00
100	big-M20 + warm start	-5.28716 · 10^6	0.05	0.00
100	big-M5 + warm start	-5.28716 · 10^6	0.05	0.00
100	ours	-5.28716 · 10^6	0.30	0.00
500	big-M50 + warm start	-5.31842 · 10^6	2.93	0.00
500	big-M20 + warm start	-5.31842 · 10^6	2.99	0.00
500	big-M5 + warm start	-5.31842 · 10^6	1.86	0.00
500	ours	-5.31842 · 10^6	0.41	0.00
1000	big-M50 + warm start	-5.26964 · 10^6	15.10	0.00
1000	big-M20 + warm start	-5.26964 · 10^6	14.76	0.00
1000	big-M5 + warm start	-5.26964 · 10^6	11.79	0.00
1000	ours	-5.26964 · 10^6	0.51	0.00
3000	big-M50 + warm start	-5.27658 · 10^6	315.30	0.00
3000	big-M20 + warm start	-5.27658 · 10^6	390.22	0.00
3000	big-M5 + warm start	-5.27658 · 10^6	249.80	0.00
3000	ours	-5.27658 · 10^6	1.76	0.00
5000	big-M50 + warm start	-5.27116 · 10^6	1100.87	0.00
5000	big-M20 + warm start	-5.27116 · 10^6	1014.36	0.00
5000	big-M5 + warm start	-5.27116 · 10^6	3627.13	1.00
5000	ours	-5.27116 · 10^6	4.32	0.00
Table 87: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.3, λ2 = 10.0). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	big-M50 + warm start	-1.10083 · 107	0.06	0.00
100	big-M20 + warm start	-1.10083 · 107	0.05	0.00
100	big-M5 + warm start	-1.10083 · 107	0.05	0.00
100	ours	-1.10083 · 107	0.32	0.00
500	big-M50 + warm start	-1.10517 · 107	8.51	0.00
500	big-M20 + warm start	-1.10517 · 107	6.01	0.00
500	big-M5 + warm start	-1.10517 · 107	13.97	0.00
500	ours	-1.10517 · 107	0.41	0.00
1000	big-M50 + warm start	-1.09507 · 107	14.03	0.00
1000	big-M20 + warm start	-1.09507 · 107	15.08	0.00
1000	big-M5 + warm start	-1.09507 · 107	11.81	0.00
1000	ours	-1.09507 · 107	0.55	0.00
3000	big-M50 + warm start	-1.10011 · 107	201.32	0.00
3000	big-M20 + warm start	-1.10011 · 107	292.08	0.00
3000	big-M5 + warm start	-1.10011 · 107	241.63	0.00
3000	ours	-1.10011 · 107	1.73	0.00
5000	big-M50 + warm start	-1.09965 · 107	3601.20	1.00
5000	big-M20 + warm start	-1.09965 · 107	3623.71	1.00
5000	big-M5 + warm start	-1.09965 · 107	3629.56	570166517.91
5000	ours	-1.09965 · 107	4.24	0.00
Table 88: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.5, λ2 = 10.0). All baselines use our beam search solution as a warm start. The method big-M5 has some large optimality gaps because Gurobi couldn't finish solving the root relaxation within 1h.
# of features	method	upper bound	time(s)	gap(%)
100	big-M50 + warm start	-2.43574 · 107	0.16	0.00
100	big-M20 + warm start	-2.43574 · 107	0.17	0.00
100	big-M5 + warm start	-2.43574 · 107	0.04	0.00
100	ours	-2.43574 · 107	0.30	0.00
500	big-M50 + warm start	-2.44184 · 107	2.11	0.00
500	big-M20 + warm start	-2.44184 · 107	11.80	0.00
500	big-M5 + warm start	-2.44184 · 107	1.69	0.00
500	ours	-2.44184 · 107	0.44	0.00
1000	big-M50 + warm start	-2.41988 · 107	12.39	0.00
1000	big-M20 + warm start	-2.41988 · 107	13.36	0.00
1000	big-M5 + warm start	-2.41988 · 107	10.59	0.00
1000	ours	-2.41988 · 107	0.52	0.00
3000	big-M50 + warm start	-2.43712 · 107	3600.43	0.59
3000	big-M20 + warm start	-2.43712 · 107	3600.08	0.59
3000	big-M5 + warm start	-2.43712 · 107	3600.72	0.56
3000	ours	-2.43712 · 107	1.81	0.00
5000	big-M50 + warm start	-2.43693 · 107	3624.24	1.00
5000	big-M20 + warm start	-2.43693 · 107	3603.69	1.00
5000	big-M5 + warm start	-2.43693 · 107	3646.36	600320013.98
5000	ours	-2.43693 · 107	4.08	0.00
Table 89: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.7, λ2 = 10.0). All baselines use our beam search solution as a warm start. The method big-M5 has some large optimality gaps because Gurobi couldn't finish solving the root relaxation within 1h.
# of features	method	upper bound	time(s)	gap(%)
100	big-M50 + warm start	-9.11025 · 107	0.04	0.00
100	big-M20 + warm start	-9.11025 · 107	0.04	0.00
100	big-M5 + warm start	-9.11025 · 107	0.04	0.00
100	ours	-9.11025 · 107	0.32	0.00
500	big-M50 + warm start	-9.12046 · 107	36.63	0.00
500	big-M20 + warm start	-9.12046 · 107	28.85	0.00
500	big-M5 + warm start	-9.12046 · 107	18.35	0.00
500	ours	-9.12046 · 107	0.38	0.00
1000	big-M50 + warm start	-9.04074 · 107	3600.19	0.06
1000	big-M20 + warm start	-9.04074 · 107	3600.33	0.20
1000	big-M5 + warm start	-9.04074 · 107	1545.53	0.00
1000	ours	-9.04074 · 107	0.52	0.00
3000	big-M50 + warm start	-9.12760 · 107	3606.60	0.59
3000	big-M20 + warm start	-9.12760 · 107	3603.80	0.59
3000	big-M5 + warm start	-9.12760 · 107	3605.53	222998156.52
3000	ours	-9.12760 · 107	20.84	0.00
5000	big-M50 + warm start	-9.12915 · 107	3606.47	1.00
5000	big-M20 + warm start	-9.12915 · 107	3602.91	1.00
5000	big-M5 + warm start	-9.12915 · 107	3634.46	618134564.44
5000	ours	-9.12915 · 107	57.41	0.00
Table 90: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.9$ ,  $\lambda_2 = 10.0$ ). All baselines use our beam search solution as a warm start. The method big-M5 has some large optimality gaps because Gurobi couldn't finish solving the root relaxation within 1h.
G.3.7 Synthetic 2 Benchmark with  $\lambda_{2} = 0.001$  with No Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-6.34585 · 10^4	3600.07	19.16
3000	big-M20	-6.34585 · 10^4	3600.07	18.11
3000	big-M5	-6.34585 · 10^4	2252.12	0.00
3000	ours	-6.34585 · 10^4	3600.49	19.45
4000	big-M50	-8.64294 · 10^4	3600.07	15.32
4000	big-M20	-8.64294 · 10^4	3600.09	14.37
4000	big-M5	-8.64294 · 10^4	1536.42	0.00
4000	ours	-8.64294 · 10^4	20.25	0.00
5000	big-M50	-1.09576 · 10^5	3600.81	10.25
5000	big-M20	-1.09576 · 10^5	3600.09	11.72
5000	big-M5	-1.09576 · 10^5	1154.40	0.00
5000	ours	-1.09576 · 10^5	20.51	0.00
6000	big-M50	-1.27643 · 10^5	3600.08	9.93
6000	big-M20	-1.27643 · 10^5	3600.59	9.92
6000	big-M5	-1.27643 · 10^5	1018.24	0.00
6000	ours	-1.27643 · 10^5	1.70	0.00
7000	big-M50	-1.50614 · 10^5	3600.09	8.29
7000	big-M20	-1.50614 · 10^5	3600.07	5.79
7000	big-M5	-1.50614 · 10^5	851.91	0.00
7000	ours	-1.50614 · 10^5	1.67	0.00
Table 91: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 0.001).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-1.59509 · 10^5	3600.08	19.07
3000	big-M20	-1.59509 · 10^5	3600.67	17.35
3000	big-M5	-1.59509 · 10^5	3600.06	6.40
3000	ours	-1.59509 · 10^5	3600.67	19.53
4000	big-M50	-2.17574 · 10^5	3600.07	14.78
4000	big-M20	-2.17574 · 10^5	3600.14	14.34
4000	big-M5	-2.17574 · 10^5	3600.40	4.63
4000	ours	-2.17574 · 10^5	100.13	0.00
5000	big-M50	-2.71986 · 10^5	3600.07	11.22
5000	big-M20	-2.71986 · 10^5	3600.07	11.12
5000	big-M5	-2.71986 · 10^5	2568.73	0.00
5000	ours	-2.71986 · 10^5	20.80	0.00
6000	big-M50	-3.16761 · 10^5	3600.09	9.96
6000	big-M20	-3.16761 · 10^5	3600.06	9.95
6000	big-M5	-3.16761 · 10^5	1834.08	0.00
6000	ours	-3.16761 · 10^5	20.70	0.00
7000	big-M50	-3.72610 · 10^5	3601.19	8.33
7000	big-M20	-3.72610 · 10^5	3600.40	8.13
7000	big-M5	-3.72610 · 10^5	1682.96	0.00
7000	ours	-3.72610 · 10^5	1.98	0.00
Table 92: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 0.001).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-3.33604 · 10^5	3600.08	18.79
3000	big-M20	-3.31111 · 10^5	3600.31	16.99
3000	big-M5	-3.33604 · 10^5	3600.23	7.53
3000	ours	-3.33604 · 10^5	3600.58	19.57
4000	big-M50	-4.53261 · 10^5	3600.07	14.98
4000	big-M20	-4.53261 · 10^5	3600.48	13.50
4000	big-M5	-4.53261 · 10^5	3600.17	6.23
4000	ours	-4.53261 · 10^5	3600.25	0.63
5000	big-M50	-5.63007 · 10^5	3601.19	11.68
5000	big-M20	-5.63007 · 10^5	3600.05	11.25
5000	big-M5	-5.63007 · 10^5	3600.07	3.85
5000	ours	-5.63007 · 10^5	36.97	0.00
6000	big-M50	-6.57799 · 10^5	3600.34	9.98
6000	big-M20	-6.57799 · 10^5	3600.43	9.36
6000	big-M5	-6.57799 · 10^5	3600.05	2.87
6000	ours	-6.57799 · 10^5	20.75	0.00
7000	big-M50	-7.72267 · 10^5	3600.11	8.33
7000	big-M20	-7.72267 · 10^5	3600.96	7.85
7000	big-M5	-7.72267 · 10^5	3600.06	2.40
7000	ours	-7.72267 · 10^5	21.02	0.00
Table 93: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 0.001).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-7.41187 · 10^5	3600.05	18.15
3000	big-M20	-7.41187 · 10^5	3600.20	14.15
3000	big-M5	-7.41187 · 10^5	3600.22	5.94
3000	ours	-7.41187 · 10^5	3601.14	19.60
4000	big-M50	-1.00103 · 10^6	3600.07	15.19
4000	big-M20	-1.00287 · 10^6	3600.21	11.81
4000	big-M5	-1.00287 · 10^6	3600.11	5.06
4000	ours	-1.00287 · 10^6	3600.37	1.88
5000	big-M50	-1.24027 · 10^6	3601.10	11.88
5000	big-M20	-1.24027 · 10^6	3601.01	10.26
5000	big-M5	-1.24027 · 10^6	3600.19	4.34
5000	ours	-1.24027 · 10^6	466.42	0.00
6000	big-M50	-1.45417 · 10^6	3600.11	9.78
6000	big-M20	-1.45417 · 10^6	3600.08	9.11
6000	big-M5	-1.45417 · 10^6	3600.19	3.94
6000	ours	-1.45417 · 10^6	57.67	0.00
7000	big-M50	-1.70483 · 10^6	3601.55	8.34
7000	big-M20	-1.70483 · 10^6	3600.10	7.77
7000	big-M5	-1.70483 · 10^6	3600.38	3.39
7000	ours	-1.70483 · 10^6	21.42	0.00
Table 94: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.7, λ2 = 0.001).
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50	-2.78521 · 10^6	3600.15	15.90
3000	big-M20	-2.78521 · 10^6	3600.05	10.37
3000	big-M5	-2.78521 · 10^6	3600.36	3.69
3000	ours	-2.78521 · 10^6	3600.01	19.60
4000	big-M50	-3.75202 · 10^6	3600.06	13.33
4000	big-M20	-3.75203 · 10^6	3600.28	8.64
4000	big-M5	-3.75132 · 10^6	3600.37	3.18
4000	ours	-3.75203 · 10^6	3600.42	3.50
5000	big-M50	-4.61877 · 10^6	3600.19	11.37
5000	big-M20	-4.61877 · 10^6	3600.05	7.62
5000	big-M5	-4.61877 · 10^6	3600.06	2.72
5000	ours	-4.61877 · 10^6	3600.57	0.93
6000	big-M50	-5.43826 · 10^6	3600.06	9.77
6000	big-M20	-5.43826 · 10^6	3600.07	6.92
6000	big-M5	-5.43826 · 10^6	3600.15	2.50
6000	ours	-5.43826 · 10^6	3600.65	0.33
7000	big-M50	-6.36767 · 10^6	3600.38	8.22
7000	big-M20	-6.36767 · 10^6	3600.05	6.06
7000	big-M5	-6.36767 · 10^6	3600.21	2.23
7000	ours	-6.36767 · 10^6	3600.78	0.11
Table 95: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 0.001).
G.3.8 Synthetic 2 Benchmark with  $\lambda_{2} = 0.001$  with Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-6.34585 · 10^4	3601.20	19.16
3000	big-M20 + warm start	-6.34585 · 10^4	3600.06	18.11
3000	big-M5 + warm start	-6.34585 · 10^4	2435.86	0.00
3000	ours	-6.34585 · 10^4	3600.49	19.45
4000	big-M50 + warm start	-8.64294 · 10^4	3600.87	15.32
4000	big-M20 + warm start	-8.64294 · 10^4	3600.12	14.37
4000	big-M5 + warm start	-8.64294 · 10^4	1618.73	0.00
4000	ours	-8.64294 · 10^4	20.25	0.00
5000	big-M50 + warm start	-1.09576 · 10^5	3601.86	10.25
5000	big-M20 + warm start	-1.09576 · 10^5	3601.59	11.72
5000	big-M5 + warm start	-1.09576 · 10^5	1530.18	0.00
5000	ours	-1.09576 · 10^5	20.51	0.00
6000	big-M50 + warm start	-1.27643 · 10^5	3601.14	9.93
6000	big-M20 + warm start	-1.27643 · 10^5	3600.64	9.82
6000	big-M5 + warm start	-1.27643 · 10^5	972.83	0.00
6000	ours	-1.27643 · 10^5	1.70	0.00
7000	big-M50 + warm start	-1.50614 · 10^5	3600.13	8.29
7000	big-M20 + warm start	-1.50614 · 10^5	3600.94	5.81
7000	big-M5 + warm start	-1.50614 · 10^5	824.89	0.00
7000	ours	-1.50614 · 10^5	1.67	0.00
Table 96: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 0.001). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-1.59509 · 10^5	3600.60	19.07
3000	big-M20 + warm start	-1.59509 · 10^5	3600.07	17.35
3000	big-M5 + warm start	-1.59509 · 10^5	3600.05	6.40
3000	ours	-1.59509 · 10^5	3600.67	19.53
4000	big-M50 + warm start	-2.17574 · 10^5	3600.10	14.78
4000	big-M20 + warm start	-2.17574 · 10^5	3600.68	14.34
4000	big-M5 + warm start	-2.17574 · 10^5	3600.04	4.63
4000	ours	-2.17574 · 10^5	100.13	0.00
5000	big-M50 + warm start	-2.71986 · 10^5	3600.06	11.22
5000	big-M20 + warm start	-2.71986 · 10^5	3601.05	11.12
5000	big-M5 + warm start	-2.71986 · 10^5	2571.91	0.00
5000	ours	-2.71986 · 10^5	20.80	0.00
6000	big-M50 + warm start	-3.16761 · 10^5	3600.07	9.96
6000	big-M20 + warm start	-3.16761 · 10^5	3600.65	9.95
6000	big-M5 + warm start	-3.16761 · 10^5	1861.02	0.00
6000	ours	-3.16761 · 10^5	20.70	0.00
7000	big-M50 + warm start	-3.72610 · 10^5	3600.08	8.33
7000	big-M20 + warm start	-3.72610 · 10^5	3600.41	8.13
7000	big-M5 + warm start	-3.72610 · 10^5	1752.62	0.00
7000	ours	-3.72610 · 10^5	1.98	0.00
Table 97: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 0.001). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-3.33604 · 10^5	3600.17	18.79
3000	big-M20 + warm start	-3.33604 · 10^5	3600.12	16.11
3000	big-M5 + warm start	-3.33604 · 10^5	3600.20	7.53
3000	ours	-3.33604 · 10^5	3600.58	19.57
4000	big-M50 + warm start	-4.53261 · 10^5	3600.17	14.98
4000	big-M20 + warm start	-4.53261 · 10^5	3600.77	13.50
4000	big-M5 + warm start	-4.53261 · 10^5	3600.23	6.23
4000	ours	-4.53261 · 10^5	3600.25	0.63
5000	big-M50 + warm start	-5.63007 · 10^5	3600.29	11.70
5000	big-M20 + warm start	-5.63007 · 10^5	3600.07	11.25
5000	big-M5 + warm start	-5.63007 · 10^5	3600.32	3.38
5000	ours	-5.63007 · 10^5	36.97	0.00
6000	big-M50 + warm start	-6.57799 · 10^5	3600.94	9.98
6000	big-M20 + warm start	-6.57799 · 10^5	3600.28	9.36
6000	big-M5 + warm start	-6.57799 · 10^5	3600.06	2.89
6000	ours	-6.57799 · 10^5	20.75	0.00
7000	big-M50 + warm start	-7.72267 · 10^5	3600.55	8.33
7000	big-M20 + warm start	-7.72267 · 10^5	3600.07	7.85
7000	big-M5 + warm start	-7.72267 · 10^5	3600.65	2.40
7000	ours	-7.72267 · 10^5	21.02	0.00
Table 98: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 0.001). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-7.41187 · 10^5	3600.76	18.15
3000	big-M20 + warm start	-7.41187 · 10^5	3600.29	14.15
3000	big-M5 + warm start	-7.41187 · 10^5	3600.04	5.94
3000	ours	-7.41187 · 10^5	3601.14	19.60
4000	big-M50 + warm start	-1.00287 · 10^6	3600.07	14.98
4000	big-M20 + warm start	-1.00287 · 10^6	3600.10	11.81
4000	big-M5 + warm start	-1.00287 · 10^6	3600.04	5.06
4000	ours	-1.00287 · 10^6	3600.37	1.88
5000	big-M50 + warm start	-1.24027 · 10^6	3600.05	11.88
5000	big-M20 + warm start	-1.24027 · 10^6	3600.49	10.26
5000	big-M5 + warm start	-1.24027 · 10^6	3600.24	4.34
5000	ours	-1.24027 · 10^6	466.42	0.00
6000	big-M50 + warm start	-1.45417 · 10^6	3600.45	9.78
6000	big-M20 + warm start	-1.45417 · 10^6	3600.26	9.06
6000	big-M5 + warm start	-1.45417 · 10^6	3600.07	3.94
6000	ours	-1.45417 · 10^6	57.67	0.00
7000	big-M50 + warm start	-1.70483 · 10^6	3601.38	8.34
7000	big-M20 + warm start	-1.70483 · 10^6	3600.06	7.77
7000	big-M5 + warm start	-1.70483 · 10^6	3600.05	3.39
7000	ours	-1.70483 · 10^6	21.42	0.00
Table 99: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.7, λ2 = 0.001). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	big-M50 + warm start	-2.78521 · 10^6	3600.74	15.90
3000	big-M20 + warm start	-2.78521 · 10^6	3600.24	10.37
3000	big-M5 + warm start	-2.78521 · 10^6	3600.32	3.69
3000	ours	-2.78521 · 10^6	3600.01	19.60
4000	big-M50 + warm start	-3.75203 · 10^6	3600.07	13.33
4000	big-M20 + warm start	-3.75203 · 10^6	3600.24	8.64
4000	big-M5 + warm start	-3.75203 · 10^6	3600.16	3.16
4000	ours	-3.75203 · 10^6	3600.42	3.50
5000	big-M50 + warm start	-4.61877 · 10^6	3601.36	11.37
5000	big-M20 + warm start	-4.61877 · 10^6	3600.23	7.62
5000	big-M5 + warm start	-4.61877 · 10^6	3600.07	2.72
5000	ours	-4.61877 · 10^6	3600.57	0.93
6000	big-M50 + warm start	-5.43826 · 10^6	3600.22	9.77
6000	big-M20 + warm start	-5.43826 · 10^6	3601.07	6.92
6000	big-M5 + warm start	-5.43826 · 10^6	3600.43	2.50
6000	ours	-5.43826 · 10^6	3600.65	0.33
7000	big-M50 + warm start	-6.36767 · 10^6	3600.12	8.22
7000	big-M20 + warm start	-6.36767 · 10^6	3600.57	6.06
7000	big-M5 + warm start	-6.36767 · 10^6	3600.04	2.23
7000	ours	-6.36767 · 10^6	3600.78	0.11
Table 100: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 0.001). All baselines use our beam search solution as a warm start.
G.3.9 Synthetic 2 Benchmark with  $\lambda_{2} = 0.1$  with No Warmstart