SlowGuess/NeurIPS_2023_OCR huggingface dataset:
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-2.36231 · 107	10	3732.54	23.59
100	SubsetSelectCIO + warm start	-2.32147 · 107	10	3623.89	25.77
100	L0BnB	-2.43574 · 107	10	2671.45	0.00
100	MSK persp.	-2.39754 · 107	10	3601.78	1.61
100	MSK persp. + warm start	-2.43574 · 107	10	3604.40	0.01
100	Convex MSK opt. persp.	-2.43574 · 107	10	0.13	0.00
100	ours	-2.43574 · 107	10	0.30	0.00
500	SubsetSelectCIO	-2.27569 · 107	10	3601.02	28.83
500	SubsetSelectCIO + warm start	-2.32447 · 107	10	3624.13	26.12
500	L0BnB	-2.44184 · 107	10	3822.46	0.00
500	MSK persp.	-	-	3605.48	-
500	MSK persp. + warm start	-2.44184 · 107	10	3603.64	0.00
500	Convex MSK opt. persp.	-2.44184 · 107	10	7.40	0.01
500	ours	-2.44184 · 107	10	0.44	0.00
1000	SubsetSelectCIO	-2.18726 · 107	7	2623.29	0.00
1000	SubsetSelectCIO + warm start	-2.32752 · 107	10	4344.29	24.76
1000	L0BnB	-2.41993 · 107	11	3983.89	0.31
1000	MSK persp.	-	-	3606.66	-
1000	MSK persp. + warm start	-2.41988 · 107	10	3608.58	0.00
1000	Convex MSK opt. persp.	-2.41988 · 107	10	63.01	0.01
1000	ours	-2.41988 · 107	10	0.52	0.00
3000	SubsetSelectCIO	-2.34295 · 107	10	3767.28	24.75
3000	SubsetSelectCIO + warm start	-	-	3600.00	-
3000	L0BnB	-2.42598 · 107	9	7332.12	0.00
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-2.43712 · 107	10	4135.58	0.00
3000	Convex MSK opt. persp.	-2.43712 · 107	10	1968.86	0.02
3000	ours	-2.43712 · 107	10	1.81	0.00
5000	SubsetSelectCIO	-	-	3600.00	-
5000	SubsetSelectCIO + warm start	-	-	3600.00	-
5000	L0BnB	-2.43693 · 107	10	4661.71	23.68
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-2.43693 · 107	10	3769.92	0.06
5000	ours	-2.43693 · 107	10	4.08	0.00
Table 134: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.7$ ,  $\lambda_2 = 10.0$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-8.97109 · 10^7	10	3602.39	21.73
100	SubsetSelectCIO + warm start	-9.05496 · 10^7	10	3626.15	20.60
100	L0BnB	-9.11184 · 10^7	93	7102.10	0.04
100	MSK persp.	-8.99265 · 10^7	10	3600.18	1.32
100	MSK persp. + warm start	-9.11025 · 10^7	10	3600.47	0.01
100	Convex MSK opt. persp.	-9.11025 · 10^7	10	0.13	0.00
100	ours	-9.11025 · 10^7	10	0.32	0.00
500	SubsetSelectCIO	-9.00037 · 10^7	10	4088.17	21.67
500	SubsetSelectCIO + warm start	-9.02231 · 10^7	10	3817.48	21.37
500	L0BnB	-9.11085 · 10^7	11	843.53	6.21
500	MSK persp.	-	-	3604.01	-
500	MSK persp. + warm start	-9.12046 · 10^7	10	3604.99	0.00
500	Convex MSK opt. persp.	-9.12046 · 10^7	10	7.57	0.01
500	ours	-9.12046 · 10^7	10	0.38	0.00
1000	SubsetSelectCIO	-	-	3600.00	-
1000	SubsetSelectCIO + warm start	-	-	3600.00	-
1000	L0BnB	-8.23898 · 10^7	1	7369.81	9.08
1000	MSK persp.	-	-	3604.50	-
1000	MSK persp. + warm start	-9.04074 · 10^7	10	3688.88	0.00
1000	Convex MSK opt. persp.	-9.04074 · 10^7	10	62.54	0.01
1000	ours	-9.04074 · 10^7	10	0.52	0.00
3000	SubsetSelectCIO	-	-	3600.00	-
3000	SubsetSelectCIO + warm start	-	-	3600.00	-
3000	L0BnB	-9.12760 · 10^7	10	1883.50	6.26
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-9.12760 · 10^7	10	3764.99	0.00
3000	Convex MSK opt. persp.	-9.12760 · 10^7	10	1999.52	0.02
3000	ours	-9.12760 · 10^7	10	20.84	0.00
5000	SubsetSelectCIO	-	-	3600.00	-
5000	SubsetSelectCIO + warm start	-	-	3600.00	-
5000	L0BnB	-8.32064 · 10^7	1	7664.70	9.32
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-9.12915 · 10^7	10	8218.76	0.00
5000	Convex MSK opt. persp.	-9.12915 · 10^7	10	3654.46	3.94
5000	ours	-9.12915 · 10^7	10	57.41	0.00
Table 135: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.9$ ,  $\lambda_2 = 10.0$ ).
G.4.4 Synthetic 2 Benchmark with  $\lambda_{2} = 0.001$
# of samples	method	upper bound	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-1.26607 · 104	2	1.93	0.00
3000	SubsetSelectCIO + warm start	-1.26607 · 104	2	2.02	0.00
3000	L0BnB	-6.34585 · 104	10	1200.55	0.00
3000	MSK persp.	-5.27912 · 104	10	3601.00	43.41
3000	MSK persp. + warm start	-6.34585 · 104	10	3601.81	19.30
3000	Convex MSK opt. persp.	-4.55044 · 104	10	1573.44	66.10
3000	ours	-6.34585 · 104	10	3600.49	19.45
4000	SubsetSelectCIO	-1.88907 · 104	2	1.90	0.00
4000	SubsetSelectCIO + warm start	-1.88907 · 104	2	2.06	0.00
4000	L0BnB	-8.64294 · 104	10	579.39	0.00
4000	MSK persp.	-8.64294 · 104	10	3602.16	15.37
4000	MSK persp. + warm start	-8.64294 · 104	10	3600.84	15.37
4000	Convex MSK opt. persp.	-8.64294 · 104	10	1398.74	2.40
4000	ours	-8.64294 · 104	10	20.25	0.00
5000	SubsetSelectCIO	-2.49366 · 104	2	2.08	0.00
5000	SubsetSelectCIO + warm start	-2.49366 · 104	2	1.96	0.00
5000	L0BnB	-1.09576 · 105	10	1292.85	0.00
5000	MSK persp.	-1.09576 · 105	10	3601.60	11.91
5000	MSK persp. + warm start	-1.09576 · 105	10	3600.55	11.91
5000	Convex MSK opt. persp.	-1.09576 · 105	10	1496.25	1.03
5000	ours	-1.09576 · 105	10	20.51	0.00
6000	SubsetSelectCIO	-	-	3600.00	-
6000	SubsetSelectCIO + warm start	-2.83063 · 104	2	2.22	0.00
6000	L0BnB	-1.27643 · 105	10	1379.36	0.00
6000	MSK persp.	-	-	3600.07	-
6000	MSK persp. + warm start	-1.27643 · 105	10	3600.30	9.93
6000	Convex MSK opt. persp.	-1.27643 · 105	10	1318.81	0.62
6000	ours	-1.27643 · 105	10	1.70	0.00
7000	SubsetSelectCIO	-3.24781 · 104	2	2.06	0.00
7000	SubsetSelectCIO + warm start	-3.24781 · 104	2	2.22	0.00
7000	L0BnB	-1.50614 · 105	10	1105.58	0.00
7000	MSK persp.	-2.07297 · 104	3	3600.06	687.08
7000	MSK persp. + warm start	-1.50614 · 105	10	3599.93	8.33
7000	Convex MSK opt. persp.	-1.50614 · 105	10	1455.27	0.45
7000	ours	-1.50614 · 105	10	1.67	0.00
Table 136: 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	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-7.20755 · 10^4	2	1.99	0.00
3000	SubsetSelectCIO + warm start	-7.20755 · 10^4	2	2.08	0.00
3000	L0BnB	-1.59509 · 10^5	10	3023.58	0.00
3000	MSK persp.	-1.48826 · 10^5	10	3601.42	27.95
3000	MSK persp. + warm start	-1.59509 · 10^5	10	3601.71	19.38
3000	Convex MSK opt. persp.	-1.13813 · 10^5	10	1627.92	67.04
3000	ours	-1.59509 · 10^5	10	3600.67	19.53
4000	SubsetSelectCIO	-1.03099 · 10^5	2	2.07	0.00
4000	SubsetSelectCIO + warm start	-1.03099 · 10^5	2	1.81	0.00
4000	L0BnB	-2.17574 · 10^5	10	3396.85	0.00
4000	MSK persp.	-2.17574 · 10^5	10	3600.26	15.27
4000	MSK persp. + warm start	-2.17574 · 10^5	10	3601.91	15.27
4000	Convex MSK opt. persp.	-2.17574 · 10^5	10	1509.12	2.41
4000	ours	-2.17574 · 10^5	10	100.13	0.00
5000	SubsetSelectCIO	-1.30339 · 10^5	2	1.98	0.00
5000	SubsetSelectCIO + warm start	-1.30339 · 10^5	2	2.01	0.00
5000	L0BnB	-2.71986 · 10^5	10	531.44	0.00
5000	MSK persp.	-2.71986 · 10^5	10	3600.74	12.00
5000	MSK persp. + warm start	-2.71986 · 10^5	10	3600.26	12.00
5000	Convex MSK opt. persp.	-2.71986 · 10^5	10	1514.31	1.04
5000	ours	-2.71986 · 10^5	10	20.80	0.00
6000	SubsetSelectCIO	-1.50202 · 10^5	2	2.02	0.00
6000	SubsetSelectCIO + warm start	-1.50202 · 10^5	2	1.96	0.00
6000	L0BnB	-3.16761 · 10^5	10	3098.88	0.00
6000	MSK persp.	-7.83273 · 10^4	1	3603.79	344.72
6000	MSK persp. + warm start	-3.16761 · 10^5	10	3604.35	9.96
6000	Convex MSK opt. persp.	-3.16761 · 10^5	10	1445.46	0.62
6000	ours	-3.16761 · 10^5	10	20.70	0.00
7000	SubsetSelectCIO	-1.75217 · 10^5	2	2.12	0.00
7000	SubsetSelectCIO + warm start	-1.75217 · 10^5	2	2.12	0.00
7000	L0BnB	-3.72610 · 10^5	10	3130.56	0.00
7000	MSK persp.	-	-	3599.99	-
7000	MSK persp. + warm start	-3.72610 · 10^5	10	3600.24	8.33
7000	Convex MSK opt. persp.	-3.72610 · 10^5	10	1520.22	0.45
7000	ours	-3.72610 · 10^5	10	1.98	0.00
Table 137: Comparison of time and optimality gap on the synthetic datasets  $(\mathrm{p} = 3000, \mathrm{k} = 10, \rho = 0.3, \lambda_2 = 0.001)$ .
# of samples	method	upper bound	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-2.19279 · 105	2	1.96	0.00
3000	SubsetSelectCIO + warm start	-2.19279 · 105	2	1.85	0.00
3000	L0BnB	-3.27442 · 105	8	7379.57	5.16
3000	MSK persp.	-2.83751 · 105	10	3601.66	40.41
3000	MSK persp. + warm start	-3.33604 · 105	10	3601.38	19.43
3000	Convex MSK opt. persp.	-2.77416 · 105	10	1878.24	43.38
3000	ours	-3.33604 · 105	10	3600.58	19.57
4000	SubsetSelectCIO	-3.03679 · 105	2	1.87	0.00
4000	SubsetSelectCIO + warm start	-3.03679 · 105	2	1.89	0.00
4000	L0BnB	-4.53261 · 105	10	4149.90	32.94
4000	MSK persp.	-4.53261 · 105	10	3602.72	15.24
4000	MSK persp. + warm start	-4.53261 · 105	10	3599.91	15.24
4000	Convex MSK opt. persp.	-4.53261 · 105	10	1445.07	2.53
4000	ours	-4.53261 · 105	10	3600.25	0.63
5000	SubsetSelectCIO	-4.12520 · 105	3	1.79	0.00
5000	SubsetSelectCIO + warm start	-4.12520 · 105	3	2.03	0.00
5000	L0BnB	-5.63007 · 105	10	3522.00	0.00
5000	MSK persp.	-5.63007 · 105	10	3600.76	12.04
5000	MSK persp. + warm start	-5.63007 · 105	10	3603.42	12.04
5000	Convex MSK opt. persp.	-5.63007 · 105	10	1618.41	1.04
5000	ours	-5.63007 · 105	10	36.97	0.00
6000	SubsetSelectCIO	-4.42065 · 105	2	1.94	0.00
6000	SubsetSelectCIO + warm start	-4.42065 · 105	2	2.12	0.00
6000	L0BnB	-6.57799 · 105	10	3327.45	0.00
6000	MSK persp.	-	-	3605.66	-
6000	MSK persp. + warm start	-6.57799 · 105	10	3605.89	9.98
6000	Convex MSK opt. persp.	-6.57799 · 105	10	1523.01	0.62
6000	ours	-6.57799 · 105	10	20.75	0.00
7000	SubsetSelectCIO	-5.16405 · 105	2	2.11	0.00
7000	SubsetSelectCIO + warm start	-5.16405 · 105	2	2.14	0.00
7000	L0BnB	-7.72267 · 105	10	3567.05	0.00
7000	MSK persp.	-	-	3600.27	-
7000	MSK persp. + warm start	-7.72267 · 105	10	3600.18	8.33
7000	Convex MSK opt. persp.	-7.72267 · 105	10	1681.35	0.45
7000	ours	-7.72267 · 105	10	21.02	0.00
Table 138: Comparison of time and optimality gap on the synthetic datasets  $(\mathrm{p} = 3000, \mathrm{k} = 10, \rho = 0.5, \lambda_2 = 0.001)$ .
# of samples	method	upper bound	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-6.06571 · 10^5	2	1.88	0.00
3000	SubsetSelectCIO + warm start	-6.06571 · 10^5	2	1.88	0.00
3000	L0BnB	-5.52986 · 10^5	1	7308.45	34.21
3000	MSK persp.	-6.89430 · 10^5	10	3600.74	28.43
3000	MSK persp. + warm start	-7.41187 · 10^5	10	3601.61	19.46
3000	Convex MSK opt. persp.	-6.84471 · 10^5	10	1652.72	29.14
3000	ours	-7.41187 · 10^5	10	3601.14	19.60
4000	SubsetSelectCIO	-8.24785 · 10^5	2	1.85	0.00
4000	SubsetSelectCIO + warm start	-8.24785 · 10^5	2	1.86	0.00
4000	L0BnB	-7.53893 · 10^5	1	7293.04	32.01
4000	MSK persp.	-9.95041 · 10^5	10	3601.67	16.12
4000	MSK persp. + warm start	-1.00287 · 10^6	10	3602.00	15.21
4000	Convex MSK opt. persp.	-1.00287 · 10^6	10	1442.72	2.82
4000	ours	-1.00287 · 10^6	10	3600.37	1.88
5000	SubsetSelectCIO	-1.02490 · 10^6	2	2.05	0.00
5000	SubsetSelectCIO + warm start	-1.02490 · 10^6	2	1.92	0.00
5000	L0BnB	-1.23555 · 10^6	9	7387.59	4.90
5000	MSK persp.	-1.24027 · 10^6	10	3604.54	12.07
5000	MSK persp. + warm start	-1.24027 · 10^6	10	3600.77	12.07
5000	Convex MSK opt. persp.	-1.24027 · 10^6	10	1654.86	1.06
5000	ours	-1.24027 · 10^6	10	466.42	0.00
6000	SubsetSelectCIO	-1.20146 · 10^6	2	1.96	0.00
6000	SubsetSelectCIO + warm start	-1.20146 · 10^6	2	2.05	0.00
6000	L0BnB	-1.44860 · 10^6	9	7265.23	4.30
6000	MSK persp.	-1.14100 · 10^6	2	3603.46	40.19
6000	MSK persp. + warm start	-1.45417 · 10^6	10	3604.24	10.00
6000	Convex MSK opt. persp.	-1.45417 · 10^6	10	1667.19	0.63
6000	ours	-1.45417 · 10^6	10	57.67	0.00
7000	SubsetSelectCIO	-1.46875 · 10^6	3	1.86	0.00
7000	SubsetSelectCIO + warm start	-1.46875 · 10^6	3	2.09	0.00
7000	L0BnB	-1.69804 · 10^6	9	7320.83	2.92
7000	MSK persp.	-1.32733 · 10^6	2	3600.13	39.15
7000	MSK persp. + warm start	-1.70483 · 10^6	10	3600.05	8.34
7000	Convex MSK opt. persp.	-1.70483 · 10^6	10	1760.17	0.45
7000	ours	-1.70483 · 10^6	10	21.42	0.00
Table 139: 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	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-2.63352 · 10^6	2	1.90	0.00
3000	SubsetSelectCIO + warm start	-2.63352 · 10^6	2	2.04	0.00
3000	L0BnB	-2.55637 · 10^6	1	7236.54	18.84
3000	MSK persp.	-2.72590 · 10^6	10	3599.87	22.06
3000	MSK persp. + warm start	-2.72590 · 10^6	10	3600.80	22.06
3000	Convex MSK opt. persp.	-2.72998 · 10^6	10	2311.76	21.67
3000	ours	-2.78521 · 10^6	10	3600.01	19.60
4000	SubsetSelectCIO	-3.38912 · 10^6	1	1.79	0.00
4000	SubsetSelectCIO + warm start	-3.38912 · 10^6	1	1.89	0.00
4000	L0BnB	-3.45201 · 10^6	1	7291.93	15.48
4000	MSK persp.	-	-	3609.13	-
4000	MSK persp. + warm start	-3.75203 · 10^6	10	3605.54	15.12
4000	Convex MSK opt. persp.	-3.75202 · 10^6	10	1481.77	3.23
4000	ours	-3.75203 · 10^6	10	3600.42	3.50
5000	SubsetSelectCIO	-4.37679 · 10^6	2	1.99	0.00
5000	SubsetSelectCIO + warm start	-4.37679 · 10^6	2	2.07	0.00
5000	L0BnB	-4.23095 · 10^6	1	7316.72	13.65
5000	MSK persp.	-4.60534 · 10^6	10	3602.67	12.43
5000	MSK persp. + warm start	-4.61877 · 10^6	10	3601.35	12.11
5000	Convex MSK opt. persp.	-4.61877 · 10^6	10	1537.61	1.22
5000	ours	-4.61877 · 10^6	10	3600.57	0.93
6000	SubsetSelectCIO	-5.15897 · 10^6	2	2.23	0.00
6000	SubsetSelectCIO + warm start	-5.15897 · 10^6	2	2.24	0.00
6000	L0BnB	-4.97742 · 10^6	1	7330.86	13.05
6000	MSK persp.	-	-	3603.01	-
6000	MSK persp. + warm start	-5.43826 · 10^6	10	3602.04	0.00
6000	Convex MSK opt. persp.	-5.43826 · 10^6	10	1652.62	0.65
6000	ours	-5.43826 · 10^6	10	3600.65	0.33
7000	SubsetSelectCIO	-6.02832 · 10^6	2	1.89	0.00
7000	SubsetSelectCIO + warm start	-6.02832 · 10^6	2	2.27	0.00
7000	L0BnB	-5.82321 · 10^6	1	7356.27	11.85
7000	MSK persp.	-	-	3600.06	-
7000	MSK persp. + warm start	-6.36767 · 10^6	10	3600.21	8.35
7000	Convex MSK opt. persp.	-6.36767 · 10^6	10	1821.91	0.45
7000	ours	-6.36767 · 10^6	10	3600.78	0.11
Table 140: Comparison of time and optimality gap on the synthetic datasets  $(\mathrm{p} = 3000, \mathrm{k} = 10, \rho = 0.9, \lambda_2 = 0.001)$ .
G.4.5 Synthetic 2 Benchmark with  $\lambda_{2} = 0.1$
# of samples	method	upper bound	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-1.83147 · 10^4	10	4229.80	313.93
3000	SubsetSelectCIO + warm start	-5.70956 · 10^4	10	3602.11	32.77
3000	L0BnB	-6.34575 · 10^4	10	1201.41	0.00
3000	MSK persp.	-6.34575 · 10^4	10	3601.57	17.90
3000	MSK persp. + warm start	-6.34575 · 10^4	10	3600.12	17.89
3000	Convex MSK opt. persp.	-6.34575 · 10^4	10	1444.07	17.85
3000	ours	-6.34575 · 10^4	10	3600.33	19.31
4000	SubsetSelectCIO	-3.10931 · 10^4	10	3872.26	235.85
4000	SubsetSelectCIO + warm start	-7.28764 · 10^4	10	4101.05	43.29
4000	L0BnB	-8.64284 · 10^4	10	566.25	0.00
4000	MSK persp.	-8.64284 · 10^4	10	3602.29	15.03
4000	MSK persp. + warm start	-8.64284 · 10^4	10	3601.14	15.05
4000	Convex MSK opt. persp.	-8.64284 · 10^4	10	1458.63	2.40
4000	ours	-8.64284 · 10^4	10	20.81	0.00
5000	SubsetSelectCIO	-4.06480 · 10^4	10	3939.75	222.71
5000	SubsetSelectCIO + warm start	-9.16380 · 10^4	10	3804.66	43.14
5000	L0BnB	-1.09575 · 10^5	10	1314.94	0.00
5000	MSK persp.	-1.09575 · 10^5	10	3603.48	11.78
5000	MSK persp. + warm start	-1.09575 · 10^5	10	3602.79	11.78
5000	Convex MSK opt. persp.	-1.09575 · 10^5	10	1478.60	1.03
5000	ours	-1.09575 · 10^5	10	20.65	0.00
6000	SubsetSelectCIO	-	-	3600.00	-
6000	SubsetSelectCIO + warm start	-1.21549 · 10^5	10	3850.54	26.10
6000	L0BnB	-1.27642 · 10^5	10	1536.59	0.00
6000	MSK persp.	-	-	3606.50	-
6000	MSK persp. + warm start	-1.27642 · 10^5	10	3603.98	9.85
6000	Convex MSK opt. persp.	-1.27642 · 10^5	10	1303.39	0.62
6000	ours	-1.27642 · 10^5	10	1.70	0.00
7000	SubsetSelectCIO	-5.54763 · 10^4	10	4067.63	224.61
7000	SubsetSelectCIO + warm start	-1.42693 · 10^5	10	3615.01	26.20
7000	L0BnB	-1.50613 · 10^5	10	1064.72	0.00
7000	MSK persp.	-	-	3599.90	-
7000	MSK persp. + warm start	-1.50613 · 10^5	10	3600.13	8.28
7000	Convex MSK opt. persp.	-1.50613 · 10^5	10	1405.96	0.45
7000	ours	-1.50613 · 10^5	10	1.67	0.00
Table 141: Comparison of time and optimality gap on the synthetic datasets  $(\mathrm{p} = 3000, \mathrm{k} = 10, \rho = 0.1, \lambda_2 = 0.1)$ .
# of samples	method	upper bound	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-1.13982 · 105	10	3755.86	67.28
3000	SubsetSelectCIO + warm start	-1.59508 · 105	10	3857.37	19.54
3000	L0BnB	-1.59508 · 105	10	1491.90	0.00
3000	MSK persp.	-1.59508 · 105	10	3600.12	17.99
3000	MSK persp. + warm start	-1.59508 · 105	10	3600.37	17.99
3000	Convex MSK opt. persp.	-1.59508 · 105	10	1454.84	17.99
3000	ours	-1.59508 · 105	10	3600.27	19.40
4000	SubsetSelectCIO	-	-	3600.00	-
4000	SubsetSelectCIO + warm start	-2.17573 · 105	10	3711.79	20.70
4000	L0BnB	-2.17573 · 105	10	3399.35	0.00
4000	MSK persp.	-2.17573 · 105	10	3601.17	14.95
4000	MSK persp. + warm start	-2.17573 · 105	10	3602.25	14.95
4000	Convex MSK opt. persp.	-2.17573 · 105	10	1510.77	2.40
4000	ours	-2.17573 · 105	10	99.53	0.00
5000	SubsetSelectCIO	-1.92108 · 105	10	3602.76	69.69
5000	SubsetSelectCIO + warm start	-2.71985 · 105	10	3733.49	19.85
5000	L0BnB	-2.71985 · 105	10	532.76	0.00
5000	MSK persp.	-2.71985 · 105	10	3601.74	11.88
5000	MSK persp. + warm start	-2.71985 · 105	10	3602.31	11.88
5000	Convex MSK opt. persp.	-2.71985 · 105	10	1576.92	1.04
5000	ours	-2.71985 · 105	10	21.01	0.00
6000	SubsetSelectCIO	-2.03206 · 105	10	4031.62	87.30
6000	SubsetSelectCIO + warm start	-3.10827 · 105	10	3780.98	22.45
6000	L0BnB	-3.16760 · 105	10	3082.87	0.00
6000	MSK persp.	-	-	3600.17	-
6000	MSK persp. + warm start	-3.16760 · 105	10	3599.98	9.90
6000	Convex MSK opt. persp.	-3.16760 · 105	10	1487.94	0.62
6000	ours	-3.16760 · 105	10	20.91	0.00
7000	SubsetSelectCIO	-2.44730 · 105	10	3992.94	82.05
7000	SubsetSelectCIO + warm start	-3.37947 · 105	10	4164.11	31.83
7000	L0BnB	-3.72609 · 105	10	3126.74	0.00
7000	MSK persp.	-	-	3600.16	-
7000	MSK persp. + warm start	-3.72609 · 105	10	3604.63	8.28
7000	Convex MSK opt. persp.	-3.72609 · 105	10	1555.38	0.45
7000	ours	-3.72609 · 105	10	1.73	0.00
Table 142: Comparison of time and optimality gap on the synthetic datasets  $(\mathrm{p} = 3000, \mathrm{k} = 10, \rho = 0.3, \lambda_2 = 0.1)$ .
# of samples	method	upper bound	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-2.79591 · 10^5	9	3686.43	42.67
3000	SubsetSelectCIO + warm start	-3.11853 · 10^5	10	3738.09	27.91
3000	L0BnB	-3.27441 · 10^5	8	7347.92	4.34
3000	MSK persp.	-3.33603 · 10^5	10	3600.83	18.03
3000	MSK persp. + warm start	-3.33603 · 10^5	10	3600.62	18.03
3000	Convex MSK opt. persp.	-3.18968 · 10^5	10	1341.73	23.47
3000	ours	-3.33603 · 10^5	10	3600.68	19.44
4000	SubsetSelectCIO	-	-	3600.00	-
4000	SubsetSelectCIO + warm start	-4.44307 · 10^5	10	3651.78	23.07
4000	L0BnB	-4.53260 · 10^5	10	4150.43	32.94
4000	MSK persp.	-4.53260 · 10^5	10	3602.80	14.94
4000	MSK persp. + warm start	-4.53260 · 10^5	10	3601.24	14.94
4000	Convex MSK opt. persp.	-4.53260 · 10^5	10	1444.17	2.53
4000	ours	-4.53260 · 10^5	10	3600.41	0.62
5000	SubsetSelectCIO	-4.69039 · 10^5	8	3951.25	43.95
5000	SubsetSelectCIO + warm start	-5.44684 · 10^5	9	4020.24	23.96
5000	L0BnB	-5.63006 · 10^5	10	3516.88	0.00
5000	MSK persp.	-5.63006 · 10^5	10	3603.31	11.92
5000	MSK persp. + warm start	-5.63006 · 10^5	10	3601.57	11.92
5000	Convex MSK opt. persp.	-5.63006 · 10^5	10	1572.75	1.04
5000	ours	-5.63006 · 10^5	10	37.26	0.00
6000	SubsetSelectCIO	-2.80001 · 10^5	1	2.28	0.00
6000	SubsetSelectCIO + warm start	-6.44963 · 10^5	10	3724.51	22.59
6000	L0BnB	-6.57798 · 10^5	10	3317.18	0.00
6000	MSK persp.	-	-	3604.37	-
6000	MSK persp. + warm start	-6.57798 · 10^5	10	3606.26	9.92
6000	Convex MSK opt. persp.	-6.57798 · 10^5	10	1657.28	0.62
6000	ours	-6.57798 · 10^5	10	20.64	0.00
7000	SubsetSelectCIO	-3.48812 · 10^5	1	2.12	0.00
7000	SubsetSelectCIO + warm start	-3.45235 · 10^5	1	2.06	0.00
7000	L0BnB	-7.72266 · 10^5	10	3589.79	0.00
7000	MSK persp.	-	-	3599.94	-
7000	MSK persp. + warm start	-7.72266 · 10^5	10	3600.05	8.29
7000	Convex MSK opt. persp.	-7.72266 · 10^5	10	1603.99	0.45
7000	ours	-7.72266 · 10^5	10	20.91	0.00
Table 143: Comparison of time and optimality gap on the synthetic datasets  $(\mathrm{p} = 3000, \mathrm{k} = 10, \rho = 0.5, \lambda_2 = 0.1)$ .
# of samples	method	upper bound	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-6.90645 · 10^5	10	4096.61	28.35
3000	SubsetSelectCIO + warm start	-7.31489 · 10^5	10	3621.77	21.19
3000	L0BnB	-5.52981 · 10^5	1	7310.65	34.21
3000	MSK persp.	-7.39018 · 10^5	10	3600.41	18.44
3000	MSK persp. + warm start	-7.41186 · 10^5	10	3600.40	18.10
3000	Convex MSK opt. persp.	-7.07443 · 10^5	10	1698.29	23.74
3000	ours	-7.41186 · 10^5	10	3600.59	19.46
4000	SubsetSelectCIO	-9.43345 · 10^5	10	3677.02	28.22
4000	SubsetSelectCIO + warm start	-9.93033 · 10^5	10	3684.08	21.81
4000	L0BnB	-7.53887 · 10^5	1	7333.72	31.99
4000	MSK persp.	-9.98256 · 10^5	10	3600.92	15.46
4000	MSK persp. + warm start	-1.00287 · 10^6	10	3602.18	14.92
4000	Convex MSK opt. persp.	-1.00287 · 10^6	10	1574.91	2.82
4000	ours	-1.00287 · 10^6	10	3600.08	1.88
5000	SubsetSelectCIO	-	-	3600.00	-
5000	SubsetSelectCIO + warm start	-1.22972 · 10^6	10	3997.81	21.00
5000	L0BnB	-1.23555 · 10^6	9	7414.89	5.14
5000	MSK persp.	-1.24027 · 10^6	10	3601.11	11.96
5000	MSK persp. + warm start	-1.24027 · 10^6	10	3601.56	11.96
5000	Convex MSK opt. persp.	-1.24027 · 10^6	10	1638.36	1.06
5000	ours	-1.24027 · 10^6	10	469.43	0.00
6000	SubsetSelectCIO	-9.41611 · 10^5	1	59.04	0.00
6000	SubsetSelectCIO + warm start	-9.78724 · 10^5	1	2.14	0.00
6000	L0BnB	-1.44860 · 10^6	9	7352.27	4.07
6000	MSK persp.	-	-	3602.12	-
6000	MSK persp. + warm start	-1.45416 · 10^6	10	3600.04	9.94
6000	Convex MSK opt. persp.	-1.45416 · 10^6	10	1653.65	0.63
6000	ours	-1.45416 · 10^6	10	59.16	0.00
7000	SubsetSelectCIO	-1.14528 · 10^6	1	2.13	0.00
7000	SubsetSelectCIO + warm start	-1.14528 · 10^6	1	2.11	0.00
7000	L0BnB	-1.69804 · 10^6	9	7503.37	3.09
7000	MSK persp.	-	-	3600.28	-
7000	MSK persp. + warm start	-1.70483 · 10^6	10	3600.18	8.30
7000	Convex MSK opt. persp.	-1.70483 · 10^6	10	1715.12	0.45
7000	ours	-1.70483 · 10^6	10	21.12	0.00
Table 144: Comparison of time and optimality gap on the synthetic datasets  $(\mathrm{p} = 3000, \mathrm{k} = 10, \rho = 0.7, \lambda_2 = 0.1)$ .
# of samples	method	upper bound	support size	time(s)	gap(%)
3000	SubsetSelectCIO	-2.70507 · 106	5	3696.31	23.15
3000	SubsetSelectCIO + warm start	-2.76623 · 106	7	4030.96	20.43
3000	L0BnB	-2.55637 · 106	1	7339.86	18.82
3000	MSK persp.	-2.75384 · 106	10	3600.44	19.50
3000	MSK persp. + warm start	-2.75384 · 106	10	3600.07	19.50
3000	Convex MSK opt. persp.	-2.72647 · 106	10	1999.54	20.67
3000	ours	-2.78521 · 106	10	3600.23	19.46
4000	SubsetSelectCIO	-3.65950 · 106	9	3841.51	23.56
4000	SubsetSelectCIO + warm start	-3.74433 · 106	10	3649.94	20.76
4000	L0BnB	-3.45200 · 106	1	7481.26	15.51
4000	MSK persp.	-3.72627 · 106	10	3601.44	15.63
4000	MSK persp. + warm start	-3.75203 · 106	10	3601.81	14.84
4000	Convex MSK opt. persp.	-3.75202 · 106	10	1543.65	3.23
4000	ours	-3.75203 · 106	10	3601.10	3.50
5000	SubsetSelectCIO	-	-	3600.00	-
5000	SubsetSelectCIO + warm start	-4.61877 · 106	10	3681.64	20.02
5000	L0BnB	-4.23094 · 106	1	7531.59	13.78
5000	MSK persp.	-4.60533 · 106	10	3605.23	12.32
5000	MSK persp. + warm start	-4.61877 · 106	10	3601.43	11.99
5000	Convex MSK opt. persp.	-4.61877 · 106	10	1633.96	1.22
5000	ours	-4.61877 · 106	10	3600.40	0.93
6000	SubsetSelectCIO	-5.38482 · 106	10	3762.10	21.45
6000	SubsetSelectCIO + warm start	-5.42848 · 106	10	3891.96	20.47
6000	L0BnB	-4.97741 · 106	1	7211.19	12.81
6000	MSK persp.	-4.81401 · 106	1	3601.53	24.21
6000	MSK persp. + warm start	-5.43826 · 106	10	3601.18	9.95
6000	Convex MSK opt. persp.	-5.43826 · 106	10	1535.09	0.65
6000	ours	-5.43826 · 106	10	3600.26	0.33
7000	SubsetSelectCIO	-5.66568 · 106	1	2.34	0.00
7000	SubsetSelectCIO + warm start	-5.66568 · 106	1	2.16	0.00
7000	L0BnB	-5.82321 · 106	1	7336.27	11.85
7000	MSK persp.	-	-	3600.04	-
7000	MSK persp. + warm start	-6.36767 · 106	10	3599.94	8.31
7000	Convex MSK opt. persp.	-6.36767 · 106	10	1779.25	0.45
7000	ours	-6.36767 · 106	10	3600.58	0.11
Table 145: Comparison of time and optimality gap on the synthetic datasets  $(\mathrm{p} = 3000, \mathrm{k} = 10, \rho = 0.9, \lambda_2 = 0.1)$ .