SlowGuess/NeurIPS_2023_OCR huggingface dataset:
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-2.52792 · 10^6	2	2.52	0.00
100	SubsetSelectCIO + warm start	-2.52792 · 10^6	2	2.71	0.00
100	LOBnB	-5.28726 · 10^6	10	3182.66	0.00
100	MSK persp.	-5.28726 · 10^6	10	3600.08	0.01
100	MSK persp. + warm start	-5.28726 · 10^6	10	3600.67	0.01
100	Convex MSK opt. persp.	-5.28726 · 10^6	10	0.11	0.00
100	ours	-5.28726 · 10^6	10	0.29	0.00
500	SubsetSelectCIO	-2.52985 · 10^6	2	3.43	0.00
500	SubsetSelectCIO + warm start	-2.52985 · 10^6	2	3.26	0.00
500	LOBnB	-5.31852 · 10^6	10	3977.43	1.65
500	MSK persp.	-5.31852 · 10^6	10	3631.98	0.10
500	MSK persp. + warm start	-5.31852 · 10^6	10	3607.49	0.10
500	Convex MSK opt. persp.	-5.31852 · 10^6	10	7.77	0.01
500	ours	-5.31852 · 10^6	10	0.41	0.00
1000	SubsetSelectCIO	-2.50201 · 10^6	2	4.67	0.00
1000	SubsetSelectCIO + warm start	-2.50201 · 10^6	2	4.99	0.00
1000	LOBnB	-5.26974 · 10^6	10	4224.81	2.82
1000	MSK persp.	-	-	3607.94	-
1000	MSK persp. + warm start	-5.26974 · 10^6	10	3601.79	0.20
1000	Convex MSK opt. persp.	-5.26974 · 10^6	10	54.48	0.01
1000	ours	-5.26974 · 10^6	10	0.52	0.00
3000	SubsetSelectCIO	-2.51861 · 10^6	2	8.73	0.00
3000	SubsetSelectCIO + warm start	-2.51861 · 10^6	2	8.50	0.00
3000	LOBnB	-5.27668 · 10^6	10	4945.04	0.35
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-	-	3600.00	-
3000	Convex MSK opt. persp.	-5.27668 · 10^6	10	1724.71	0.02
3000	ours	-5.27668 · 10^6	10	1.75	0.00
5000	SubsetSelectCIO	-2.49167 · 10^6	2	7.60	0.00
5000	SubsetSelectCIO + warm start	-2.49167 · 10^6	2	13.32	0.00
5000	LOBnB	-5.27126 · 10^6	10	5589.24	0.49
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-5.27126 · 10^6	10	3660.81	0.05
5000	ours	-5.27126 · 10^6	10	4.32	0.00
Table 122: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.3$ ,  $\lambda_2 = 0.001$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-7.42601 · 10^6	2	2.18	0.00
100	SubsetSelectCIO + warm start	-7.42601 · 10^6	2	2.61	0.00
100	L0BnB	-1.10084 · 10^7	10	3464.13	0.00
100	MSK persp.	-1.10084 · 10^7	10	3605.37	0.01
100	MSK persp. + warm start	-1.10084 · 10^7	10	3602.75	0.01
100	Convex MSK opt. persp.	-1.10084 · 10^7	10	0.14	0.00
100	ours	-1.10084 · 10^7	10	0.29	0.00
500	SubsetSelectCIO	-7.42485 · 10^6	2	3.33	0.00
500	SubsetSelectCIO + warm start	-7.42485 · 10^6	2	3.59	0.00
500	L0BnB	-1.10518 · 10^7	10	4143.57	0.77
500	MSK persp.	-1.10518 · 10^7	10	3606.04	0.10
500	MSK persp. + warm start	-1.10518 · 10^7	10	3600.96	0.10
500	Convex MSK opt. persp.	-1.10518 · 10^7	10	7.89	0.01
500	ours	-1.10518 · 10^7	10	0.40	0.00
1000	SubsetSelectCIO	-7.35715 · 10^6	2	3.71	0.00
1000	SubsetSelectCIO + warm start	-7.35715 · 10^6	2	4.39	0.00
1000	L0BnB	-1.09508 · 10^7	10	1931.45	0.65
1000	MSK persp.	-	-	3708.29	-
1000	MSK persp. + warm start	-1.09508 · 10^7	10	3692.66	0.00
1000	Convex MSK opt. persp.	-1.09508 · 10^7	10	56.04	0.01
1000	ours	-1.09508 · 10^7	10	0.59	0.00
3000	SubsetSelectCIO	-7.42176 · 10^6	2	8.32	0.00
3000	SubsetSelectCIO + warm start	-7.42176 · 10^6	2	8.23	0.00
3000	L0BnB	-1.10012 · 10^7	10	5903.24	1.27
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-1.10012 · 10^7	10	4174.40	0.00
3000	Convex MSK opt. persp.	-1.10012 · 10^7	10	1848.58	0.02
3000	ours	-1.10012 · 10^7	10	1.74	0.00
5000	SubsetSelectCIO	-7.37395 · 10^6	2	8.42	0.00
5000	SubsetSelectCIO + warm start	-7.37395 · 10^6	2	11.87	0.00
5000	L0BnB	-1.09966 · 10^7	10	6029.87	1.11
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-1.09966 · 10^7	10	6700.63	0.00
5000	Convex MSK opt. persp.	-1.09966 · 10^7	10	3657.70	0.06
5000	ours	-1.09966 · 10^7	10	4.00	0.00
Table 123: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.5$ ,  $\lambda_2 = 0.001$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-2.01567 · 10^7	2	2.50	0.00
100	SubsetSelectCIO + warm start	-2.01567 · 10^7	2	2.64	0.00
100	L0BnB	-2.43575 · 10^7	10	2716.56	0.00
100	MSK persp.	-2.43575 · 10^7	10	3602.86	0.01
100	MSK persp. + warm start	-2.43575 · 10^7	10	3600.41	0.01
100	Convex MSK opt. persp.	-2.43575 · 10^7	10	0.12	0.00
100	ours	-2.43575 · 10^7	10	0.32	0.00
500	SubsetSelectCIO	-2.01420 · 10^7	2	3.26	0.00
500	SubsetSelectCIO + warm start	-2.01420 · 10^7	2	3.40	0.00
500	L0BnB	-2.44185 · 10^7	10	3747.19	0.00
500	MSK persp.	-2.24367 · 10^7	10	3608.22	8.94
500	MSK persp. + warm start	-2.44185 · 10^7	10	3623.31	0.10
500	Convex MSK opt. persp.	-2.44185 · 10^7	10	8.07	0.01
500	ours	-2.44185 · 10^7	10	0.46	0.00
1000	SubsetSelectCIO	-1.99762 · 10^7	2	3.67	0.00
1000	SubsetSelectCIO + warm start	-1.99762 · 10^7	2	4.71	0.00
1000	L0BnB	-2.41994 · 10^7	11	4179.51	0.01
1000	MSK persp.	-	-	3620.44	-
1000	MSK persp. + warm start	-2.41989 · 10^7	10	3686.70	0.20
1000	Convex MSK opt. persp.	-2.41989 · 10^7	10	57.29	0.01
1000	ours	-2.41989 · 10^7	10	0.52	0.00
3000	SubsetSelectCIO	-2.01741 · 10^7	2	9.04	0.00
3000	SubsetSelectCIO + warm start	-2.01741 · 10^7	2	9.24	0.00
3000	L0BnB	-2.42599 · 10^7	9	7445.35	0.00
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-2.43713 · 10^7	10	3888.61	0.00
3000	Convex MSK opt. persp.	-2.43713 · 10^7	10	1929.94	0.02
3000	ours	-2.43713 · 10^7	10	1.77	0.00
5000	SubsetSelectCIO	-2.00985 · 10^7	2	13.61	0.00
5000	SubsetSelectCIO + warm start	-2.00985 · 10^7	2	12.21	0.00
5000	L0BnB	-2.43694 · 10^7	10	4485.99	23.68
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-2.43694 · 10^7	10	3748.60	0.06
5000	ours	-2.43694 · 10^7	10	4.15	0.00
Table 124: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.7$ ,  $\lambda_2 = 0.001$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-8.64519 · 107	2	2.27	0.00
100	SubsetSelectCIO + warm start	-8.64519 · 107	2	2.79	0.00
100	L0BnB	-9.11185 · 107	93	6954.00	0.04
100	MSK persp.	-8.96852 · 107	10	3601.43	1.59
100	MSK persp. + warm start	-9.11026 · 107	10	3601.28	0.01
100	Convex MSK opt. persp.	-9.11026 · 107	10	0.13	0.00
100	ours	-9.11026 · 107	10	0.30	0.00
500	SubsetSelectCIO	-8.63791 · 107	2	2.89	0.00
500	SubsetSelectCIO + warm start	-8.63791 · 107	2	3.47	0.00
500	L0BnB	-9.10885 · 107	9	7562.05	0.14
500	MSK persp.	-	-	3605.36	-
500	MSK persp. + warm start	-9.12047 · 107	10	3630.75	0.10
500	Convex MSK opt. persp.	-9.12047 · 107	10	7.26	0.01
500	ours	-9.12047 · 107	10	0.38	0.00
1000	SubsetSelectCIO	-8.56943 · 107	2	4.58	0.00
1000	SubsetSelectCIO + warm start	-8.56943 · 107	2	4.73	0.00
1000	L0BnB	-8.23906 · 107	1	7300.29	9.17
1000	MSK persp.	-	-	3646.29	-
1000	MSK persp. + warm start	-9.04075 · 107	10	3613.37	0.00
1000	Convex MSK opt. persp.	-9.04075 · 107	10	67.12	0.01
1000	ours	-9.04075 · 107	10	0.52	0.00
3000	SubsetSelectCIO	-8.66243 · 107	2	8.50	0.00
3000	SubsetSelectCIO + warm start	-8.66243 · 107	2	8.12	0.00
3000	L0BnB	-9.12781 · 107	12	1706.18	7.80
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-9.12761 · 107	10	3998.60	0.00
3000	Convex MSK opt. persp.	-9.12761 · 107	10	2038.29	0.02
3000	ours	-9.12761 · 107	10	20.74	0.00
5000	SubsetSelectCIO	-8.64926 · 107	2	12.63	0.00
5000	SubsetSelectCIO + warm start	-8.64926 · 107	2	11.94	0.00
5000	L0BnB	-8.32072 · 107	1	7472.42	9.32
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-9.12916 · 107	10	3722.78	3.21
5000	ours	-9.12916 · 107	10	58.29	0.00
Table 125: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.9$ ,  $\lambda_2 = 0.001$ ).
G.4.2 Synthetic 1 Benchmark with  $\lambda_{2} = 0.1$
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-	-	3600.00	-
100	SubsetSelectCIO + warm start	-	-	3600.00	-
100	L0BnB	-2.10878 · 10^6	10	1415.23	0.00
100	MSK persp.	-2.10878 · 10^6	10	928.86	0.00
100	MSK persp. + warm start	-2.10878 · 10^6	10	912.78	0.00
100	Convex MSK opt. persp.	-2.10878 · 10^6	10	0.10	0.00
100	ours	-2.10878 · 10^6	10	0.36	0.00
500	SubsetSelectCIO	-1.19239 · 10^5	1	3.17	0.00
500	SubsetSelectCIO + warm start	-1.19239 · 10^5	1	3.39	0.00
500	L0BnB	-2.12833 · 10^6	10	1973.90	0.00
500	MSK persp.	-2.12830 · 10^6	10	3606.82	0.10
500	MSK persp. + warm start	-2.12833 · 10^6	10	3604.01	0.10
500	Convex MSK opt. persp.	-2.12833 · 10^6	10	6.49	0.01
500	ours	-2.12833 · 10^6	10	0.37	0.00
1000	SubsetSelectCIO	-4.71433 · 10^5	2	4.47	0.00
1000	SubsetSelectCIO + warm start	-4.71433 · 10^5	2	4.68	0.00
1000	L0BnB	-2.11005 · 10^6	10	2101.63	0.00
1000	MSK persp.	-	-	3602.15	-
1000	MSK persp. + warm start	-2.11005 · 10^6	10	3600.98	0.20
1000	Convex MSK opt. persp.	-2.11005 · 10^6	10	58.36	0.01
1000	ours	-2.11005 · 10^6	10	0.57	0.00
3000	SubsetSelectCIO	-4.72135 · 10^5	2	8.81	0.00
3000	SubsetSelectCIO + warm start	-4.72135 · 10^5	2	8.98	0.00
3000	L0BnB	-2.10230 · 10^6	10	2515.21	0.00
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-2.10230 · 10^6	10	3660.82	0.00
3000	Convex MSK opt. persp.	-2.10230 · 10^6	10	1765.70	0.02
3000	ours	-2.10230 · 10^6	10	1.79	0.00
5000	SubsetSelectCIO	-4.62647 · 10^5	2	12.26	0.00
5000	SubsetSelectCIO + warm start	-4.62647 · 10^5	2	12.06	0.00
5000	L0BnB	-2.09635 · 10^6	10	2666.21	3.42
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-2.09635 · 10^6	10	3685.08	0.06
5000	ours	-2.09635 · 10^6	10	4.20	0.00
Table 126: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.1$ ,  $\lambda_2 = 0.1$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-2.52792 · 10^6	2	2.32	0.00
100	SubsetSelectCIO + warm start	-2.52792 · 10^6	2	2.46	0.00
100	L0BnB	-5.28726 · 10^6	10	3182.16	0.00
100	MSK persp.	-5.28724 · 10^6	10	3601.53	0.01
100	MSK persp. + warm start	-5.28726 · 10^6	10	3600.02	0.01
100	Convex MSK opt. persp.	-5.28726 · 10^6	10	0.10	0.00
100	ours	-5.28726 · 10^6	10	0.27	0.00
500	SubsetSelectCIO	-2.52985 · 10^6	2	3.54	0.00
500	SubsetSelectCIO + warm start	-2.52985 · 10^6	2	3.26	0.00
500	L0BnB	-5.31852 · 10^6	10	3918.77	1.65
500	MSK persp.	-5.31852 · 10^6	10	3601.88	0.10
500	MSK persp. + warm start	-5.31852 · 10^6	10	3602.35	0.10
500	Convex MSK opt. persp.	-5.31852 · 10^6	10	8.26	0.01
500	ours	-5.31852 · 10^6	10	0.41	0.00
1000	SubsetSelectCIO	-2.50201 · 10^6	2	4.49	0.00
1000	SubsetSelectCIO + warm start	-2.50201 · 10^6	2	4.12	0.00
1000	L0BnB	-5.26974 · 10^6	10	4052.64	1.67
1000	MSK persp.	-	-	3600.90	-
1000	MSK persp. + warm start	-5.26974 · 10^6	10	3600.10	0.20
1000	Convex MSK opt. persp.	-5.26974 · 10^6	10	55.50	0.01
1000	ours	-5.26974 · 10^6	10	0.52	0.00
3000	SubsetSelectCIO	-2.51861 · 10^6	2	5.94	0.00
3000	SubsetSelectCIO + warm start	-2.51861 · 10^6	2	8.61	0.00
3000	L0BnB	-5.27668 · 10^6	10	4807.82	2.18
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-5.27668 · 10^6	10	3744.12	0.00
3000	Convex MSK opt. persp.	-5.27668 · 10^6	10	1795.54	0.02
3000	ours	-5.27668 · 10^6	10	1.84	0.00
5000	SubsetSelectCIO	-2.49167 · 10^6	2	13.36	0.00
5000	SubsetSelectCIO + warm start	-2.49167 · 10^6	2	12.61	0.00
5000	L0BnB	-5.27126 · 10^6	10	5667.77	0.00
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-5.27126 · 10^6	10	3606.96	0.05
5000	ours	-5.27126 · 10^6	10	4.42	0.00
Table 127: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.3$ ,  $\lambda_2 = 0.1$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-7.42601 · 10^6	2	2.59	0.00
100	SubsetSelectCIO + warm start	-7.42601 · 10^6	2	2.80	0.00
100	L0BnB	-1.10084 · 10^7	10	3398.37	0.00
100	MSK persp.	-1.09018 · 10^7	10	3599.98	0.99
100	MSK persp. + warm start	-1.10084 · 10^7	10	3603.80	0.01
100	Convex MSK opt. persp.	-1.10084 · 10^7	10	0.11	0.00
100	ours	-1.10084 · 10^7	10	0.32	0.00
500	SubsetSelectCIO	-7.42485 · 10^6	2	2.97	0.00
500	SubsetSelectCIO + warm start	-7.42485 · 10^6	2	3.22	0.00
500	L0BnB	-1.10518 · 10^7	10	4364.09	0.77
500	MSK persp.	-	-	3605.36	-
500	MSK persp. + warm start	-1.10518 · 10^7	10	3610.84	0.00
500	Convex MSK opt. persp.	-1.10518 · 10^7	10	7.25	0.01
500	ours	-1.10518 · 10^7	10	0.40	0.00
1000	SubsetSelectCIO	-7.35715 · 10^6	2	4.35	0.00
1000	SubsetSelectCIO + warm start	-7.35715 · 10^6	2	4.97	0.00
1000	L0BnB	-1.09508 · 10^7	10	1882.56	0.65
1000	MSK persp.	-	-	3609.88	-
1000	MSK persp. + warm start	-1.09508 · 10^7	10	3616.06	0.00
1000	Convex MSK opt. persp.	-1.09508 · 10^7	10	63.89	0.01
1000	ours	-1.09508 · 10^7	10	0.53	0.00
3000	SubsetSelectCIO	-7.42176 · 10^6	2	5.59	0.00
3000	SubsetSelectCIO + warm start	-7.42176 · 10^6	2	8.42	0.00
3000	L0BnB	-1.10012 · 10^7	10	5588.51	1.24
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-1.10012 · 10^7	10	4173.41	0.00
3000	Convex MSK opt. persp.	-1.10012 · 10^7	10	1779.87	0.02
3000	ours	-1.10012 · 10^7	10	1.67	0.00
5000	SubsetSelectCIO	-7.37395 · 10^6	2	8.72	0.00
5000	SubsetSelectCIO + warm start	-7.37395 · 10^6	2	12.24	0.00
5000	L0BnB	-1.09966 · 10^7	10	6428.17	1.09
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-1.09966 · 10^7	10	6106.85	0.00
5000	Convex MSK opt. persp.	-1.09966 · 10^7	10	3665.75	0.06
5000	ours	-1.09966 · 10^7	10	4.13	0.00
Table 128: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.5$ ,  $\lambda_2 = 0.1$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-2.01567 · 10^7	2	2.23	0.00
100	SubsetSelectCIO + warm start	-2.01567 · 10^7	2	3.02	0.00
100	L0BnB	-2.43575 · 10^7	10	2673.71	0.00
100	MSK persp.	-2.41233 · 10^7	10	3601.54	0.98
100	MSK persp. + warm start	-2.43575 · 10^7	10	3600.03	0.01
100	Convex MSK opt. persp.	-2.43575 · 10^7	10	0.14	0.00
100	ours	-2.43575 · 10^7	10	0.34	0.00
500	SubsetSelectCIO	-2.01420 · 10^7	2	3.58	0.00
500	SubsetSelectCIO + warm start	-2.01420 · 10^7	2	3.34	0.00
500	L0BnB	-2.44185 · 10^7	10	3602.76	0.00
500	MSK persp.	-2.44185 · 10^7	10	3603.68	0.10
500	MSK persp. + warm start	-2.44185 · 10^7	10	3604.92	0.10
500	Convex MSK opt. persp.	-2.44185 · 10^7	10	7.37	0.01
500	ours	-2.44185 · 10^7	10	0.44	0.00
1000	SubsetSelectCIO	-1.99762 · 10^7	2	4.47	0.00
1000	SubsetSelectCIO + warm start	-1.99762 · 10^7	2	4.56	0.00
1000	L0BnB	-2.41994 · 10^7	11	4204.09	0.01
1000	MSK persp.	-	-	3610.81	-
1000	MSK persp. + warm start	-2.41989 · 10^7	10	3609.15	0.00
1000	Convex MSK opt. persp.	-2.41989 · 10^7	10	56.82	0.01
1000	ours	-2.41989 · 10^7	10	0.53	0.00
3000	SubsetSelectCIO	-2.01741 · 10^7	2	5.42	0.00
3000	SubsetSelectCIO + warm start	-2.01741 · 10^7	2	8.40	0.00
3000	L0BnB	-2.42599 · 10^7	9	7207.55	0.00
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-2.43713 · 10^7	10	3980.12	0.00
3000	Convex MSK opt. persp.	-2.43713 · 10^7	10	1799.08	0.02
3000	ours	-2.43713 · 10^7	10	1.75	0.00
5000	SubsetSelectCIO	-2.00985 · 10^7	2	12.64	0.00
5000	SubsetSelectCIO + warm start	-2.00985 · 10^7	2	13.96	0.00
5000	L0BnB	-2.43694 · 10^7	10	4979.94	21.63
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-2.43694 · 10^7	10	3606.45	0.06
5000	ours	-2.43694 · 10^7	10	4.22	0.00
Table 129: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.7$ ,  $\lambda_2 = 0.1$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-8.11196 · 107	1	2.42	0.00
100	SubsetSelectCIO + warm start	-8.11196 · 107	1	2.56	0.00
100	L0BnB	-9.11185 · 107	93	6919.82	0.04
100	MSK persp.	-8.94241 · 107	10	3600.44	1.89
100	MSK persp. + warm start	-9.11026 · 107	10	3600.46	0.01
100	Convex MSK opt. persp.	-9.11026 · 107	10	0.18	0.00
100	ours	-9.11026 · 107	10	0.31	0.00
500	SubsetSelectCIO	-8.63791 · 107	2	3.47	0.00
500	SubsetSelectCIO + warm start	-8.63791 · 107	2	3.56	0.00
500	L0BnB	-9.10885 · 107	9	7359.92	0.14
500	MSK persp.	-	-	3601.63	-
500	MSK persp. + warm start	-9.12047 · 107	10	3602.72	0.00
500	Convex MSK opt. persp.	-9.12047 · 107	10	8.27	0.01
500	ours	-9.12047 · 107	10	0.38	0.00
1000	SubsetSelectCIO	-8.56943 · 107	2	4.83	0.00
1000	SubsetSelectCIO + warm start	-8.56943 · 107	2	4.74	0.00
1000	L0BnB	-8.23906 · 107	1	7326.90	9.06
1000	MSK persp.	-	-	3641.35	-
1000	MSK persp. + warm start	-9.04075 · 107	10	3611.36	0.00
1000	Convex MSK opt. persp.	-9.04075 · 107	10	61.25	0.01
1000	ours	-9.04075 · 107	10	0.52	0.00
3000	SubsetSelectCIO	-8.66243 · 107	2	8.37	0.00
3000	SubsetSelectCIO + warm start	-8.66243 · 107	2	9.68	0.00
3000	L0BnB	-9.12781 · 107	12	2205.35	5.43
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-9.12761 · 107	10	3661.26	0.00
3000	Convex MSK opt. persp.	-9.12761 · 107	10	2087.06	0.02
3000	ours	-9.12761 · 107	10	20.88	0.00
5000	SubsetSelectCIO	-8.64926 · 107	2	12.48	0.00
5000	SubsetSelectCIO + warm start	-8.64926 · 107	2	12.73	0.00
5000	L0BnB	-8.32072 · 107	1	7201.18	9.32
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-9.12916 · 107	10	8314.78	0.00
5000	Convex MSK opt. persp.	-9.12916 · 107	10	3691.61	3.40
5000	ours	-9.12916 · 107	10	58.07	0.00
Table 130: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.9$ ,  $\lambda_2 = 0.1$ ).
G.4.3 Synthetic 1 Benchmark with  $\lambda_{2} = 10.0$
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-1.62628 · 10^6	10	3693.06	55.44
100	SubsetSelectCIO + warm start	-1.62806 · 10^6	10	3672.85	55.27
100	L0BnB	-2.10868 · 10^6	10	1432.58	0.00
100	MSK persp.	-2.10868 · 10^6	10	926.33	0.00
100	MSK persp. + warm start	-2.10868 · 10^6	10	919.61	0.00
100	Convex MSK opt. persp.	-2.10868 · 10^6	10	0.10	0.00
100	ours	-2.10868 · 10^6	10	0.30	0.00
500	SubsetSelectCIO	-1.33331 · 10^6	10	3776.44	91.62
500	SubsetSelectCIO + warm start	-4.87330 · 10^5	7	1654.44	0.00
500	L0BnB	-2.12823 · 10^6	10	1929.01	0.00
500	MSK persp.	-2.12822 · 10^6	10	3601.80	0.09
500	MSK persp. + warm start	-2.12823 · 10^6	10	3609.69	0.09
500	Convex MSK opt. persp.	-2.12823 · 10^6	10	7.05	0.01
500	ours	-2.12823 · 10^6	10	0.38	0.00
1000	SubsetSelectCIO	-7.92745 · 10^5	10	3709.35	219.40
1000	SubsetSelectCIO + warm start	-1.33080 · 10^6	10	3702.48	90.26
1000	L0BnB	-2.10995 · 10^6	10	2118.04	0.00
1000	MSK persp.	-	-	3601.31	-
1000	MSK persp. + warm start	-2.10995 · 10^6	10	3600.48	0.20
1000	Convex MSK opt. persp.	-2.10995 · 10^6	10	54.57	0.01
1000	ours	-2.10995 · 10^6	10	0.57	0.00
3000	SubsetSelectCIO	-7.89940 · 10^5	10	3633.47	219.21
3000	SubsetSelectCIO + warm start	-1.31406 · 10^6	10	4260.43	91.89
3000	L0BnB	-2.10220 · 10^6	10	2318.67	4.73
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-	-	3600.00	-
3000	Convex MSK opt. persp.	-2.10220 · 10^6	10	1669.36	0.02
3000	ours	-2.10220 · 10^6	10	1.68	0.00
5000	SubsetSelectCIO	-9.76808 · 10^5	10	3838.13	157.56
5000	SubsetSelectCIO + warm start	-1.30652 · 10^6	10	4063.53	92.56
5000	L0BnB	-2.09625 · 10^6	10	2700.89	3.42
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-2.09625 · 10^6	10	3669.10	0.06
5000	ours	-2.09625 · 10^6	10	5.09	0.00
Table 131: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.1$ ,  $\lambda_2 = 10.0$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-4.75666 · 10^6	10	3695.63	33.24
100	SubsetSelectCIO + warm start	-3.72965 · 10^6	10	1008.71	0.00
100	L0BnB	-5.28716 · 10^6	10	3186.48	0.00
100	MSK persp.	-5.28716 · 10^6	10	3600.15	0.01
100	MSK persp. + warm start	-5.28716 · 10^6	10	3600.07	0.01
100	Convex MSK opt. persp.	-5.28716 · 10^6	10	0.16	0.00
100	ours	-5.28716 · 10^6	10	0.30	0.00
500	SubsetSelectCIO	-4.61560 · 10^6	10	3761.41	38.34
500	SubsetSelectCIO + warm start	-4.00602 · 10^6	10	3621.18	59.39
500	L0BnB	-5.31842 · 10^6	10	3958.53	1.65
500	MSK persp.	-5.31842 · 10^6	10	3604.61	0.09
500	MSK persp. + warm start	-5.31842 · 10^6	10	3604.65	0.09
500	Convex MSK opt. persp.	-5.31842 · 10^6	10	7.05	0.01
500	ours	-5.31842 · 10^6	10	0.41	0.00
1000	SubsetSelectCIO	-3.05644 · 10^6	6	3348.60	0.00
1000	SubsetSelectCIO + warm start	-3.95760 · 10^6	10	3658.16	59.79
1000	L0BnB	-5.26964 · 10^6	10	4043.93	1.62
1000	MSK persp.	-	-	3600.44	-
1000	MSK persp. + warm start	-5.26964 · 10^6	10	3600.14	0.20
1000	Convex MSK opt. persp.	-5.26964 · 10^6	10	54.61	0.01
1000	ours	-5.26964 · 10^6	10	0.51	0.00
3000	SubsetSelectCIO	-4.74347 · 10^6	10	3927.16	33.42
3000	SubsetSelectCIO + warm start	-	-	3600.00	-
3000	L0BnB	-5.27658 · 10^6	10	4906.10	0.35
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-	-	3600.00	-
3000	Convex MSK opt. persp.	-5.27658 · 10^6	10	1962.60	0.02
3000	ours	-5.27658 · 10^6	10	1.76	0.00
5000	SubsetSelectCIO	-	-	3600.00	-
5000	SubsetSelectCIO + warm start	-	-	3600.00	-
5000	L0BnB	-	-	7200.00	-
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-5.27116 · 10^6	10	3651.58	0.05
5000	ours	-5.27116 · 10^6	10	4.32	0.00
Table 132: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.3$ ,  $\lambda_2 = 10.0$ ).
# of features	method	upper bound	support size	time(s)	gap(%)
100	SubsetSelectCIO	-1.00851 · 107	10	3728.97	30.84
100	SubsetSelectCIO + warm start	-1.02880 · 107	10	3654.11	28.26
100	L0BnB	-1.10083 · 107	10	3417.76	0.26
100	MSK persp.	-1.10083 · 107	10	1286.52	0.00
100	MSK persp. + warm start	-1.10083 · 107	10	1786.69	0.00
100	Convex MSK opt. persp.	-1.10083 · 107	10	0.16	0.00
100	ours	-1.10083 · 107	10	0.32	0.00
500	SubsetSelectCIO	-9.42612 · 106	10	3685.22	40.77
500	SubsetSelectCIO + warm start	-9.92278 · 106	10	3685.15	33.72
500	L0BnB	-1.10517 · 107	10	4308.04	0.77
500	MSK persp.	-1.10517 · 107	10	3601.09	0.09
500	MSK persp. + warm start	-1.10517 · 107	10	3602.10	0.09
500	Convex MSK opt. persp.	-1.10517 · 107	10	7.72	0.01
500	ours	-1.10517 · 107	10	0.41	0.00
1000	SubsetSelectCIO	-1.00437 · 107	10	3649.62	30.84
1000	SubsetSelectCIO + warm start	-6.62765 · 106	2	3147.50	0.00
1000	L0BnB	-1.09507 · 107	10	1834.54	0.65
1000	MSK persp.	-	-	3600.46	-
1000	MSK persp. + warm start	-1.09507 · 107	10	3601.83	0.20
1000	Convex MSK opt. persp.	-1.09507 · 107	10	57.71	0.01
1000	ours	-1.09507 · 107	10	0.55	0.00
3000	SubsetSelectCIO	-	-	3600.00	-
3000	SubsetSelectCIO + warm start	-	-	3600.00	-
3000	L0BnB	-1.10011 · 107	10	5263.44	1.10
3000	MSK persp.	-	-	3600.00	-
3000	MSK persp. + warm start	-1.10011 · 107	10	4076.24	0.00
3000	Convex MSK opt. persp.	-1.10011 · 107	10	1538.48	0.02
3000	ours	-1.10011 · 107	10	1.73	0.00
5000	SubsetSelectCIO	-	-	3600.00	-
5000	SubsetSelectCIO + warm start	-	-	3600.00	-
5000	L0BnB	-1.09965 · 107	10	6222.74	1.11
5000	MSK persp.	-	-	3600.00	-
5000	MSK persp. + warm start	-	-	3600.00	-
5000	Convex MSK opt. persp.	-1.09965 · 107	10	3742.63	0.05
5000	ours	-1.09965 · 107	10	4.24	0.00
Table 133: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.5$ ,  $\lambda_2 = 10.0$ ).