SlowGuess/NeurIPS_2023_OCR huggingface dataset:
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-2.10878 · 10^6	0.05	0.00
100	big-M50 + warm start	-2.10878 · 10^6	0.08	0.00
100	persp. + warm start	-2.10878 · 10^6	0.08	0.00
100	eig. persp. + warm start	-2.10878 · 10^6	0.01	0.00
100	ours	-2.10878 · 10^6	0.29	0.00
500	SOS1 + warm start	-2.12833 · 10^6	6.31	0.00
500	big-M50 + warm start	-2.12833 · 10^6	37.62	0.00
500	persp. + warm start	-2.12833 · 10^6	19.60	0.00
500	eig. persp. + warm start	-2.12833 · 10^6	0.61	0.00
500	ours	-2.12833 · 10^6	0.37	0.00
1000	SOS1 + warm start	-2.11005 · 10^6	77.82	0.00
1000	big-M50 + warm start	-2.11005 · 10^6	28.51	0.00
1000	persp. + warm start	-2.11005 · 10^6	1148.21	0.00
1000	eig. persp. + warm start	-2.11005 · 10^6	4.08	0.00
1000	ours	-2.11005 · 10^6	0.58	0.00
3000	SOS1 + warm start	-2.10230 · 10^6	2100.58	0.00
3000	big-M50 + warm start	-2.10230 · 10^6	407.92	0.00
3000	persp. + warm start	-2.10231 · 10^6	2275.23	0.00
3000	eig. persp. + warm start	-2.10230 · 10^6	102.19	0.00
3000	ours	-2.10230 · 10^6	1.73	0.00
5000	SOS1 + warm start	-2.09635 · 10^6	3601.23	0.95
5000	big-M50 + warm start	-2.09635 · 10^6	1380.38	0.00
5000	persp. + warm start	-2.09637 · 10^6	3600.68	1.00
5000	eig. persp. + warm start	-2.09635 · 10^6	483.38	0.00
5000	ours	-2.09635 · 10^6	4.22	0.00
Table 6: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.1$ ,  $\lambda_2 = 0.001$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-5.28726 · 10^6	0.05	0.00
100	big-M50 + warm start	-5.28726 · 10^6	0.26	0.00
100	persp. + warm start	-5.28726 · 10^6	0.07	0.00
100	eig. persp. + warm start	-5.28726 · 10^6	0.02	0.00
100	ours	-5.28726 · 10^6	0.29	0.00
500	SOS1 + warm start	-5.31852 · 10^6	9.61	0.00
500	big-M50 + warm start	-5.31852 · 10^6	3.00	0.00
500	persp. + warm start	-5.31852 · 10^6	8.06	0.00
500	eig. persp. + warm start	-5.31852 · 10^6	0.63	0.00
500	ours	-5.31852 · 10^6	0.41	0.00
1000	SOS1 + warm start	-5.26974 · 10^6	88.36	0.00
1000	big-M50 + warm start	-5.26974 · 10^6	13.79	0.00
1000	persp. + warm start	-5.26974 · 10^6	54.26	0.00
1000	eig. persp. + warm start	-5.26974 · 10^6	4.18	0.00
1000	ours	-5.26974 · 10^6	0.52	0.00
3000	SOS1 + warm start	-5.27668 · 10^6	2740.42	0.00
3000	big-M50 + warm start	-5.27668 · 10^6	313.22	0.00
3000	persp. + warm start	-5.27668 · 10^6	144.09	0.00
3000	eig. persp. + warm start	-5.27668 · 10^6	101.95	0.00
3000	ours	-5.27668 · 10^6	1.75	0.00
5000	SOS1 + warm start	-5.27126 · 10^6	3600.51	1.00
5000	big-M50 + warm start	-5.27126 · 10^6	1705.37	0.00
5000	persp. + warm start	-5.27126 · 10^6	543.39	0.00
5000	eig. persp. + warm start	-5.27126 · 10^6	485.74	0.00
5000	ours	-5.27126 · 10^6	4.32	0.00
Table 7: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.3$ ,  $\lambda_2 = 0.001$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-1.10084 · 107	0.05	0.00
100	big-M50 + warm start	-1.10084 · 107	0.06	0.00
100	persp. + warm start	-1.10084 · 107	0.09	0.00
100	eig. persp. + warm start	-1.10084 · 107	0.02	0.00
100	ours	-1.10084 · 107	0.29	0.00
500	SOS1 + warm start	-1.10518 · 107	7.68	0.00
500	big-M50 + warm start	-1.10518 · 107	11.59	0.00
500	persp. + warm start	-1.10518 · 107	3600.02	0.08
500	eig. persp. + warm start	-1.10518 · 107	0.63	0.00
500	ours	-1.10518 · 107	0.40	0.00
1000	SOS1 + warm start	-1.09508 · 107	90.16	0.00
1000	big-M50 + warm start	-1.09508 · 107	21.16	0.00
1000	persp. + warm start	-1.09508 · 107	9.85	0.00
1000	eig. persp. + warm start	-1.09508 · 107	4.04	0.00
1000	ours	-1.09508 · 107	0.59	0.00
3000	SOS1 + warm start	-1.10012 · 107	3044.33	0.00
3000	big-M50 + warm start	-1.10012 · 107	243.42	0.00
3000	persp. + warm start	-1.10012 · 107	141.68	0.00
3000	eig. persp. + warm start	-1.10012 · 107	100.50	0.00
3000	ours	-1.10012 · 107	1.74	0.00
5000	SOS1 + warm start	-1.09966 · 107	3603.69	0.99
5000	big-M50 + warm start	-1.09966 · 107	3600.95	1.00
5000	persp. + warm start	-1.09966 · 107	3603.33	1.00
5000	eig. persp. + warm start	-1.09966 · 107	445.26	0.00
5000	ours	-1.09966 · 107	4.00	0.00
Table 8: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.5, λ2 = 0.001). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-2.43575 · 10^7	0.05	0.00
100	big-M50 + warm start	-2.43575 · 10^7	0.05	0.00
100	persp. + warm start	-2.43575 · 10^7	0.19	0.00
100	eig. persp. + warm start	-2.43575 · 10^7	0.02	0.00
100	ours	-2.43575 · 10^7	0.32	0.00
500	SOS1 + warm start	-2.44185 · 10^7	8.94	0.00
500	big-M50 + warm start	-2.44185 · 10^7	4.02	0.00
500	persp. + warm start	-2.44185 · 10^7	271.52	0.00
500	eig. persp. + warm start	-2.44185 · 10^7	0.60	0.00
500	ours	-2.44185 · 10^7	0.46	0.00
1000	SOS1 + warm start	-2.41989 · 10^7	83.96	0.00
1000	big-M50 + warm start	-2.41989 · 10^7	11.63	0.00
1000	persp. + warm start	-2.41989 · 10^7	3600.13	0.20
1000	eig. persp. + warm start	-2.41989 · 10^7	4.02	0.00
1000	ours	-2.41989 · 10^7	0.52	0.00
3000	SOS1 + warm start	-2.43713 · 10^7	3601.04	0.59
3000	big-M50 + warm start	-2.43713 · 10^7	3608.31	0.59
3000	persp. + warm start	-2.43713 · 10^7	3600.34	0.59
3000	eig. persp. + warm start	-2.43713 · 10^7	110.94	0.00
3000	ours	-2.43713 · 10^7	1.77	0.00
5000	SOS1 + warm start	-2.43694 · 10^7	3600.18	1.00
5000	big-M50 + warm start	-2.43694 · 10^7	3670.94	1.00
5000	persp. + warm start	-2.43694 · 10^7	3600.18	1.00
5000	eig. persp. + warm start	-2.43694 · 10^7	442.81	0.00
5000	ours	-2.43694 · 10^7	4.15	0.00
Table 9: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.7, λ2 = 0.001). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-9.11026 · 107	0.04	0.00
100	big-M50 + warm start	-9.11026 · 107	0.03	0.00
100	persp. + warm start	-9.11026 · 107	0.26	0.00
100	eig. persp. + warm start	-9.11026 · 107	0.02	0.00
100	ours	-9.11026 · 107	0.30	0.00
500	SOS1 + warm start	-9.12047 · 107	48.24	0.00
500	big-M50 + warm start	-9.12047 · 107	34.86	0.00
500	persp. + warm start	-9.12047 · 107	3600.02	0.09
500	eig. persp. + warm start	-9.12047 · 107	0.59	0.00
500	ours	-9.12047 · 107	0.38	0.00
1000	SOS1 + warm start	-9.04075 · 107	1090.45	0.00
1000	big-M50 + warm start	-9.04075 · 107	3600.01	0.08
1000	persp. + warm start	-9.04075 · 107	3600.08	0.20
1000	eig. persp. + warm start	-9.04075 · 107	4.15	0.00
1000	ours	-9.04075 · 107	0.52	0.00
3000	SOS1 + warm start	-9.12761 · 107	3600.07	0.59
3000	big-M50 + warm start	-9.12761 · 107	3612.88	0.59
3000	persp. + warm start	-9.12761 · 107	3600.07	0.59
3000	eig. persp. + warm start	-9.12761 · 107	100.26	0.00
3000	ours	-9.12761 · 107	20.74	0.00
5000	SOS1 + warm start	-9.12916 · 107	3601.57	1.00
5000	big-M50 + warm start	-9.12916 · 107	3622.52	1.00
5000	persp. + warm start	-9.12916 · 107	3602.87	1.00
5000	eig. persp. + warm start	-9.12916 · 107	439.82	0.00
5000	ours	-9.12916 · 107	58.29	0.00
Table 10: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.9$ ,  $\lambda_2 = 0.001$ ). All baselines use our beam search solution as a warm start.
G.2.3 Synthetic 1 Benchmark with  $\lambda_{2} = 0.1$  with No Warmstart
# of features	method	upper bound	time(s)	gap(%)
100	SOS1	-2.10878 · 10^6	3.04	0.00
100	big-M50	-2.10878 · 10^6	0.86	0.00
100	persp.	-2.10878 · 10^6	0.07	0.00
100	eig. persp.	-2.10878 · 10^6	0.02	0.00
100	ours	-2.10878 · 10^6	0.36	0.00
500	SOS1	-2.12833 · 10^6	8.13	0.00
500	big-M50	-2.12833 · 10^6	37.96	0.00
500	persp.	-2.12833 · 10^6	3.87	0.00
500	eig. persp.	-2.12833 · 10^6	0.63	0.00
500	ours	-2.12833 · 10^6	0.37	0.00
1000	SOS1	-2.11005 · 10^6	81.40	0.00
1000	big-M50	-2.11005 · 10^6	1007.27	0.00
1000	persp.	-2.11005 · 10^6	1195.64	0.00
1000	eig. persp.	-2.11005 · 10^6	4.07	0.00
1000	ours	-2.11005 · 10^6	0.57	0.00
3000	SOS1	-2.10230 · 10^6	3601.25	0.59
3000	big-M50	-7.99504 · 10^5	3608.36	164.51
3000	persp.	-2.10230 · 10^6	2055.12	0.00
3000	eig. persp.	-2.10230 · 10^6	103.29	0.00
3000	ours	-2.10230 · 10^6	1.79	0.00
5000	SOS1	-1.98939 · 10^6	3600.15	6.43
5000	big-M50	-9.69753 · 10^5	3619.50	118.35
5000	persp.	-2.09635 · 10^6	3600.72	1.00
5000	eig. persp.	-2.09635 · 10^6	584.03	0.00
5000	ours	-2.09635 · 10^6	4.20	0.00
Table 11: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.1, λ2 = 0.1).
# of features	method	upper bound	time(s)	gap(%)
100	SOS1	-5.28726 · 10^6	3.37	0.00
100	big-M50	-4.60269 · 10^6	1.83	0.00
100	persp.	-5.28726 · 10^6	0.07	0.00
100	eig. persp.	-5.28726 · 10^6	0.11	0.00
100	ours	-5.28726 · 10^6	0.27	0.00
500	SOS1	-5.31852 · 10^6	10.00	0.00
500	big-M50	-5.31827 · 10^6	85.52	0.00
500	persp.	-5.31852 · 10^6	8.41	0.00
500	eig. persp.	-5.31852 · 10^6	1.07	0.00
500	ours	-5.31852 · 10^6	0.41	0.00
1000	SOS1	-5.26974 · 10^6	88.50	0.00
1000	big-M50	-5.26974 · 10^6	1576.75	0.00
1000	persp.	-5.26974 · 10^6	91.53	0.00
1000	eig. persp.	-5.26974 · 10^6	4.11	0.00
1000	ours	-5.26974 · 10^6	0.52	0.00
3000	SOS1	-5.16793 · 10^6	3600.25	2.71
3000	big-M50	-3.73049 · 10^6	3603.09	42.28
3000	persp.	-5.27668 · 10^6	3600.09	0.59
3000	eig. persp.	-5.27668 · 10^6	103.60	0.00
3000	ours	-5.27668 · 10^6	1.84	0.00
5000	SOS1	-5.16087 · 10^6	3600.13	3.16
5000	big-M50	-3.98069 · 10^6	3606.79	33.75
5000	persp.	-5.27126 · 10^6	3601.72	1.00
5000	eig. persp.	-5.27126 · 10^6	611.45	0.00
5000	ours	-5.27126 · 10^6	4.42	0.00
Table 12: 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	time(s)	gap(%)
100	SOS1	-1.10084 · 107	2.97	0.00
100	big-M50	-1.10084 · 107	0.44	0.00
100	persp.	-1.10084 · 107	0.96	0.00
100	eig. persp.	-1.10084 · 107	0.13	0.00
100	ours	-1.10084 · 107	0.32	0.00
500	SOS1	-1.10518 · 107	9.66	0.00
500	big-M50	-1.10515 · 107	86.52	0.00
500	persp.	-1.10518 · 107	3600.03	0.08
500	eig. persp.	-1.10518 · 107	0.62	0.00
500	ours	-1.10518 · 107	0.40	0.00
1000	SOS1	-1.09508 · 107	87.74	0.00
1000	big-M50	-1.09506 · 107	1539.25	0.00
1000	persp.	-1.09508 · 107	3600.12	0.19
1000	eig. persp.	-1.09508 · 107	335.72	0.00
1000	ours	-1.09508 · 107	0.53	0.00
3000	SOS1	-1.10012 · 107	3600.36	0.51
3000	big-M50	-	3608.09	-
3000	persp.	-1.10012 · 107	3600.08	0.59
3000	eig. persp.	-1.10012 · 107	103.03	0.00
3000	ours	-1.10012 · 107	1.67	0.00
5000	SOS1	-1.08838 · 107	3602.31	2.05
5000	big-M50	-	3649.14	-
5000	persp.	-1.09966 · 107	3601.44	1.00
5000	eig. persp.	-	3758.62	-
5000	ours	-1.09966 · 107	4.13	0.00
Table 13: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.5, λ2 = 0.1).
# of features	method	upper bound	time(s)	gap(%)
100	SOS1	-2.43575 · 107	0.05	0.00
100	big-M50	-2.43573 · 107	0.41	0.00
100	persp.	-2.43575 · 107	0.43	0.00
100	eig. persp.	-2.43575 · 107	0.13	0.00
100	ours	-2.43575 · 107	0.34	0.00
500	SOS1	-2.44185 · 107	9.10	0.00
500	big-M50	-2.44167 · 107	96.59	0.00
500	persp.	-2.44185 · 107	148.21	0.00
500	eig. persp.	-2.44185 · 107	23.81	0.00
500	ours	-2.44185 · 107	0.44	0.00
1000	SOS1	-2.41989 · 107	89.99	0.00
1000	big-M50	-2.41986 · 107	1608.58	0.00
1000	persp.	-2.41989 · 107	3600.10	0.19
1000	eig. persp.	-2.41989 · 107	4.42	0.00
1000	ours	-2.41989 · 107	0.53	0.00
3000	SOS1	-2.42629 · 107	3601.14	1.04
3000	big-M50	-	3605.20	-
3000	persp.	-2.43713 · 107	3601.16	0.59
3000	eig. persp.	-	3658.27	-
3000	ours	-2.43713 · 107	1.75	0.00
5000	SOS1	-2.42551 · 107	3600.14	1.48
5000	big-M50	-2.43694 · 107	3650.52	1.00
5000	persp.	-2.43694 · 107	3606.01	1.00
5000	eig. persp.	-	3750.44	-
5000	ours	-2.43694 · 107	4.22	0.00
Table 14: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.7, λ2 = 0.1).
# of features	method	upper bound	time(s)	gap(%)
100	SOS1	-9.11026 · 107	0.05	0.00
100	big-M50	-9.11026 · 107	0.18	0.00
100	persp.	-9.11026 · 107	0.30	0.00
100	eig. persp.	-9.11026 · 107	0.13	0.00
100	ours	-9.11026 · 107	0.31	0.00
500	SOS1	-9.12047 · 107	22.15	0.00
500	big-M50	-9.12047 · 107	398.02	0.00
500	persp.	-9.12047 · 107	3600.01	0.08
500	eig. persp.	-9.12046 · 107	84.16	0.00
500	ours	-9.12047 · 107	0.38	0.00
1000	SOS1	-9.04075 · 107	1090.30	0.00
1000	big-M50	-9.04075 · 107	3600.05	0.01
1000	persp.	-9.04075 · 107	3600.08	0.20
1000	eig. persp.	-9.04074 · 107	881.89	0.00
1000	ours	-9.04075 · 107	0.52	0.00
3000	SOS1	-9.11671 · 107	3600.55	0.71
3000	big-M50	-	3619.63	-
3000	persp.	-9.12761 · 107	3600.05	0.59
3000	eig. persp.	-	3600.21	-
3000	ours	-9.12761 · 107	20.88	0.00
5000	SOS1	-9.02495 · 107	3602.25	2.17
5000	big-M50	-	3620.37	-
5000	persp.	-9.12900 · 107	3600.18	1.00
5000	eig. persp.	-	4028.26	-
5000	ours	-9.12916 · 107	58.07	0.00
Table 15: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.9, λ2 = 0.1).
G.2.4 Synthetic 1 Benchmark with  $\lambda_{2} = 0.1$  with Warmstart
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-2.10878 · 10^6	0.05	0.00
100	big-M50 + warm start	-2.10878 · 10^6	0.08	0.00
100	persp. + warm start	-2.10878 · 10^6	0.07	0.00
100	eig. persp. + warm start	-2.10878 · 10^6	0.02	0.00
100	ours	-2.10878 · 10^6	0.36	0.00
500	SOS1 + warm start	-2.12833 · 10^6	5.04	0.00
500	big-M50 + warm start	-2.12833 · 10^6	3.78	0.00
500	persp. + warm start	-2.12833 · 10^6	3.84	0.00
500	eig. persp. + warm start	-2.12833 · 10^6	0.60	0.00
500	ours	-2.12833 · 10^6	0.37	0.00
1000	SOS1 + warm start	-2.11005 · 10^6	74.29	0.00
1000	big-M50 + warm start	-2.11005 · 10^6	26.94	0.00
1000	persp. + warm start	-2.11005 · 10^6	1175.58	0.00
1000	eig. persp. + warm start	-2.11005 · 10^6	4.18	0.00
1000	ours	-2.11005 · 10^6	0.57	0.00
3000	SOS1 + warm start	-2.10230 · 10^6	2144.33	0.00
3000	big-M50 + warm start	-2.10230 · 10^6	193.85	0.00
3000	persp. + warm start	-2.10230 · 10^6	2108.24	0.00
3000	eig. persp. + warm start	-2.10230 · 10^6	110.85	0.00
3000	ours	-2.10230 · 10^6	1.79	0.00
5000	SOS1 + warm start	-2.09635 · 10^6	3600.66	0.95
5000	big-M50 + warm start	-2.09635 · 10^6	1366.14	0.00
5000	persp. + warm start	-2.09635 · 10^6	3601.28	1.00
5000	eig. persp. + warm start	-2.09635 · 10^6	460.24	0.00
5000	ours	-2.09635 · 10^6	4.20	0.00
Table 16: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.1, λ2 = 0.1). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-5.28726 · 10^6	0.05	0.00
100	big-M50 + warm start	-5.28726 · 10^6	0.07	0.00
100	persp. + warm start	-5.28726 · 10^6	0.07	0.00
100	eig. persp. + warm start	-5.28726 · 10^6	0.02	0.00
100	ours	-5.28726 · 10^6	0.27	0.00
500	SOS1 + warm start	-5.31852 · 10^6	9.77	0.00
500	big-M50 + warm start	-5.31852 · 10^6	39.88	0.00
500	persp. + warm start	-5.31852 · 10^6	8.26	0.00
500	eig. persp. + warm start	-5.31852 · 10^6	0.63	0.00
500	ours	-5.31852 · 10^6	0.41	0.00
1000	SOS1 + warm start	-5.26974 · 10^6	89.01	0.00
1000	big-M50 + warm start	-5.26974 · 10^6	14.33	0.00
1000	persp. + warm start	-5.26974 · 10^6	91.61	0.00
1000	eig. persp. + warm start	-5.26974 · 10^6	4.09	0.00
1000	ours	-5.26974 · 10^6	0.52	0.00
3000	SOS1 + warm start	-5.27668 · 10^6	2743.09	0.00
3000	big-M50 + warm start	-5.27668 · 10^6	317.41	0.00
3000	persp. + warm start	-5.27668 · 10^6	139.45	0.00
3000	eig. persp. + warm start	-5.27668 · 10^6	111.91	0.00
3000	ours	-5.27668 · 10^6	1.84	0.00
5000	SOS1 + warm start	-5.27126 · 10^6	3600.80	1.00
5000	big-M50 + warm start	-5.27126 · 10^6	1658.42	0.00
5000	persp. + warm start	-5.27126 · 10^6	546.50	0.00
5000	eig. persp. + warm start	-5.27126 · 10^6	454.67	0.00
5000	ours	-5.27126 · 10^6	4.42	0.00
Table 17: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.3$ ,  $\lambda_2 = 0.1$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-1.10084 · 107	0.05	0.00
100	big-M50 + warm start	-1.10084 · 107	0.06	0.00
100	persp. + warm start	-1.10084 · 107	0.90	0.00
100	eig. persp. + warm start	-1.10084 · 107	0.02	0.00
100	ours	-1.10084 · 107	0.32	0.00
500	SOS1 + warm start	-1.10518 · 107	9.26	0.00
500	big-M50 + warm start	-1.10518 · 107	2.28	0.00
500	persp. + warm start	-1.10518 · 107	3600.02	0.08
500	eig. persp. + warm start	-1.10518 · 107	0.64	0.00
500	ours	-1.10518 · 107	0.40	0.00
1000	SOS1 + warm start	-1.09508 · 107	89.44	0.00
1000	big-M50 + warm start	-1.09508 · 107	14.11	0.00
1000	persp. + warm start	-1.09508 · 107	10.09	0.00
1000	eig. persp. + warm start	-1.09508 · 107	4.05	0.00
1000	ours	-1.09508 · 107	0.53	0.00
3000	SOS1 + warm start	-1.10012 · 107	3046.03	0.00
3000	big-M50 + warm start	-1.10012 · 107	222.56	0.00
3000	persp. + warm start	-1.10012 · 107	141.64	0.00
3000	eig. persp. + warm start	-1.10012 · 107	110.14	0.00
3000	ours	-1.10012 · 107	1.67	0.00
5000	SOS1 + warm start	-1.09966 · 107	3600.73	0.99
5000	big-M50 + warm start	-1.09966 · 107	3606.43	1.00
5000	persp. + warm start	-1.09966 · 107	3600.38	1.00
5000	eig. persp. + warm start	-1.09966 · 107	455.87	0.00
5000	ours	-1.09966 · 107	4.13	0.00
Table 18: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.5$ ,  $\lambda_2 = 0.1$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-2.43575 · 107	0.05	0.00
100	big-M50 + warm start	-2.43575 · 107	0.06	0.00
100	persp. + warm start	-2.43575 · 107	0.43	0.00
100	eig. persp. + warm start	-2.43575 · 107	0.02	0.00
100	ours	-2.43575 · 107	0.34	0.00
500	SOS1 + warm start	-2.44185 · 107	9.20	0.00
500	big-M50 + warm start	-2.44185 · 107	4.00	0.00
500	persp. + warm start	-2.44185 · 107	146.41	0.00
500	eig. persp. + warm start	-2.44185 · 107	0.62	0.00
500	ours	-2.44185 · 107	0.44	0.00
1000	SOS1 + warm start	-2.41989 · 107	88.02	0.00
1000	big-M50 + warm start	-2.41989 · 107	12.62	0.00
1000	persp. + warm start	-2.41989 · 107	1117.62	0.00
1000	eig. persp. + warm start	-2.41989 · 107	4.05	0.00
1000	ours	-2.41989 · 107	0.53	0.00
3000	SOS1 + warm start	-2.43713 · 107	3600.80	0.59
3000	big-M50 + warm start	-2.43713 · 107	3600.43	0.59
3000	persp. + warm start	-2.43713 · 107	3600.68	0.59
3000	eig. persp. + warm start	-2.43713 · 107	101.46	0.00
3000	ours	-2.43713 · 107	1.75	0.00
5000	SOS1 + warm start	-2.43694 · 107	3600.86	1.00
5000	big-M50 + warm start	-2.43694 · 107	3616.71	1.00
5000	persp. + warm start	-2.43694 · 107	3603.18	1.00
5000	eig. persp. + warm start	-2.43694 · 107	455.57	0.00
5000	ours	-2.43694 · 107	4.22	0.00
Table 19: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.7$ ,  $\lambda_2 = 0.1$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-9.11026 · 107	0.04	0.00
100	big-M50 + warm start	-9.11026 · 107	0.04	0.00
100	persp. + warm start	-9.11026 · 107	0.29	0.00
100	eig. persp. + warm start	-9.11026 · 107	0.02	0.00
100	ours	-9.11026 · 107	0.31	0.00
500	SOS1 + warm start	-9.12047 · 107	52.72	0.00
500	big-M50 + warm start	-9.12047 · 107	14.34	0.00
500	persp. + warm start	-9.12047 · 107	166.73	0.00
500	eig. persp. + warm start	-9.12047 · 107	0.60	0.00
500	ours	-9.12047 · 107	0.38	0.00
1000	SOS1 + warm start	-9.04075 · 107	1068.59	0.00
1000	big-M50 + warm start	-9.04075 · 107	3600.16	0.01
1000	persp. + warm start	-9.04075 · 107	3600.01	0.20
1000	eig. persp. + warm start	-9.04075 · 107	4.17	0.00
1000	ours	-9.04075 · 107	0.52	0.00
3000	SOS1 + warm start	-9.12761 · 107	3600.39	0.59
3000	big-M50 + warm start	-9.12761 · 107	3611.02	0.59
3000	persp. + warm start	-9.12761 · 107	3600.23	0.59
3000	eig. persp. + warm start	-9.12761 · 107	101.65	0.00
3000	ours	-9.12761 · 107	20.88	0.00
5000	SOS1 + warm start	-9.12916 · 107	3603.57	1.00
5000	big-M50 + warm start	-9.12916 · 107	3606.64	1.00
5000	persp. + warm start	-9.12916 · 107	3601.56	1.00
5000	eig. persp. + warm start	-9.12916 · 107	454.63	0.00
5000	ours	-9.12916 · 107	58.07	0.00
Table 20: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.9$ ,  $\lambda_2 = 0.1$ ). All baselines use our beam search solution as a warm start.
G.2.5 Synthetic 1 Benchmark with  $\lambda_{2} = 10.0$  with No Warmstart
# of features	method	upper bound	time(s)	gap(%)
100	SOS1	-2.10868 · 10^6	3.37	0.00
100	big-M50	-2.10868 · 10^6	0.28	0.00
100	persp.	-2.10868 · 10^6	2.06	0.00
100	eig. persp.	-2.10868 · 10^6	0.02	0.00
100	ours	-2.10868 · 10^6	0.30	0.00
500	SOS1	-2.12823 · 10^6	6.58	0.00
500	big-M50	-2.12819 · 10^6	42.85	0.00
500	persp.	-2.12823 · 10^6	143.68	0.00
500	eig. persp.	-2.12823 · 10^6	0.65	0.00
500	ours	-2.12823 · 10^6	0.38	0.00
1000	SOS1	-2.10995 · 10^6	70.57	0.00
1000	big-M50	-2.10995 · 10^6	1072.15	0.00
1000	persp.	-2.10995 · 10^6	452.04	0.00
1000	eig. persp.	-2.10995 · 10^6	4.22	0.00
1000	ours	-2.10995 · 10^6	0.57	0.00
3000	SOS1	-1.87788 · 10^6	3600.81	12.60
3000	big-M50	-7.92671 · 10^5	3605.01	166.77
3000	persp.	-2.10220 · 10^6	3600.04	0.55
3000	eig. persp.	-2.10220 · 10^6	102.51	0.00
3000	ours	-2.10220 · 10^6	1.68	0.00
5000	SOS1	-1.98929 · 10^6	3600.16	6.43
5000	big-M50	-7.89596 · 10^5	3617.52	168.16
5000	persp.	-2.09584 · 10^6	3601.36	1.00
5000	eig. persp.	-2.09625 · 10^6	475.37	0.00
5000	ours	-2.09625 · 10^6	5.09	0.00
Table 21: 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	SOS1	-5.28716 · 10^6	2.98	0.00
100	big-M50	-5.28713 · 10^6	0.40	0.00
100	persp.	-5.28716 · 10^6	0.77	0.00
100	eig. persp.	-5.28716 · 10^6	0.10	0.00
100	ours	-5.28716 · 10^6	0.30	0.00
500	SOS1	-5.31842 · 10^6	10.06	0.00
500	big-M50	-5.31819 · 10^6	89.34	0.00
500	persp.	-5.31841 · 10^6	8.73	0.00
500	eig. persp.	-5.31842 · 10^6	0.60	0.00
500	ours	-5.31842 · 10^6	0.41	0.00
1000	SOS1	-5.26964 · 10^6	87.65	0.00
1000	big-M50	-5.26964 · 10^6	1869.66	0.00
1000	persp.	-5.26964 · 10^6	97.81	0.00
1000	eig. persp.	-5.26964 · 10^6	226.32	0.00
1000	ours	-5.26964 · 10^6	0.51	0.00
3000	SOS1	-5.27658 · 10^6	3600.46	0.55
3000	big-M50	-3.96610 · 10^6	3600.32	33.83
3000	persp.	-5.27658 · 10^6	3600.06	0.58
3000	eig. persp.	-	3615.07	-
3000	ours	-5.27658 · 10^6	1.76	0.00
5000	SOS1	-5.16077 · 10^6	3603.93	3.16
5000	big-M50	-3.96757 · 10^6	3607.78	34.19
5000	persp.	-5.27116 · 10^6	3600.43	1.00
5000	eig. persp.	-5.27116 · 10^6	614.83	0.00
5000	ours	-5.27116 · 10^6	4.32	0.00
Table 22: 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	time(s)	gap(%)
100	SOS1	-1.10083 · 107	2.83	0.00
100	big-M50	-1.10083 · 107	0.44	0.00
100	persp.	-1.10083 · 107	0.46	0.00
100	eig. persp.	-1.10083 · 107	0.13	0.00
100	ours	-1.10083 · 107	0.32	0.00
500	SOS1	-1.10517 · 107	11.73	0.00
500	big-M50	-1.10515 · 107	91.37	0.00
500	persp.	-1.10517 · 107	3600.01	0.08
500	eig. persp.	-1.10517 · 107	0.63	0.00
500	ours	-1.10517 · 107	0.41	0.00
1000	SOS1	-1.09507 · 107	90.27	0.00
1000	big-M50	-1.09505 · 107	1534.37	0.00
1000	persp.	-1.09507 · 107	3600.04	0.19
1000	eig. persp.	-1.09507 · 107	3.92	0.00
1000	ours	-1.09507 · 107	0.55	0.00
3000	SOS1	-1.10011 · 107	3600.92	0.51
3000	big-M50	-	3603.26	-
3000	persp.	-1.10011 · 107	3601.24	0.58
3000	eig. persp.	-	3646.60	-
3000	ours	-1.10011 · 107	1.73	0.00
5000	SOS1	-1.08825 · 107	3600.12	2.06
5000	big-M50	-9.62941 · 106	3608.37	15.34
5000	persp.	-1.09965 · 107	3602.42	1.00
5000	eig. persp.	-	3600.21	-
5000	ours	-1.09965 · 107	4.24	0.00
Table 23: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.5, λ2 = 10.0).