SlowGuess/NeurIPS_2023_OCR huggingface dataset:
# of features	method	upper bound	time(s)	gap(%)
100	SOS1	-2.43574 · 107	0.05	0.00
100	big-M50	-2.43574 · 107	0.42	0.00
100	persp.	-2.43574 · 107	0.55	0.00
100	eig. persp.	-2.43574 · 107	0.12	0.00
100	ours	-2.43574 · 107	0.30	0.00
500	SOS1	-2.44184 · 107	10.39	0.00
500	big-M50	-2.44178 · 107	85.12	0.00
500	persp.	-2.44184 · 107	248.24	0.00
500	eig. persp.	-2.44184 · 107	0.61	0.00
500	ours	-2.44184 · 107	0.44	0.00
1000	SOS1	-2.41988 · 107	91.02	0.00
1000	big-M50	-2.41986 · 107	1415.19	0.00
1000	persp.	-2.41988 · 107	3600.04	0.16
1000	eig. persp.	-2.41988 · 107	9.28	0.00
1000	ours	-2.41988 · 107	0.52	0.00
3000	SOS1	-2.43712 · 107	3601.13	0.59
3000	big-M50	-	3615.34	-
3000	persp.	-2.43712 · 107	3600.26	0.58
3000	eig. persp.	-	3669.39	-
3000	ours	-2.43712 · 107	1.81	0.00
5000	SOS1	-2.42550 · 107	3600.13	1.48
5000	big-M50	-2.43693 · 107	3607.06	1.00
5000	persp.	-2.43693 · 107	3601.61	1.00
5000	eig. persp.	-2.43693 · 107	619.11	0.00
5000	ours	-2.43693 · 107	4.08	0.00
Table 24: 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	SOS1	-9.11025 · 107	0.05	0.00
100	big-M50	-9.11025 · 107	0.17	0.00
100	persp.	-9.11025 · 107	0.29	0.00
100	eig. persp.	-9.11018 · 107	11.35	0.00
100	ours	-9.11025 · 107	0.32	0.00
500	SOS1	-9.12046 · 107	13.62	0.00
500	big-M50	-9.12046 · 107	117.86	0.00
500	persp.	-9.12046 · 107	997.83	0.00
500	eig. persp.	-9.12046 · 107	84.29	0.00
500	ours	-9.12046 · 107	0.38	0.00
1000	SOS1	-9.04074 · 107	1201.62	0.00
1000	big-M50	-9.04074 · 107	3600.04	0.01
1000	persp.	-9.04074 · 107	3600.03	0.20
1000	eig. persp.	-9.04074 · 107	343.13	0.00
1000	ours	-9.04074 · 107	0.52	0.00
3000	SOS1	-9.12760 · 107	3601.23	0.59
3000	big-M50	-	3611.90	-
3000	persp.	-9.12755 · 107	3601.40	0.59
3000	eig. persp.	-	3625.51	-
3000	ours	-9.12760 · 107	20.84	0.00
5000	SOS1	-8.98304 · 107	3600.14	2.65
5000	big-M50	-	3615.45	-
5000	persp.	-9.12241 · 107	3600.13	1.07
5000	eig. persp.	-	3819.54	-
5000	ours	-9.12915 · 107	57.41	0.00
Table 25: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10, ρ = 0.9, λ2 = 10.0).
G.2.6 Synthetic 1 Benchmark with  $\lambda_{2} = 10.0$  with Warmstart
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-2.10868 · 10^6	0.05	0.00
100	big-M50 + warm start	-2.10868 · 10^6	0.25	0.00
100	persp. + warm start	-2.10868 · 10^6	2.05	0.00
100	eig. persp. + warm start	-2.10868 · 10^6	0.01	0.00
100	ours	-2.10868 · 10^6	0.30	0.00
500	SOS1 + warm start	-2.12823 · 10^6	6.54	0.00
500	big-M50 + warm start	-2.12823 · 10^6	3.76	0.00
500	persp. + warm start	-2.12823 · 10^6	144.06	0.00
500	eig. persp. + warm start	-2.12823 · 10^6	0.61	0.00
500	ours	-2.12823 · 10^6	0.38	0.00
1000	SOS1 + warm start	-2.10995 · 10^6	67.85	0.00
1000	big-M50 + warm start	-2.10995 · 10^6	23.00	0.00
1000	persp. + warm start	-2.10995 · 10^6	436.41	0.00
1000	eig. persp. + warm start	-2.10995 · 10^6	4.18	0.00
1000	ours	-2.10995 · 10^6	0.57	0.00
3000	SOS1 + warm start	-2.10220 · 10^6	2171.96	0.00
3000	big-M50 + warm start	-2.10220 · 10^6	241.10	0.00
3000	persp. + warm start	-2.10220 · 10^6	3571.06	0.00
3000	eig. persp. + warm start	-2.10220 · 10^6	99.06	0.00
3000	ours	-2.10220 · 10^6	1.68	0.00
5000	SOS1 + warm start	-2.09625 · 10^6	3602.96	0.95
5000	big-M50 + warm start	-2.09625 · 10^6	1309.75	0.00
5000	persp. + warm start	-2.09625 · 10^6	3601.45	0.98
5000	eig. persp. + warm start	-2.09625 · 10^6	442.61	0.00
5000	ours	-2.09625 · 10^6	5.09	0.00
Table 26: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.1$ ,  $\lambda_2 = 10.0$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-5.28716 · 10^6	0.05	0.00
100	big-M50 + warm start	-5.28716 · 10^6	0.25	0.00
100	persp. + warm start	-5.28716 · 10^6	0.89	0.00
100	eig. persp. + warm start	-5.28716 · 10^6	0.02	0.00
100	ours	-5.28716 · 10^6	0.30	0.00
500	SOS1 + warm start	-5.31842 · 10^6	7.34	0.00
500	big-M50 + warm start	-5.31842 · 10^6	2.93	0.00
500	persp. + warm start	-5.31842 · 10^6	8.30	0.00
500	eig. persp. + warm start	-5.31842 · 10^6	0.62	0.00
500	ours	-5.31842 · 10^6	0.41	0.00
1000	SOS1 + warm start	-5.26964 · 10^6	84.96	0.00
1000	big-M50 + warm start	-5.26964 · 10^6	15.10	0.00
1000	persp. + warm start	-5.26964 · 10^6	97.38	0.00
1000	eig. persp. + warm start	-5.26964 · 10^6	4.10	0.00
1000	ours	-5.26964 · 10^6	0.51	0.00
3000	SOS1 + warm start	-5.27658 · 10^6	2809.30	0.00
3000	big-M50 + warm start	-5.27658 · 10^6	315.30	0.00
3000	persp. + warm start	-5.27658 · 10^6	152.21	0.00
3000	eig. persp. + warm start	-5.27658 · 10^6	100.20	0.00
3000	ours	-5.27658 · 10^6	1.76	0.00
5000	SOS1 + warm start	-5.27116 · 10^6	3601.86	0.99
5000	big-M50 + warm start	-5.27116 · 10^6	1100.87	0.00
5000	persp. + warm start	-5.27116 · 10^6	633.11	0.00
5000	eig. persp. + warm start	-5.27116 · 10^6	460.01	0.00
5000	ours	-5.27116 · 10^6	4.32	0.00
Table 27: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.3$ ,  $\lambda_2 = 10.0$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-1.10083 · 107	0.05	0.00
100	big-M50 + warm start	-1.10083 · 107	0.06	0.00
100	persp. + warm start	-1.10083 · 107	0.46	0.00
100	eig. persp. + warm start	-1.10083 · 107	0.02	0.00
100	ours	-1.10083 · 107	0.32	0.00
500	SOS1 + warm start	-1.10517 · 107	10.26	0.00
500	big-M50 + warm start	-1.10517 · 107	8.51	0.00
500	persp. + warm start	-1.10517 · 107	3600.01	0.08
500	eig. persp. + warm start	-1.10517 · 107	0.64	0.00
500	ours	-1.10517 · 107	0.41	0.00
1000	SOS1 + warm start	-1.09507 · 107	88.05	0.00
1000	big-M50 + warm start	-1.09507 · 107	14.03	0.00
1000	persp. + warm start	-1.09507 · 107	10.25	0.00
1000	eig. persp. + warm start	-1.09507 · 107	4.13	0.00
1000	ours	-1.09507 · 107	0.55	0.00
3000	SOS1 + warm start	-1.10011 · 107	3008.66	0.00
3000	big-M50 + warm start	-1.10011 · 107	201.32	0.00
3000	persp. + warm start	-1.10011 · 107	153.29	0.00
3000	eig. persp. + warm start	-1.10011 · 107	102.13	0.00
3000	ours	-1.10011 · 107	1.73	0.00
5000	SOS1 + warm start	-1.09965 · 107	3601.18	0.99
5000	big-M50 + warm start	-1.09965 · 107	3601.20	1.00
5000	persp. + warm start	-1.09965 · 107	3602.12	1.00
5000	eig. persp. + warm start	-1.09965 · 107	499.65	0.00
5000	ours	-1.09965 · 107	4.24	0.00
Table 28: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.5$ ,  $\lambda_2 = 10.0$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-2.43574 · 10^7	0.05	0.00
100	big-M50 + warm start	-2.43574 · 10^7	0.16	0.00
100	persp. + warm start	-2.43574 · 10^7	0.51	0.00
100	eig. persp. + warm start	-2.43574 · 10^7	0.02	0.00
100	ours	-2.43574 · 10^7	0.30	0.00
500	SOS1 + warm start	-2.44184 · 10^7	8.95	0.00
500	big-M50 + warm start	-2.44184 · 10^7	2.11	0.00
500	persp. + warm start	-2.44184 · 10^7	215.86	0.00
500	eig. persp. + warm start	-2.44184 · 10^7	0.61	0.00
500	ours	-2.44184 · 10^7	0.44	0.00
1000	SOS1 + warm start	-2.41988 · 10^7	86.91	0.00
1000	big-M50 + warm start	-2.41988 · 10^7	12.39	0.00
1000	persp. + warm start	-2.41988 · 10^7	3600.03	0.08
1000	eig. persp. + warm start	-2.41988 · 10^7	4.14	0.00
1000	ours	-2.41988 · 10^7	0.52	0.00
3000	SOS1 + warm start	-2.43712 · 10^7	3601.60	0.59
3000	big-M50 + warm start	-2.43712 · 10^7	3600.43	0.59
3000	persp. + warm start	-2.43712 · 10^7	3601.09	0.58
3000	eig. persp. + warm start	-2.43712 · 10^7	113.06	0.00
3000	ours	-2.43712 · 10^7	1.81	0.00
5000	SOS1 + warm start	-2.43693 · 10^7	3601.29	1.00
5000	big-M50 + warm start	-2.43693 · 10^7	3624.24	1.00
5000	persp. + warm start	-2.43693 · 10^7	3600.19	1.00
5000	eig. persp. + warm start	-2.43693 · 10^7	457.54	0.00
5000	ours	-2.43693 · 10^7	4.08	0.00
Table 29: Comparison of time and optimality gap on the synthetic datasets (n = 100000, k = 10,  $\rho = 0.7$ ,  $\lambda_2 = 10.0$ ). All baselines use our beam search solution as a warm start.
# of features	method	upper bound	time(s)	gap(%)
100	SOS1 + warm start	-9.11025 · 107	0.04	0.00
100	big-M50 + warm start	-9.11025 · 107	0.04	0.00
100	persp. + warm start	-9.11025 · 107	0.29	0.00
100	eig. persp. + warm start	-9.11025 · 107	0.03	0.00
100	ours	-9.11025 · 107	0.32	0.00
500	SOS1 + warm start	-9.12046 · 107	54.77	0.00
500	big-M50 + warm start	-9.12046 · 107	36.63	0.00
500	persp. + warm start	-9.12046 · 107	306.17	0.00
500	eig. persp. + warm start	-9.12046 · 107	0.59	0.00
500	ours	-9.12046 · 107	0.38	0.00
1000	SOS1 + warm start	-9.04074 · 107	1178.82	0.00
1000	big-M50 + warm start	-9.04074 · 107	3600.19	0.06
1000	persp. + warm start	-9.04074 · 107	3600.04	0.19
1000	eig. persp. + warm start	-9.04074 · 107	4.15	0.00
1000	ours	-9.04074 · 107	0.52	0.00
3000	SOS1 + warm start	-9.12760 · 107	3600.52	0.59
3000	big-M50 + warm start	-9.12760 · 107	3606.60	0.59
3000	persp. + warm start	-9.12760 · 107	3600.05	0.59
3000	eig. persp. + warm start	-9.12760 · 107	103.22	0.00
3000	ours	-9.12760 · 107	20.84	0.00
5000	SOS1 + warm start	-9.12915 · 107	3602.40	1.00
5000	big-M50 + warm start	-9.12915 · 107	3606.47	1.00
5000	persp. + warm start	-9.12915 · 107	3600.16	1.00
5000	eig. persp. + warm start	-9.12915 · 107	490.05	0.00
5000	ours	-9.12915 · 107	57.41	0.00
Table 30: 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.
G.2.7 Synthetic 2 Benchmark with  $\lambda_{2} = 0.001$  with No Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-5.42178 · 10^4	3600.10	39.82
3000	big-M50	-6.34585 · 10^4	3600.07	19.16
3000	persp.	-4.05327 · 10^4	3600.06	87.02
3000	eig. persp.	-5.42702 · 10^4	3600.07	39.69
3000	ours	-6.34585 · 10^4	3600.49	19.45
4000	SOS1	-8.64294 · 10^4	3601.19	15.37
4000	big-M50	-8.64294 · 10^4	3600.07	15.32
4000	persp.	-8.29421 · 10^4	3600.06	20.22
4000	eig. persp.	-8.29301 · 10^4	3600.08	6.80
4000	ours	-8.64294 · 10^4	20.25	0.00
5000	SOS1	-9.85324 · 10^4	3600.10	24.46
5000	big-M50	-1.09576 · 10^5	3600.81	10.25
5000	persp.	-1.09576 · 10^5	3600.07	11.91
5000	eig. persp.	-1.09576 · 10^5	3600.07	0.05
5000	ours	-1.09576 · 10^5	20.51	0.00
6000	SOS1	-1.15043 · 10^5	3600.09	21.96
6000	big-M50	-1.27643 · 10^5	3600.08	9.93
6000	persp.	-1.27643 · 10^5	3600.14	9.93
6000	eig. persp.	-1.27643 · 10^5	3600.06	12.64
6000	ours	-1.27643 · 10^5	1.70	0.00
7000	SOS1	-1.50614 · 10^5	3600.86	8.32
7000	big-M50	-1.50614 · 10^5	3600.09	8.29
7000	persp.	-1.50614 · 10^5	3600.21	8.33
7000	eig. persp.	-1.50614 · 10^5	106.73	0.00
7000	ours	-1.50614 · 10^5	1.67	0.00
Table 31: 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	SOS1	-1.50509 · 10^5	3601.34	26.68
3000	big-M50	-1.59509 · 10^5	3600.08	19.07
3000	persp.	-1.48091 · 10^5	3600.99	28.74
3000	eig. persp.	-1.35700 · 10^5	3600.07	40.51
3000	ours	-1.59509 · 10^5	3600.67	19.53
4000	SOS1	-2.09237 · 10^5	3600.08	19.87
4000	big-M50	-2.17574 · 10^5	3600.07	14.78
4000	persp.	-2.17574 · 10^5	3600.06	14.91
4000	eig. persp.	-2.02936 · 10^5	3600.06	23.67
4000	ours	-2.17574 · 10^5	100.13	0.00
5000	SOS1	-2.71986 · 10^5	3600.29	12.00
5000	big-M50	-2.71986 · 10^5	3600.07	11.22
5000	persp.	-2.71986 · 10^5	3600.40	11.98
5000	eig. persp.	-2.71986 · 10^5	2152.71	0.00
5000	ours	-2.71986 · 10^5	20.80	0.00
6000	SOS1	-3.16761 · 10^5	3600.70	9.96
6000	big-M50	-3.16761 · 10^5	3600.09	9.96
6000	persp.	-3.16761 · 10^5	3600.06	9.96
6000	eig. persp.	-3.16761 · 10^5	1968.93	0.00
6000	ours	-3.16761 · 10^5	20.70	0.00
7000	SOS1	-3.72610 · 10^5	3601.25	8.33
7000	big-M50	-3.72610 · 10^5	3601.19	8.33
7000	persp.	-3.72610 · 10^5	3600.14	8.33
7000	eig. persp.	-3.72610 · 10^5	2906.29	0.00
7000	ours	-3.72610 · 10^5	1.98	0.00
Table 32: 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	SOS1	-3.24748 · 105	3600.06	22.83
3000	big-M50	-3.33604 · 105	3600.08	18.79
3000	persp.	-3.03254 · 105	3600.06	31.54
3000	eig. persp.	-3.04485 · 105	3600.80	31.01
3000	ours	-3.33604 · 105	3600.58	19.57
4000	SOS1	-4.49670 · 105	3600.55	16.16
4000	big-M50	-4.53261 · 105	3600.07	14.98
4000	persp.	-4.50269 · 105	3600.06	15.99
4000	eig. persp.	-4.30194 · 105	3600.06	21.59
4000	ours	-4.53261 · 105	3600.25	0.63
5000	SOS1	-5.57342 · 105	3600.12	13.18
5000	big-M50	-5.63007 · 105	3601.19	11.68
5000	persp.	-5.63007 · 105	3600.57	11.75
5000	eig. persp.	-5.63007 · 105	3600.05	0.29
5000	ours	-5.63007 · 105	36.97	0.00
6000	SOS1	-6.51710 · 105	3600.08	11.01
6000	big-M50	-6.57799 · 105	3600.34	9.98
6000	persp.	-6.57799 · 105	3600.15	9.97
6000	eig. persp.	-6.57799 · 105	1685.67	0.00
6000	ours	-6.57799 · 105	20.75	0.00
7000	SOS1	-7.46135 · 105	3600.71	12.13
7000	big-M50	-7.72267 · 105	3600.11	8.33
7000	persp.	-7.72267 · 105	3600.50	8.32
7000	eig. persp.	-7.72267 · 105	3600.27	10.37
7000	ours	-7.72267 · 105	21.02	0.00
Table 33: 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	SOS1	-7.34855 · 10^5	3600.26	20.63
3000	big-M50	-7.41187 · 10^5	3600.05	18.15
3000	persp.	-6.86643 · 10^5	3600.74	29.10
3000	eig. persp.	-7.02765 · 10^5	3600.06	26.14
3000	ours	-7.41187 · 10^5	3601.14	19.60
4000	SOS1	-1.00137 · 10^6	3601.49	15.39
4000	big-M50	-1.00103 · 10^6	3600.07	15.19
4000	persp.	-9.96124 · 10^5	3600.10	15.98
4000	eig. persp.	-9.90959 · 10^5	3600.06	17.00
4000	ours	-1.00287 · 10^6	3600.37	1.88
5000	SOS1	-1.23058 · 10^6	3600.25	12.96
5000	big-M50	-1.24027 · 10^6	3601.10	11.88
5000	persp.	-1.23565 · 10^6	3601.00	12.49
5000	eig. persp.	-1.22403 · 10^6	3600.05	2.31
5000	ours	-1.24027 · 10^6	466.42	0.00
6000	SOS1	-1.44944 · 10^6	3600.09	10.36
6000	big-M50	-1.45417 · 10^6	3600.11	9.78
6000	persp.	-1.45417 · 10^6	3600.55	9.82
6000	eig. persp.	-1.45417 · 10^6	3430.19	0.00
6000	ours	-1.45417 · 10^6	57.67	0.00
7000	SOS1	-1.68757 · 10^6	3600.20	9.45
7000	big-M50	-1.70483 · 10^6	3601.55	8.34
7000	persp.	-1.70483 · 10^6	3600.96	8.13
7000	eig. persp.	-1.70483 · 10^6	3432.42	0.00
7000	ours	-1.70483 · 10^6	21.42	0.00
Table 34: 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	SOS1	-2.77903 · 10^6	3600.07	19.87
3000	big-M50	-2.78521 · 10^6	3600.15	15.90
3000	persp.	-2.76075 · 10^6	3600.05	20.66
3000	eig. persp.	-2.74468 · 10^6	3600.81	21.37
3000	ours	-2.78521 · 10^6	3600.01	19.60
4000	SOS1	-3.75203 · 10^6	3600.12	15.10
4000	big-M50	-3.75202 · 10^6	3600.06	13.33
4000	persp.	-3.72754 · 10^6	3600.07	15.88
4000	eig. persp.	-3.72921 · 10^6	3600.51	16.56
4000	ours	-3.75203 · 10^6	3600.42	3.50
5000	SOS1	-4.61204 · 10^6	3600.15	12.27
5000	big-M50	-4.61877 · 10^6	3600.19	11.37
5000	persp.	-4.60317 · 10^6	3600.36	12.48
5000	eig. persp.	-4.60063 · 10^6	3600.26	13.32
5000	ours	-4.61877 · 10^6	3600.57	0.93
6000	SOS1	-5.41709 · 10^6	3600.27	10.45
6000	big-M50	-5.43826 · 10^6	3600.06	9.77
6000	persp.	-5.41956 · 10^6	3600.06	10.39
6000	eig. persp.	-5.40204 · 10^6	3600.06	11.80
6000	ours	-5.43826 · 10^6	3600.65	0.33
7000	SOS1	-6.35797 · 10^6	3600.07	8.51
7000	big-M50	-6.36767 · 10^6	3600.38	8.22
7000	persp.	-6.35116 · 10^6	3600.24	8.62
7000	eig. persp.	-6.36156 · 10^6	3610.76	0.53
7000	ours	-6.36767 · 10^6	3600.78	0.11
Table 35: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 0.001).
G.2.8 Synthetic 2 Benchmark with  $\lambda_{2} = 0.001$  with Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1 + warm start	-6.34585 · 10^4	3600.96	19.46
3000	big-M50 + warm start	-6.34585 · 10^4	3601.20	19.16
3000	persp. + warm start	-6.34585 · 10^4	3600.10	19.45
3000	eig. persp. + warm start	-6.34585 · 10^4	3600.13	19.46
3000	ours	-6.34585 · 10^4	3600.49	19.45
4000	SOS1 + warm start	-8.64294 · 10^4	3600.56	15.37
4000	big-M50 + warm start	-8.64294 · 10^4	3600.87	15.32
4000	persp. + warm start	-8.64294 · 10^4	3600.07	14.33
4000	eig. persp. + warm start	-8.64294 · 10^4	3600.12	1.64
4000	ours	-8.64294 · 10^4	20.25	0.00
5000	SOS1 + warm start	-1.09576 · 10^5	3600.38	11.91
5000	big-M50 + warm start	-1.09576 · 10^5	3601.86	10.25
5000	persp. + warm start	-1.09576 · 10^5	3600.07	11.91
5000	eig. persp. + warm start	-1.09576 · 10^5	3389.68	0.00
5000	ours	-1.09576 · 10^5	20.51	0.00
6000	SOS1 + warm start	-1.27643 · 10^5	3600.46	9.92
6000	big-M50 + warm start	-1.27643 · 10^5	3601.14	9.93
6000	persp. + warm start	-1.27643 · 10^5	3600.05	9.93
6000	eig. persp. + warm start	-1.27643 · 10^5	3600.07	12.65
6000	ours	-1.27643 · 10^5	1.70	0.00
7000	SOS1 + warm start	-1.50614 · 10^5	3600.07	5.84
7000	big-M50 + warm start	-1.50614 · 10^5	3600.13	8.29
7000	persp. + warm start	-1.50614 · 10^5	3600.17	8.33
7000	eig. persp. + warm start	-1.50614 · 10^5	101.37	0.00
7000	ours	-1.50614 · 10^5	1.67	0.00
Table 36: 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	SOS1 + warm start	-1.59509 · 10^5	3600.74	19.53
3000	big-M50 + warm start	-1.59509 · 10^5	3600.60	19.07
3000	persp. + warm start	-1.59509 · 10^5	3601.09	19.53
3000	eig. persp. + warm start	-1.59509 · 10^5	3601.11	19.54
3000	ours	-1.59509 · 10^5	3600.67	19.53
4000	SOS1 + warm start	-2.17574 · 10^5	3600.07	15.28
4000	big-M50 + warm start	-2.17574 · 10^5	3600.10	14.78
4000	persp. + warm start	-2.17574 · 10^5	3601.12	14.91
4000	eig. persp. + warm start	-2.17574 · 10^5	3600.11	15.34
4000	ours	-2.17574 · 10^5	100.13	0.00
5000	SOS1 + warm start	-2.71986 · 10^5	3600.44	12.00
5000	big-M50 + warm start	-2.71986 · 10^5	3600.06	11.22
5000	persp. + warm start	-2.71986 · 10^5	3600.14	11.35
5000	eig. persp. + warm start	-2.71986 · 10^5	2118.55	0.00
5000	ours	-2.71986 · 10^5	20.80	0.00
6000	SOS1 + warm start	-3.16761 · 10^5	3600.73	9.96
6000	big-M50 + warm start	-3.16761 · 10^5	3600.07	9.96
6000	persp. + warm start	-3.16761 · 10^5	3601.28	9.96
6000	eig. persp. + warm start	-3.16761 · 10^5	3600.50	11.79
6000	ours	-3.16761 · 10^5	20.70	0.00
7000	SOS1 + warm start	-3.72610 · 10^5	3600.61	8.33
7000	big-M50 + warm start	-3.72610 · 10^5	3600.08	8.33
7000	persp. + warm start	-3.72610 · 10^5	3600.08	8.33
7000	eig. persp. + warm start	-3.72610 · 10^5	3600.81	9.45
7000	ours	-3.72610 · 10^5	1.98	0.00
Table 37: 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	SOS1 + warm start	-3.33604 · 10^5	3600.69	19.57
3000	big-M50 + warm start	-3.33604 · 10^5	3600.17	18.79
3000	persp. + warm start	-3.33604 · 10^5	3600.06	19.57
3000	eig. persp. + warm start	-3.33604 · 10^5	3600.06	19.58
3000	ours	-3.33604 · 10^5	3600.58	19.57
4000	SOS1 + warm start	-4.53261 · 10^5	3600.07	15.24
4000	big-M50 + warm start	-4.53261 · 10^5	3600.17	14.98
4000	persp. + warm start	-4.53261 · 10^5	3600.08	15.23
4000	eig. persp. + warm start	-4.53261 · 10^5	3600.09	15.41
4000	ours	-4.53261 · 10^5	3600.25	0.63
5000	SOS1 + warm start	-5.63007 · 10^5	3600.25	12.04
5000	big-M50 + warm start	-5.63007 · 10^5	3600.29	11.70
5000	persp. + warm start	-5.63007 · 10^5	3600.76	11.75
5000	eig. persp. + warm start	-5.63007 · 10^5	3600.39	0.37
5000	ours	-5.63007 · 10^5	36.97	0.00
6000	SOS1 + warm start	-6.57799 · 10^5	3600.29	9.98
6000	big-M50 + warm start	-6.57799 · 10^5	3600.94	9.98
6000	persp. + warm start	-6.57799 · 10^5	3601.41	9.57
6000	eig. persp. + warm start	-6.57799 · 10^5	1420.54	0.00
6000	ours	-6.57799 · 10^5	20.75	0.00
7000	SOS1 + warm start	-7.72267 · 10^5	3600.06	8.33
7000	big-M50 + warm start	-7.72267 · 10^5	3600.55	8.33
7000	persp. + warm start	-7.72267 · 10^5	3600.39	7.86
7000	eig. persp. + warm start	-7.72267 · 10^5	3600.81	10.37
7000	ours	-7.72267 · 10^5	21.02	0.00
Table 38: 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	SOS1 + warm start	-7.41187 · 10^5	3600.08	19.60
3000	big-M50 + warm start	-7.41187 · 10^5	3600.76	18.15
3000	persp. + warm start	-7.41187 · 10^5	3600.32	19.60
3000	eig. persp. + warm start	-7.41187 · 10^5	3600.07	19.60
3000	ours	-7.41187 · 10^5	3601.14	19.60
4000	SOS1 + warm start	-1.00287 · 10^6	3600.07	15.22
4000	big-M50 + warm start	-1.00287 · 10^6	3600.07	14.98
4000	persp. + warm start	-1.00287 · 10^6	3600.06	15.20
4000	eig. persp. + warm start	-1.00287 · 10^6	3600.07	15.61
4000	ours	-1.00287 · 10^6	3600.37	1.88
5000	SOS1 + warm start	-1.24027 · 10^6	3600.06	12.08
5000	big-M50 + warm start	-1.24027 · 10^6	3600.05	11.88
5000	persp. + warm start	-1.24027 · 10^6	3600.07	12.07
5000	eig. persp. + warm start	-1.24027 · 10^6	3600.20	0.97
5000	ours	-1.24027 · 10^6	466.42	0.00
6000	SOS1 + warm start	-1.45417 · 10^6	3601.20	10.00
6000	big-M50 + warm start	-1.45417 · 10^6	3600.45	9.78
6000	persp. + warm start	-1.45417 · 10^6	3600.04	9.82
6000	eig. persp. + warm start	-1.45417 · 10^6	2727.13	0.00
6000	ours	-1.45417 · 10^6	57.67	0.00
7000	SOS1 + warm start	-1.70483 · 10^6	3600.69	8.34
7000	big-M50 + warm start	-1.70483 · 10^6	3601.38	8.34
7000	persp. + warm start	-1.70483 · 10^6	3601.09	8.13
7000	eig. persp. + warm start	-1.70483 · 10^6	1517.88	0.00
7000	ours	-1.70483 · 10^6	21.42	0.00
Table 39: 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	SOS1 + warm start	-2.78521 · 10^6	3600.34	19.60
3000	big-M50 + warm start	-2.78521 · 10^6	3600.74	15.90
3000	persp. + warm start	-2.78521 · 10^6	3600.06	19.60
3000	eig. persp. + warm start	-2.78521 · 10^6	3600.44	19.61
3000	ours	-2.78521 · 10^6	3600.01	19.60
4000	SOS1 + warm start	-3.75203 · 10^6	3600.06	15.10
4000	big-M50 + warm start	-3.75203 · 10^6	3600.07	13.33
4000	persp. + warm start	-3.75203 · 10^6	3600.06	15.12
4000	eig. persp. + warm start	-3.75203 · 10^6	3600.29	15.85
4000	ours	-3.75203 · 10^6	3600.42	3.50
5000	SOS1 + warm start	-4.61877 · 10^6	3600.07	12.11
5000	big-M50 + warm start	-4.61877 · 10^6	3601.36	11.37
5000	persp. + warm start	-4.61877 · 10^6	3600.06	12.10
5000	eig. persp. + warm start	-4.61877 · 10^6	3604.20	12.88
5000	ours	-4.61877 · 10^6	3600.57	0.93
6000	SOS1 + warm start	-5.43826 · 10^6	3600.07	10.02
6000	big-M50 + warm start	-5.43826 · 10^6	3600.22	9.77
6000	persp. + warm start	-5.43826 · 10^6	3601.43	10.01
6000	eig. persp. + warm start	-5.43826 · 10^6	3600.95	0.85
6000	ours	-5.43826 · 10^6	3600.65	0.33
7000	SOS1 + warm start	-6.36767 · 10^6	3600.06	8.35
7000	big-M50 + warm start	-6.36767 · 10^6	3600.12	8.22
7000	persp. + warm start	-6.36767 · 10^6	3600.06	8.34
7000	eig. persp. + warm start	-6.36767 · 10^6	3600.28	0.35
7000	ours	-6.36767 · 10^6	3600.78	0.11
Table 40: 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.2.9 Synthetic 2 Benchmark with  $\lambda_{2} = 0.1$  with No Warmstart