SlowGuess/NeurIPS_2023_OCR huggingface dataset:
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-6.34575 · 10^4	3600.08	19.36
3000	big-M50	-6.34575 · 10^4	3600.07	19.14
3000	persp.	-4.81335 · 10^4	3600.14	57.24
3000	eig. persp.	-4.80471 · 10^4	3600.06	57.76
3000	ours	-6.34575 · 10^4	3600.33	19.31
4000	SOS1	-8.64284 · 10^4	3600.20	15.37
4000	big-M50	-8.64284 · 10^4	3600.23	15.33
4000	persp.	-8.29378 · 10^4	3600.27	20.19
4000	eig. persp.	-8.64284 · 10^4	3600.49	1.61
4000	ours	-8.64284 · 10^4	20.81	0.00
5000	SOS1	-1.09575 · 10^5	3600.96	11.91
5000	big-M50	-1.09575 · 10^5	3600.08	11.60
5000	persp.	-1.09575 · 10^5	3600.07	11.90
5000	eig. persp.	-1.09575 · 10^5	3600.18	0.23
5000	ours	-1.09575 · 10^5	20.65	0.00
6000	SOS1	-1.27642 · 10^5	3600.09	9.92
6000	big-M50	-1.27642 · 10^5	3601.02	9.92
6000	persp.	-1.27642 · 10^5	3600.93	9.92
6000	eig. persp.	-1.27642 · 10^5	3600.74	12.65
6000	ours	-1.27642 · 10^5	1.70	0.00
7000	SOS1	-1.50613 · 10^5	3600.12	8.32
7000	big-M50	-1.50613 · 10^5	3600.45	8.32
7000	persp.	-1.50613 · 10^5	3600.20	8.29
7000	eig. persp.	-1.50613 · 10^5	113.48	0.00
7000	ours	-1.50613 · 10^5	1.67	0.00
Table 41: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 0.1).
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-1.52666 · 10^5	3600.76	24.80
3000	big-M50	-1.59508 · 10^5	3601.50	19.07
3000	persp.	-1.28190 · 10^5	3600.48	48.57
3000	eig. persp.	-1.34180 · 10^5	3600.42	42.09
3000	ours	-1.59508 · 10^5	3600.27	19.40
4000	SOS1	-2.04336 · 10^5	3600.51	22.74
4000	big-M50	-2.17573 · 10^5	3600.23	14.78
4000	persp.	-2.17573 · 10^5	3600.79	14.88
4000	eig. persp.	-2.09228 · 10^5	3600.20	19.95
4000	ours	-2.17573 · 10^5	99.53	0.00
5000	SOS1	-2.66411 · 10^5	3600.59	14.34
5000	big-M50	-2.71985 · 10^5	3600.09	11.22
5000	persp.	-2.71985 · 10^5	3600.21	11.96
5000	eig. persp.	-2.71985 · 10^5	2019.93	0.00
5000	ours	-2.71985 · 10^5	21.01	0.00
6000	SOS1	-3.16760 · 10^5	3600.27	9.96
6000	big-M50	-3.16760 · 10^5	3600.07	9.96
6000	persp.	-3.16760 · 10^5	3600.07	9.96
6000	eig. persp.	-3.16760 · 10^5	1196.41	0.00
6000	ours	-3.16760 · 10^5	20.91	0.00
7000	SOS1	-3.55314 · 10^5	3600.32	13.60
7000	big-M50	-3.72609 · 10^5	3600.80	8.33
7000	persp.	-3.72609 · 10^5	3600.06	8.31
7000	eig. persp.	-3.72609 · 10^5	3600.07	9.46
7000	ours	-3.72609 · 10^5	1.73	0.00
Table 42: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 0.1).
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-3.23979 · 105	3601.13	23.04
3000	big-M50	-3.33603 · 105	3601.02	18.78
3000	persp.	-2.94042 · 105	3600.41	35.54
3000	eig. persp.	-3.07151 · 105	3600.07	29.86
3000	ours	-3.33603 · 105	3600.68	19.44
4000	SOS1	-4.35290 · 105	3600.08	20.00
4000	big-M50	-4.53260 · 105	3600.12	14.95
4000	persp.	-4.50184 · 105	3600.07	15.99
4000	eig. persp.	-4.41599 · 105	3600.06	18.45
4000	ours	-4.53260 · 105	3600.41	0.62
5000	SOS1	-5.63006 · 105	3600.09	12.04
5000	big-M50	-5.63006 · 105	3600.07	11.69
5000	persp.	-5.63006 · 105	3600.16	11.74
5000	eig. persp.	-5.63006 · 105	3600.43	0.46
5000	ours	-5.63006 · 105	37.26	0.00
6000	SOS1	-6.57798 · 105	3600.98	9.98
6000	big-M50	-6.57798 · 105	3600.53	9.98
6000	persp.	-6.57798 · 105	3600.13	9.96
6000	eig. persp.	-6.57798 · 105	1737.91	0.00
6000	ours	-6.57798 · 105	20.64	0.00
7000	SOS1	-7.46134 · 105	3600.11	12.13
7000	big-M50	-7.72266 · 105	3600.96	8.33
7000	persp.	-7.72266 · 105	3600.23	8.31
7000	eig. persp.	-7.72266 · 105	3601.08	10.36
7000	ours	-7.72266 · 105	20.91	0.00
Table 43: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 0.1).
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-7.36308 · 105	3601.07	20.30
3000	big-M50	-7.41186 · 105	3600.75	18.15
3000	persp.	-7.11307 · 105	3600.38	24.54
3000	eig. persp.	-7.14103 · 105	3600.06	24.13
3000	ours	-7.41186 · 105	3600.59	19.46
4000	SOS1	-1.00107 · 106	3600.08	15.42
4000	big-M50	-1.00287 · 106	3600.07	14.98
4000	persp.	-9.98563 · 105	3600.06	15.67
4000	eig. persp.	-9.88486 · 105	3600.06	17.29
4000	ours	-1.00287 · 106	3600.08	1.88
5000	SOS1	-1.23516 · 106	3600.08	12.54
5000	big-M50	-1.24027 · 106	3600.07	11.88
5000	persp.	-1.23565 · 106	3600.08	12.47
5000	eig. persp.	-1.22304 · 106	3600.46	14.19
5000	ours	-1.24027 · 106	469.43	0.00
6000	SOS1	-1.45416 · 106	3600.07	10.00
6000	big-M50	-1.45416 · 106	3601.17	9.78
6000	persp.	-1.45416 · 106	3601.30	9.82
6000	eig. persp.	-1.45416 · 106	3600.05	0.03
6000	ours	-1.45416 · 106	59.16	0.00
7000	SOS1	-1.69945 · 106	3600.08	8.68
7000	big-M50	-1.70483 · 106	3602.06	8.34
7000	persp.	-1.70483 · 106	3600.92	8.13
7000	eig. persp.	-1.70483 · 106	1537.97	0.00
7000	ours	-1.70483 · 106	21.12	0.00
Table 44: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.7, λ2 = 0.1).
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-2.77826 · 10^6	3600.41	19.82
3000	big-M50	-2.78468 · 10^6	3600.31	15.93
3000	persp.	-2.76199 · 10^6	3600.06	20.50
3000	eig. persp.	-2.76520 · 10^6	3600.06	20.47
3000	ours	-2.78521 · 10^6	3600.23	19.46
4000	SOS1	-3.74434 · 10^6	3600.35	15.33
4000	big-M50	-3.75203 · 10^6	3600.08	13.35
4000	persp.	-3.73180 · 10^6	3600.06	15.73
4000	eig. persp.	-3.74150 · 10^6	3600.69	16.17
4000	ours	-3.75203 · 10^6	3601.10	3.50
5000	SOS1	-4.60905 · 10^6	3600.07	12.34
5000	big-M50	-4.61877 · 10^6	3600.08	11.40
5000	persp.	-4.54788 · 10^6	3600.79	13.84
5000	eig. persp.	-4.59650 · 10^6	3600.06	13.42
5000	ours	-4.61877 · 10^6	3600.40	0.93
6000	SOS1	-5.40939 · 10^6	3600.07	10.60
6000	big-M50	-5.43826 · 10^6	3600.23	9.79
6000	persp.	-5.42336 · 10^6	3600.65	10.30
6000	eig. persp.	-5.43107 · 10^6	3600.99	11.20
6000	ours	-5.43826 · 10^6	3600.26	0.33
7000	SOS1	-6.33762 · 10^6	3600.07	8.86
7000	big-M50	-6.36767 · 10^6	3600.07	8.22
7000	persp.	-6.26888 · 10^6	3600.06	10.04
7000	eig. persp.	-6.34367 · 10^6	3600.05	10.10
7000	ours	-6.36767 · 10^6	3600.58	0.11
Table 45: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 0.1).
G.2.10 Synthetic 2 Benchmark with  $\lambda_{2} = 0.1$  with Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1 + warm start	-6.34575 · 10^4	3600.07	19.36
3000	big-M50 + warm start	-6.34575 · 10^4	3600.12	19.14
3000	persp. + warm start	-6.34575 · 10^4	3600.14	19.27
3000	eig. persp. + warm start	-6.34575 · 10^4	3600.13	19.45
3000	ours	-6.34575 · 10^4	3600.33	19.31
4000	SOS1 + warm start	-8.64284 · 10^4	3600.36	15.36
4000	big-M50 + warm start	-8.64284 · 10^4	3600.05	15.33
4000	persp. + warm start	-8.64284 · 10^4	3600.52	14.30
4000	eig. persp. + warm start	-8.64284 · 10^4	3600.17	1.64
4000	ours	-8.64284 · 10^4	20.81	0.00
5000	SOS1 + warm start	-1.09575 · 10^5	3600.99	11.91
5000	big-M50 + warm start	-1.09575 · 10^5	3601.88	11.60
5000	persp. + warm start	-1.09575 · 10^5	3600.14	11.90
5000	eig. persp. + warm start	-1.09575 · 10^5	2971.48	0.00
5000	ours	-1.09575 · 10^5	20.65	0.00
6000	SOS1 + warm start	-1.27642 · 10^5	3600.07	9.92
6000	big-M50 + warm start	-1.27642 · 10^5	3600.62	9.93
6000	persp. + warm start	-1.27642 · 10^5	3600.06	9.92
6000	eig. persp. + warm start	-1.27642 · 10^5	3601.19	12.65
6000	ours	-1.27642 · 10^5	1.70	0.00
7000	SOS1 + warm start	-1.50613 · 10^5	3600.13	5.84
7000	big-M50 + warm start	-1.50613 · 10^5	3600.07	8.32
7000	persp. + warm start	-1.50613 · 10^5	3600.19	8.29
7000	eig. persp. + warm start	-1.50613 · 10^5	103.79	0.00
7000	ours	-1.50613 · 10^5	1.67	0.00
Table 46: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 0.1). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1 + warm start	-1.59508 · 10^5	3600.06	19.45
3000	big-M50 + warm start	-1.59508 · 10^5	3600.22	19.07
3000	persp. + warm start	-1.59508 · 10^5	3600.06	19.40
3000	eig. persp. + warm start	-1.59508 · 10^5	3600.14	19.52
3000	ours	-1.59508 · 10^5	3600.27	19.40
4000	SOS1 + warm start	-2.17573 · 10^5	3600.40	15.28
4000	big-M50 + warm start	-2.17573 · 10^5	3600.10	14.78
4000	persp. + warm start	-2.17573 · 10^5	3600.06	14.88
4000	eig. persp. + warm start	-2.17573 · 10^5	3600.10	15.34
4000	ours	-2.17573 · 10^5	99.53	0.00
5000	SOS1 + warm start	-2.71985 · 10^5	3601.08	12.00
5000	big-M50 + warm start	-2.71985 · 10^5	3600.07	11.22
5000	persp. + warm start	-2.71985 · 10^5	3600.06	11.34
5000	eig. persp. + warm start	-2.71985 · 10^5	1642.59	0.00
5000	ours	-2.71985 · 10^5	21.01	0.00
6000	SOS1 + warm start	-3.16760 · 10^5	3600.83	9.96
6000	big-M50 + warm start	-3.16760 · 10^5	3600.07	9.96
6000	persp. + warm start	-3.16760 · 10^5	3600.05	9.96
6000	eig. persp. + warm start	-3.16760 · 10^5	3600.06	11.77
6000	ours	-3.16760 · 10^5	20.91	0.00
7000	SOS1 + warm start	-3.72609 · 10^5	3600.67	8.33
7000	big-M50 + warm start	-3.72609 · 10^5	3601.58	8.33
7000	persp. + warm start	-3.72609 · 10^5	3601.01	8.31
7000	eig. persp. + warm start	-3.72609 · 10^5	3600.55	9.45
7000	ours	-3.72609 · 10^5	1.73	0.00
Table 47: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 0.1). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1 + warm start	-3.33603 · 10^5	3600.06	19.49
3000	big-M50 + warm start	-3.33603 · 10^5	3601.53	18.78
3000	persp. + warm start	-3.33603 · 10^5	3600.07	19.47
3000	eig. persp. + warm start	-3.33603 · 10^5	3600.06	19.56
3000	ours	-3.33603 · 10^5	3600.68	19.44
4000	SOS1 + warm start	-4.53260 · 10^5	3600.06	15.24
4000	big-M50 + warm start	-4.53260 · 10^5	3600.07	15.04
4000	persp. + warm start	-4.53260 · 10^5	3600.07	15.20
4000	eig. persp. + warm start	-4.53260 · 10^5	3601.61	15.40
4000	ours	-4.53260 · 10^5	3600.41	0.62
5000	SOS1 + warm start	-5.63006 · 10^5	3600.47	12.04
5000	big-M50 + warm start	-5.63006 · 10^5	3601.70	11.69
5000	persp. + warm start	-5.63006 · 10^5	3600.06	11.74
5000	eig. persp. + warm start	-5.63006 · 10^5	3600.50	0.47
5000	ours	-5.63006 · 10^5	37.26	0.00
6000	SOS1 + warm start	-6.57798 · 10^5	3601.18	9.98
6000	big-M50 + warm start	-6.57798 · 10^5	3600.18	9.98
6000	persp. + warm start	-6.57798 · 10^5	3600.06	9.56
6000	eig. persp. + warm start	-6.57798 · 10^5	1376.14	0.00
6000	ours	-6.57798 · 10^5	20.64	0.00
7000	SOS1 + warm start	-7.72266 · 10^5	3600.10	8.33
7000	big-M50 + warm start	-7.72266 · 10^5	3600.63	8.33
7000	persp. + warm start	-7.72266 · 10^5	3600.34	7.86
7000	eig. persp. + warm start	-7.72266 · 10^5	3600.44	10.37
7000	ours	-7.72266 · 10^5	20.91	0.00
Table 48: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 0.1). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1 + warm start	-7.41186 · 10^5	3600.06	19.51
3000	big-M50 + warm start	-7.41186 · 10^5	3601.56	18.15
3000	persp. + warm start	-7.41186 · 10^5	3600.87	19.52
3000	eig. persp. + warm start	-7.41186 · 10^5	3600.05	19.59
3000	ours	-7.41186 · 10^5	3600.59	19.46
4000	SOS1 + warm start	-1.00287 · 10^6	3600.07	15.21
4000	big-M50 + warm start	-1.00287 · 10^6	3600.09	14.98
4000	persp. + warm start	-1.00287 · 10^6	3600.07	15.18
4000	eig. persp. + warm start	-1.00287 · 10^6	3600.06	15.61
4000	ours	-1.00287 · 10^6	3600.08	1.88
5000	SOS1 + warm start	-1.24027 · 10^6	3601.15	12.08
5000	big-M50 + warm start	-1.24027 · 10^6	3600.40	11.88
5000	persp. + warm start	-1.24027 · 10^6	3601.42	12.05
5000	eig. persp. + warm start	-1.24027 · 10^6	3600.19	0.97
5000	ours	-1.24027 · 10^6	469.43	0.00
6000	SOS1 + warm start	-1.45416 · 10^6	3600.09	10.00
6000	big-M50 + warm start	-1.45416 · 10^6	3600.05	9.78
6000	persp. + warm start	-1.45416 · 10^6	3600.10	9.82
6000	eig. persp. + warm start	-1.45416 · 10^6	3600.16	0.22
6000	ours	-1.45416 · 10^6	59.16	0.00
7000	SOS1 + warm start	-1.70483 · 10^6	3600.74	8.34
7000	big-M50 + warm start	-1.70483 · 10^6	3600.07	8.34
7000	persp. + warm start	-1.70483 · 10^6	3600.06	8.13
7000	eig. persp. + warm start	-1.70483 · 10^6	1417.14	0.00
7000	ours	-1.70483 · 10^6	21.12	0.00
Table 49: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.7, λ2 = 0.1). 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	3600.07	19.52
3000	big-M50 + warm start	-2.78521 · 10^6	3600.07	15.91
3000	persp. + warm start	-2.78521 · 10^6	3600.06	19.50
3000	eig. persp. + warm start	-2.78521 · 10^6	3600.06	19.60
3000	ours	-2.78521 · 10^6	3600.23	19.46
4000	SOS1 + warm start	-3.75203 · 10^6	3600.07	15.10
4000	big-M50 + warm start	-3.75203 · 10^6	3600.27	13.35
4000	persp. + warm start	-3.75203 · 10^6	3600.51	15.10
4000	eig. persp. + warm start	-3.75203 · 10^6	3600.07	15.85
4000	ours	-3.75203 · 10^6	3601.10	3.50
5000	SOS1 + warm start	-4.61877 · 10^6	3600.07	12.11
5000	big-M50 + warm start	-4.61877 · 10^6	3600.12	11.40
5000	persp. + warm start	-4.61877 · 10^6	3600.06	12.09
5000	eig. persp. + warm start	-4.61877 · 10^6	3600.35	12.88
5000	ours	-4.61877 · 10^6	3600.40	0.93
6000	SOS1 + warm start	-5.43826 · 10^6	3600.37	10.02
6000	big-M50 + warm start	-5.43826 · 10^6	3600.07	9.79
6000	persp. + warm start	-5.43826 · 10^6	3600.07	10.00
6000	eig. persp. + warm start	-5.43826 · 10^6	3600.08	11.05
6000	ours	-5.43826 · 10^6	3600.26	0.33
7000	SOS1 + warm start	-6.36767 · 10^6	3600.50	8.35
7000	big-M50 + warm start	-6.36767 · 10^6	3600.07	8.22
7000	persp. + warm start	-6.36767 · 10^6	3600.06	8.33
7000	eig. persp. + warm start	-6.36767 · 10^6	3600.05	0.38
7000	ours	-6.36767 · 10^6	3600.58	0.11
Table 50: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 0.1). All baselines use our beam search solution as a warm start.
G.2.11 Synthetic 2 Benchmark with  $\lambda_{2} = 10.0$  with No Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-6.33576 · 104	3600.08	18.27
3000	big-M50	-6.33576 · 104	3600.31	18.47
3000	persp.	-6.33576 · 104	3600.20	17.53
3000	eig. persp.	-4.46130 · 104	3601.33	69.85
3000	ours	-6.33576 · 104	3601.41	4.54
4000	SOS1	-8.63287 · 104	3600.28	15.23
4000	big-M50	-8.63287 · 104	3600.85	15.21
4000	persp.	-8.63287 · 104	3600.15	12.69
4000	eig. persp.	-8.58997 · 104	3600.10	1.74
4000	ours	-8.63287 · 104	20.33	0.00
5000	SOS1	-1.09474 · 105	3600.08	11.86
5000	big-M50	-1.09474 · 105	3600.28	11.85
5000	persp.	-1.09474 · 105	3600.80	10.44
5000	eig. persp.	-1.09474 · 105	3600.06	13.55
5000	ours	-1.09474 · 105	20.71	0.00
6000	SOS1	-1.27542 · 105	3600.19	9.90
6000	big-M50	-1.27542 · 105	3600.37	9.90
6000	persp.	-1.27542 · 105	3600.14	9.03
6000	eig. persp.	-1.27542 · 105	2609.62	0.00
6000	ours	-1.27542 · 105	1.69	0.00
7000	SOS1	-1.42695 · 105	3600.12	14.24
7000	big-M50	-1.50512 · 105	3600.56	8.31
7000	persp.	-1.50512 · 105	3600.15	7.73
7000	eig. persp.	-1.50512 · 105	107.90	0.00
7000	ours	-1.50512 · 105	1.70	0.00
Table 51: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 10.0).
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-1.52806 · 10^5	3601.20	23.64
3000	big-M50	-1.59409 · 10^5	3600.72	18.52
3000	persp.	-1.59409 · 10^5	3600.14	17.84
3000	eig. persp.	-1.25897 · 10^5	3600.88	51.38
3000	ours	-1.59409 · 10^5	3600.62	4.68
4000	SOS1	-2.17472 · 10^5	3600.70	15.13
4000	big-M50	-2.17472 · 10^5	3600.07	14.64
4000	persp.	-2.17472 · 10^5	3600.05	12.93
4000	eig. persp.	-2.03767 · 10^5	3600.20	23.07
4000	ours	-2.17472 · 10^5	86.90	0.00
5000	SOS1	-2.71885 · 10^5	3600.32	11.94
5000	big-M50	-2.71885 · 10^5	3603.74	11.16
5000	persp.	-2.71885 · 10^5	3600.05	10.84
5000	eig. persp.	-2.71885 · 10^5	1969.80	0.00
5000	ours	-2.71885 · 10^5	20.75	0.00
6000	SOS1	-3.10076 · 10^5	3600.08	12.27
6000	big-M50	-3.16660 · 10^5	3600.07	9.93
6000	persp.	-3.16660 · 10^5	3600.06	9.24
6000	eig. persp.	-3.16660 · 10^5	2020.73	0.00
6000	ours	-3.16660 · 10^5	21.05	0.00
7000	SOS1	-3.72509 · 10^5	3600.17	8.31
7000	big-M50	-3.72509 · 10^5	3601.17	8.31
7000	persp.	-3.72509 · 10^5	3600.13	7.78
7000	eig. persp.	-3.72509 · 10^5	2139.72	0.00
7000	ours	-3.72509 · 10^5	1.71	0.00
Table 52: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 10.0).
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-3.31006 · 105	3600.07	19.45
3000	big-M50	-3.33503 · 105	3600.07	18.43
3000	persp.	-3.30991 · 105	3600.07	18.99
3000	eig. persp.	-2.97162 · 105	3600.08	34.18
3000	ours	-3.33503 · 105	3600.25	6.21
4000	SOS1	-4.36975 · 105	3600.35	19.35
4000	big-M50	-4.53159 · 105	3600.16	14.84
4000	persp.	-4.50480 · 105	3600.38	12.96
4000	eig. persp.	-4.38770 · 105	3600.11	19.13
4000	ours	-4.53159 · 105	3600.95	0.35
5000	SOS1	-5.37846 · 105	3600.98	17.20
5000	big-M50	-5.62906 · 105	3600.16	11.59
5000	persp.	-5.62906 · 105	3600.07	10.92
5000	eig. persp.	-5.62906 · 105	3600.19	12.71
5000	ours	-5.62906 · 105	31.47	0.00
6000	SOS1	-6.51605 · 105	3600.45	10.98
6000	big-M50	-6.57698 · 105	3600.16	9.95
6000	persp.	-6.57698 · 105	3601.04	9.39
6000	eig. persp.	-6.57698 · 105	2550.81	0.00
6000	ours	-6.57698 · 105	21.24	0.00
7000	SOS1	-7.65939 · 105	3600.07	9.20
7000	big-M50	-7.72166 · 105	3600.12	8.31
7000	persp.	-7.72166 · 105	3600.13	7.90
7000	eig. persp.	-7.72166 · 105	3600.06	10.13
7000	ours	-7.72166 · 105	20.37	0.00
Table 53: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 10.0).
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-7.35981 · 10^5	3600.07	19.40
3000	big-M50	-7.41086 · 10^5	3600.65	17.88
3000	persp.	-7.25990 · 10^5	3600.38	20.81
3000	eig. persp.	-7.18339 · 10^5	3600.06	23.35
3000	ours	-7.41086 · 10^5	3600.48	8.32
4000	SOS1	-9.99525 · 10^5	3600.07	15.44
4000	big-M50	-1.00277 · 10^6	3600.09	14.85
4000	persp.	-9.91636 · 10^5	3601.08	14.29
4000	eig. persp.	-9.92065 · 10^5	3600.10	16.78
4000	ours	-1.00277 · 10^6	3600.62	1.66
5000	SOS1	-1.23548 · 10^6	3600.08	12.44
5000	big-M50	-1.24017 · 10^6	3600.25	11.82
5000	persp.	-1.23555 · 10^6	3600.42	11.18
5000	eig. persp.	-1.21712 · 10^6	3600.05	14.73
5000	ours	-1.24017 · 10^6	386.72	0.00
6000	SOS1	-1.42629 · 10^6	3600.81	12.11
6000	big-M50	-1.45406 · 10^6	3600.07	9.75
6000	persp.	-1.45406 · 10^6	3600.07	9.44
6000	eig. persp.	-1.45406 · 10^6	2941.57	0.00
6000	ours	-1.45406 · 10^6	54.13	0.00
7000	SOS1	-1.69139 · 10^6	3600.49	9.17
7000	big-M50	-1.70473 · 10^6	3600.81	8.32
7000	persp.	-1.70473 · 10^6	3600.06	7.85
7000	eig. persp.	-1.70473 · 10^6	1585.19	0.00
7000	ours	-1.70473 · 10^6	21.30	0.00
Table 54: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.7, λ2 = 10.0).
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1	-2.78026 · 10^6	3600.08	18.78
3000	big-M50	-2.78507 · 10^6	3600.17	15.95
3000	persp.	-2.75178 · 10^6	3600.06	20.10
3000	eig. persp.	-2.73809 · 10^6	3600.11	21.60
3000	ours	-2.78507 · 10^6	3600.78	9.79
4000	SOS1	-3.74696 · 10^6	3600.08	15.12
4000	big-M50	-3.75192 · 10^6	3600.06	13.19
4000	persp.	-3.73412 · 10^6	3600.07	14.42
4000	eig. persp.	-3.73458 · 10^6	3600.81	16.31
4000	ours	-3.75192 · 10^6	3600.32	3.23
5000	SOS1	-4.60115 · 10^6	3600.29	12.47
5000	big-M50	-4.61867 · 10^6	3600.17	11.32
5000	persp.	-4.60427 · 10^6	3600.07	11.73
5000	eig. persp.	-4.58981 · 10^6	3600.07	13.56
5000	ours	-4.61867 · 10^6	3600.30	0.89
6000	SOS1	-5.42828 · 10^6	3600.08	10.18
6000	big-M50	-5.43816 · 10^6	3600.12	9.75
6000	persp.	-5.41942 · 10^6	3600.07	9.90
6000	eig. persp.	-5.41142 · 10^6	3600.42	11.59
6000	ours	-5.43816 · 10^6	3600.22	0.32
7000	SOS1	-6.33201 · 10^6	3600.05	8.93
7000	big-M50	-6.36757 · 10^6	3600.32	8.20
7000	persp.	-6.35270 · 10^6	3600.06	8.23
7000	eig. persp.	-6.36342 · 10^6	3600.05	0.46
7000	ours	-6.36757 · 10^6	3600.06	0.10
Table 55: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.9, λ2 = 10.0).
G.2.12 Synthetic 2 Benchmark with  $\lambda_{2} = 10.0$  with Warmstart
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1 + warm start	-6.33576 · 104	3601.40	18.27
3000	big-M50 + warm start	-6.33576 · 104	3600.12	18.47
3000	persp. + warm start	-6.33576 · 104	3601.92	17.53
3000	eig. persp. + warm start	-6.33576 · 104	3600.22	19.61
3000	ours	-6.33576 · 104	3601.41	4.54
4000	SOS1 + warm start	-8.63287 · 104	3600.08	15.23
4000	big-M50 + warm start	-8.63287 · 104	3600.29	15.21
4000	persp. + warm start	-8.63287 · 104	3600.30	12.69
4000	eig. persp. + warm start	-8.63287 · 104	3600.38	1.32
4000	ours	-8.63287 · 104	20.33	0.00
5000	SOS1 + warm start	-1.09474 · 105	3600.06	11.86
5000	big-M50 + warm start	-1.09474 · 105	3600.15	11.85
5000	persp. + warm start	-1.09474 · 105	3601.20	10.44
5000	eig. persp. + warm start	-1.09474 · 105	3601.16	13.51
5000	ours	-1.09474 · 105	20.71	0.00
6000	SOS1 + warm start	-1.27542 · 105	3600.07	9.90
6000	big-M50 + warm start	-1.27542 · 105	3601.39	9.89
6000	persp. + warm start	-1.27542 · 105	3600.21	9.03
6000	eig. persp. + warm start	-1.27542 · 105	3601.11	12.70
6000	ours	-1.27542 · 105	1.69	0.00
7000	SOS1 + warm start	-1.50512 · 105	3600.28	5.83
7000	big-M50 + warm start	-1.50512 · 105	3601.02	8.31
7000	persp. + warm start	-1.50512 · 105	3600.16	7.73
7000	eig. persp. + warm start	-1.50512 · 105	104.66	0.00
7000	ours	-1.50512 · 105	1.70	0.00
Table 56: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.1, λ2 = 10.0). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1 + warm start	-1.59409 · 10^5	3601.55	18.52
3000	big-M50 + warm start	-1.59409 · 10^5	3600.67	18.52
3000	persp. + warm start	-1.59409 · 10^5	3600.26	17.84
3000	eig. persp. + warm start	-1.59409 · 10^5	3600.07	19.55
3000	ours	-1.59409 · 10^5	3600.62	4.68
4000	SOS1 + warm start	-2.17472 · 10^5	3600.06	15.13
4000	big-M50 + warm start	-2.17472 · 10^5	3600.08	14.64
4000	persp. + warm start	-2.17472 · 10^5	3600.78	12.93
4000	eig. persp. + warm start	-2.17472 · 10^5	3600.35	2.79
4000	ours	-2.17472 · 10^5	86.90	0.00
5000	SOS1 + warm start	-2.71885 · 10^5	3601.07	11.94
5000	big-M50 + warm start	-2.71885 · 10^5	3600.07	11.16
5000	persp. + warm start	-2.71885 · 10^5	3600.22	10.25
5000	eig. persp. + warm start	-2.71885 · 10^5	3109.70	0.00
5000	ours	-2.71885 · 10^5	20.75	0.00
6000	SOS1 + warm start	-3.16660 · 10^5	3600.92	9.93
6000	big-M50 + warm start	-3.16660 · 10^5	3600.06	9.93
6000	persp. + warm start	-3.16660 · 10^5	3600.04	9.24
6000	eig. persp. + warm start	-3.16660 · 10^5	3601.01	11.80
6000	ours	-3.16660 · 10^5	21.05	0.00
7000	SOS1 + warm start	-3.72509 · 10^5	3600.29	8.31
7000	big-M50 + warm start	-3.72509 · 10^5	3601.61	8.31
7000	persp. + warm start	-3.72509 · 10^5	3600.12	7.78
7000	eig. persp. + warm start	-3.72509 · 10^5	3600.07	9.46
7000	ours	-3.72509 · 10^5	1.71	0.00
Table 57: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.3, λ2 = 10.0). All baselines use our beam search solution as a warm start.
# of samples	method	upper bound	time(s)	gap(%)
3000	SOS1 + warm start	-3.33503 · 10^5	3600.95	18.55
3000	big-M50 + warm start	-3.33503 · 10^5	3600.08	18.43
3000	persp. + warm start	-3.33503 · 10^5	3600.40	18.09
3000	eig. persp. + warm start	-3.33503 · 10^5	3600.06	19.55
3000	ours	-3.33503 · 10^5	3600.25	6.21
4000	SOS1 + warm start	-4.53159 · 10^5	3600.06	15.09
4000	big-M50 + warm start	-4.53159 · 10^5	3600.06	14.84
4000	persp. + warm start	-4.53159 · 10^5	3673.11	12.29
4000	eig. persp. + warm start	-4.53159 · 10^5	3600.09	15.34
4000	ours	-4.53159 · 10^5	3600.95	0.35
5000	SOS1 + warm start	-5.62906 · 10^5	3600.07	11.98
5000	big-M50 + warm start	-5.62906 · 10^5	3600.08	11.59
5000	persp. + warm start	-5.62906 · 10^5	3600.06	10.92
5000	eig. persp. + warm start	-5.62906 · 10^5	3600.31	0.24
5000	ours	-5.62906 · 10^5	31.47	0.00
6000	SOS1 + warm start	-6.57698 · 10^5	3600.88	9.95
6000	big-M50 + warm start	-6.57698 · 10^5	3600.14	9.95
6000	persp. + warm start	-6.57698 · 10^5	3600.06	9.02
6000	eig. persp. + warm start	-6.57698 · 10^5	1703.35	0.00
6000	ours	-6.57698 · 10^5	21.24	0.00
7000	SOS1 + warm start	-7.72166 · 10^5	3600.06	8.32
7000	big-M50 + warm start	-7.72166 · 10^5	3601.52	8.31
7000	persp. + warm start	-7.72166 · 10^5	3600.16	7.90
7000	eig. persp. + warm start	-7.72166 · 10^5	3600.41	10.39
7000	ours	-7.72166 · 10^5	20.37	0.00
Table 58: Comparison of time and optimality gap on the synthetic datasets (p = 3000, k = 10, ρ = 0.5, λ2 = 10.0). All baselines use our beam search solution as a warm start.