Autoresearch in Mixed-Integer Linear and Nonlinear Programming

Yuwei Gu; Yaoxin Wu; Tong Guo; Wen Song; Zhiguang Cao

cs.AISep 30, 2026

Autoresearch in Mixed-Integer Linear and Nonlinear Programming

Authors: Yuwei Gu, Yaoxin Wu, Tong Guo, Wen Song, Zhiguang Cao

Organizations: Chengdu University of Information Technology · Eindhoven University of Technology · Nanyang University of Technology · Shandong University · Singapore Management University

Abstract

Despite recent progress in autoresearch, applying it to practical operations research problems, typically formulated as NP-hard mixed-integer linear or nonlinear programs (MILPs or MINLPs), remains challenging because effective research requires systematically managing competing ideas and long-horizon experimental trajectories. We introduce AutoMIP, a reusable agent skill for organizing long-horizon autoresearch in mixed-integer programming through idea pooling and algorithm tree search. AutoMIP maintains a persistent pool of complementary candidate ideas while organizing executable experiments into an algorithm tree, enabling the agent to preserve unexplored hypotheses, refine promising algorithms, and switch to alternative methodological directions based on historical states. On MILP and MINLP benchmark cohorts, AutoMIP achieves the highest final success rates among the evaluated autoresearch frameworks. On MIPLib, AutoMIP discovers new best solutions for 31 of 60 instances, surpassing existing autoresearch frameworks. On MINLPLib, it achieves new best solutions for 52 of 60 instances. Ablation studies further demonstrate the complementary contributions of idea pooling and algorithm tree search, highlighting the importance of jointly maintaining diverse research ideas and structured experimental trajectories for long-horizon autoresearch.

Figures & tables

Figure 1: AutoMIP workflow. AutoMIP preserves candidate hypotheses in idea pool, selects refinements or new branches (research directions) from reusable experimental states in algorithm tree, and updates idea pool and algorithm tree from experimental feedback. The workflow supports MILP and MINLP optimization, with MPS taken as an example input format.

Figure 2: Coupling a direction to its starting state. After n0 $\rightarrow$ n1 $\rightarrow$ n4, selecting P5 and reusing n0 creates the $R$ /P5 branch n3. Solid edges record ancestry; dashed arc restores historical artifacts.

Input: Exploration state $M_{t}=(T_{t},F_{t},P_{t},c_{t},A_{t})$ .
Output: Registered experiment and restored experimental state.
Read the current node $c_{t}$ , frontier nodes $F_{t}$ , candidate pool $P_{t}$ , and artifacts $A_{t}$ .
if An admissible refinement exists:
	Set $d_{t}=L$ and $p_{t}=c_{t}$ .
else if an unused candidate is available:
	Sample $q_{t}\sim\mathrm{Uniform}(U_{t})$ .

Algorithm 1 Algorithm Tree Search

Method	0–1 h			1–3 h			3–6 h			6–12 h			Total
	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token
AutoMIP	14/24	1,211	320,244	4/5	5,545	912,789	0/2	11,256	1,960,932	0/0	43,200	961,095	18/31	22,202	734,057
Codex	1/13	1,822	393,785	3/11	6,426	1,070,770	0/4	13,980	2,857,174	1/1	42,049	1,137,639	5/29	24,931	1,078,847
Loop	3/15	1,888	323,983	1/6	7,123	648,990	0/3	11,665	1,685,786	0/1	42,679	1,700,813	4/25	27,375	1,310,672
AutoEoH	0/12	2,157	606,353	3/9	7,142	1,035,190	0/4	14,175	2,556,814	0/1	43,162	2,660,954	3/26	27,626	1,999,227
EvoX	0/6	1,302	306,763	0/5	7,032	720,454	0/3	13,390	1,586,613	1/1	42,975	1,244,128	1/15	34,333	2,297,150

Table 1: Open cohort, 60 instances.

Method	0–10 min			10–20 min			20–30 min			30–60 min			Total
	solved	time (s)	token	solved	time (s)	token	solved	time (s)	token	solved	time (s)	token	solved	time (s)	token
AutoMIP	14	399	96,926	8	869	120,307	5	1,511	154,571	3	3,454	171,106	30	2,035	145,646
Codex	6	322	196,416	6	909	116,999	5	1,471	153,526	12	2,912	206,999	29	2,333	192,485
Loop	4	518	80,309	8	873	102,209	4	1,364	135,665	6	3,218	228,227	22	2,602	195,393
AutoEoH	2	478	97,654	6	923	110,134	4	1,471	142,942	7	2,943	182,870	19	2,561	170,094
EvoX	3	574	117,775	6	961	101,781	4	1,546	203,375	2	2,894	254,562	15	2,495	229,032

Table 2: Hard cohort, 60 instances and a one-hour horizon.

Figure 3: Cumulative task success from per-instance timestamps. Each step adds a successful instance; the denominator is 60 in every panel. Curves extend to 43,200 s for Open and MINLPLib and 3,600 s for Hard. The cutoffs agree with Tables 1 , 2 , and 4 , respectively.

Method	0–1 h			1–3 h			3–6 h			6–12 h			Total
	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token
AutoMIP	5/8	1,440	317,315	3/5	5,545	912,789	1/2	11,256	1,960,932	0/0	43,200	2,076,973	9/15	13,888	1,070,460
NoPool	1/5	2,019	233,353	2/4	5,643	450,314	1/2	11,460	1,168,343	0/1	42,334	2,874,266	4/12	21,829	1,558,655
NoTree	1/3	1,304	202,388	1/6	5,841	518,799	0/1	13,560	604,118	0/0	43,200	2,935,203	2/10	24,226	1,683,805

Table 3: Ablations on 20 shared instance identities.

Method	0–1 h			1–3 h			3–6 h			6–12 h			Total
	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token
AutoMIP	13/40	1,076	203,122	3/10	5,895	623,950	0/1	19,140	1,051,147	0/1	41,089	3,376,242	16/52	8,182	763,362
Codex	9/35	1,219	200,220	2/10	5,501	409,595	0/0	-	-	0/0	43,200	2,927,059	11/45	12,428	916,826
Loop	8/32	1,553	262,392	1/6	6,437	465,120	0/2	18,703	1,779,411	0/0	43,200	3,537,847	9/40	16,495	1,425,050
AutoEoH	7/25	933	193,259	1/4	6,538	546,097	0/0	-	-	0/0	43,200	3,513,110	8/29	23,145	1,932,038
EvoX	8/31	1,111	218,111	0/0	-	-	0/0	-	-	0/0	43,200	3,241,573	8/31	21,454	1,679,451

Table 4: MINLPLib, the supplied common 60-instance subset.

Method	0–1 h			1–3 h			3–6 h			6–12 h			Total
	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token	w/s	time (s)	token
Codex	6/9	1,385	312,966	2/5	5,089	944,082	1/1	11,085	1,688,996	0/0	43,200	4,641,312	9/15	13,250	1,621,633
Claude Code	2/7	1,819	305,213	2/3	6,326	1,182,345	2/2	13,433	1,679,399	0/0	43,200	4,680,348	6/12	20,209	2,324,256

Table 5: Cross-agent evaluation on 20 instances. Both agents use AutoMIP.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Mechanism	NoPool	NoTree
PoolThink and persistent candidate pool	Removed	Retained
On-demand direction generation	Retained	Not used
Random selection from unused candidates	Removed	Retained
Multiway tree and parent links	Retained	Removed
$L/R$ labels and frontier	Retained	Removed
Historical-state selection and backtracking	Retained	Removed

Appendix

Table 6: Components retained and removed in the two complementary ablations.

Figure 4: Instance-level convergence over ten trajectories per instance: five random seeds for each of the two methods. Lines show each method’s pointwise median; shaded bands show its interquartile range (25th–75th percentiles) across the five seeds.

Instance	AutoMIP			Codex			Loop
	Result	Time (s)	Token	Result	Time (s)	Token	Result	Time (s)	Token
allcolor58	YES	7,491	2,242,663	YES	11,061	3,272,799	YES	11,460	944,874
cmflsp40-36-2-10	YES	1,707	389,130	YES	5,460	558,205	YES	32,772	4,151,482
dc1l	YES	1,713	457,746	YES	8,612	3,614,436	NO	-	-
neos-1423785	YES	3,660	356,572	YES	43,200	2,819,091	YES	6,780	382,582
nsr8k	YES	2,467	670,627	YES	5,460	951,526	YES	5,580	104,652

Appendix

Table 7: Per-instance Open results, panel 1.

Instance	AutoEoH			EvoX
	Result	Time (s)	Token	Result	Time (s)	Token
allcolor58	YES	8,064	1,589,550	NO	-	-
cmflsp40-36-2-10	YES	3,399	532,571	NO	-	-
dc1l	NO	-	-	YES	13,380	839,628
neos-1423785	YES	3,761	393,219	YES	1,940	322,286
nsr8k	YES	13,800	3,076,397	NO	-	-

Appendix

Table 8: Per-instance Open results, panel 2.

Instance	AutoMIP			Codex			Loop
	Result	Time (s)	Token	Result	Time (s)	Token	Result	Time (s)	Token
2club200v15p5scn	YES	1,048	118,047	YES	3,159	178,653	YES	720	98,788
8div-n59k10	NO	-	-	NO	-	-	NO	-	187,464
8div-n59k11	NO	-	-	NO	-	255,446	NO	-	126,442
8div-n59k12	NO	-	-	NO	-	217,464	NO	-	-
academictimetablebig	YES	1,297	111,128	YES	3,113	160,989	YES	704	77,285

Appendix

Table 9: Per-instance Hard results, panel 1.

Instance	AutoEoH			EvoX
	Result	Time (s)	Token	Result	Time (s)	Token
2club200v15p5scn	YES	1,099	189,724	YES	1,660	155,633
8div-n59k10	NO	-	259,168	NO	-	-
8div-n59k11	NO	-	-	NO	-	114,174
8div-n59k12	NO	-	-	NO	-	153,108
academictimetablebig	NO	-	100,525	NO	-	-

Appendix

Table 10: Per-instance Hard results, panel 2.

Instance	AutoMIP			NoPool			NoTree
	Result	Time (s)	Token	Result	Time (s)	Token	Result	Time (s)	Token
allcolor58	YES	7,491	2,242,663	YES	10,920	1,091,491	NO	-	-
dc1l	YES	1,713	457,746	NO	-	-	NO	-	-
neos-1423785	YES	3,660	356,572	NO	-	-	YES	13,560	604,118
s82	YES	172	76,005	YES	12,000	1,245,195	YES	1,531	172,552
triptim4	NO	-	-	NO	-	-	NO	-	-

Appendix

Table 11: Per-instance Ablation results.

Instance	Codex			Claude Code
	Result	Time (s)	Token	Result	Time (s)	Token
allcolor58	YES	7,491	2,242,663	YES	9,120	2,942,663
dc1l	YES	1,713	457,746	NO	-	-
s82	YES	172	76,005	YES	15,420	2,743,724
triptim4	NO	-	-	NO	-	-
zeil	YES	11,085	1,688,996	YES	3,078	412,299

Appendix

Table 12: Per-instance CLI-transfer results.

Instance	AutoMIP			Codex			Loop
	Result	Time (s)	Token	Result	Time (s)	Token	Result	Time (s)	Token
elec100	NO	-	-	NO	-	-	NO	-	-
gabriel09	YES	1,012	108,534	YES	4,154	242,505	YES	2,294	156,847
etamac	YES	1,226	358,589	YES	611	200,631	YES	699	205,836
eq6_1	YES	660	122,384	YES	927	121,620	YES	754	316,728
emfl050	YES	1,219	215,608	YES	3,230	451,415	YES	2,678	423,937

Appendix

Table 13: Per-instance MINLPLib results, panel 1.

Instance	AutoEoH			EvoX
	Result	Time (s)	Token	Result	Time (s)	Token
elec100	NO	-	-	NO	-	-
gabriel09	YES	383	109,927	YES	692	242,582
etamac	YES	1,473	221,210	YES	1,646	272,268
eq6_1	YES	557	118,939	YES	2,508	230,076
emfl050	YES	626	175,207	YES	982	249,243

Appendix

Table 14: Per-instance MINLPLib results, panel 2.

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