HINTS: GENERATING MIP WARM STARTS VIA AN EVOLVING EXPERIENCE LIBRARY

Mixed-integer programming (MIP) solvers provide a general framework for combinatorial optimization, but their performance can depend heavily on how quickly they find a strong incumbent—the best feasible solution found so far. Specialized heuristics can produce good solutions quickly, yet they lack the systematic search and optimality guarantees of MIP solvers. We ask whether LLM-generated heuristics can combine these strengths by providing better starting solutions while leaving the solver unchanged. We propose **HINTS** (**H**euristic **IN**jection of **T**ight **S**tarts), which uses an LLM to generate a construction heuristic that produces a feasible MIP start for each instance. The start is injected into an unchanged solver, which then improves it through its native branch-and-bound search. To adapt construction strategies across heterogeneous instances, HINTS maintains a growing library of scoped experiences that record when a construction applies and how to build the corresponding start. Experiences are evaluated by the solver's progress after injection rather than by the heuristic solution alone. Across eight CO-Bench tasks and 1,611 instances, HINTS reduces the mean MIP gap across three solvers from 0.375 to 0.172 and consistently outperforms fixed textbook warm starts. With HINTS, OR-Tools further outperforms solver-only COPT and approaches solver-only Gurobi. These results show that LLM-generated warm starts can improve MIP solving without modifying the solver itself.