Step-by-Step Optimization-like Reasoning in LLMs over Expanding Search Spaces

Verifiable reward training has improved mathematical and coding reasoning, but these domains capture only part of step-by-step decision making. Many real-world tasks require finding a high-value feasible plan among many valid alternatives. We introduce $\text{OPT}\star$, a scalable family of optimization-style tasks for training and evaluating LLM step-by-step optimization-like reasoning along a complexity axis: each task provides a feasibility checker and evaluator, while a complexity parameter expands the search space without requiring new human labels. This motivates studying these tasks in two regimes: (i) solver-guided online policy optimization, which uses a solver as a value oracle for partial states and applies rank-based reward shaping to reinforce better next steps, and (ii) search-based offline RL when such solvers are unavailable. Theoretically, we relate success in large search spaces to the information a reasoner extracts per unit of search budget. Empirically, we ablate the ingredients that make search efficient on $\text{OPT}\star$ and show that training on $\text{OPT}\star$ improves step-by-step optimization-like reasoning.