Large language models (LLMs) have recently shown significant potential in various biological tasks such as protein engineering and molecule design. These tasks typically involve black-box discrete sequence optimization, where the challenge lies in generating sequences that are not only biologically feasible but also adhere to hard fine-grained constraints. However, LLMs often struggle with such constraints, especially in biological contexts where verifying candidate solutions is costly and time-consuming. In this study, we explore the possibility of employing LLMs as highly-constrained bilevel optimizers through a methodology we refer to as Language Model Optimization with Margin Expectation (LLOME). This approach combines both offline and online optimization, utilizing limited oracle evaluations to iteratively enhance the sequences generated by the LLM. We additionally propose a novel training objective -- Margin-Aligned Expectation (MargE) -- that trains the LLM to smoothly interpolate between the reward and reference distributions. Lastly, we introduce a synthetic test suite that bears strong geometric similarity to real biophysical problems and enables rapid evaluation of LLM optimizers without time-consuming lab validation. Our findings reveal that, in comparison to genetic algorithm baselines, LLMs achieve significantly lower regret solutions while requiring fewer test function evaluations. However, we also observe that LLMs exhibit moderate miscalibration, are susceptible to generator collapse, and have difficulty finding the optimal solution when no explicit ground truth rewards are available.

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