NextEval: An Agent-Based Solver and Benchmark for Derivative-Free Optimization

Derivative-free optimization (DFO) addresses problems where derivatives are unavailable or costly to obtain. This setting often arises in simulation-based and experimental applications, where evaluating their black-box objective can itself be costly. Thus, it remains a central challenge in DFO to effectively exploit accumulated observations to select informative evaluation points under a limited budget. In this light, we introduce the NextEval solver, an agent-based DFO solver, which integrates the capabilities of the large language model (LLM) to guide the sequential selection of evaluation points. Specifically, we establish a system which combines an LLM agent with a numerical environment. The environment provides tools for analyzing past evaluations and executing numerical methods, while maintaining a shared evaluation history and method-specific state to support continuation across method switches. Using observations and tool feedback, the LLM agent selects tools, configures their parameters, and allocates the evaluation budget, while numerical methods compute and evaluate candidate points. Building on this system, we also introduce NextEval Bench, a benchmark protocol for comparing agent configurations on shared DFO tasks using the same numerical tools under matched evaluation budgets and information constraints. Numerical results on low-dimensional DFO problems demonstrate that the NextEval solver outperforms the state-of-the-art DFO baselines in function-evaluation efficiency.