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

• The study targets choices among continuing a method, changing parameters, improving the sample set, or switching methods based on progress, sample geometry, and model-prediction agreement.Source: §1 · p. 1
• The agent follows a history-dependent policy in which new evaluations can change its next operation and parameters, using general-purpose LLMs without optimization-specific policy training.Source: §1 · p. 2
• NextEval Bench compares LLM–harness configurations on individual problems in a common environment and summarizes performance across the problem set.The protocol builds on OptiProfiler, which provides problem libraries, configurable test conditions, and reproducible performance and data profiles.Source: §1 · p. 2
• NextEval combines an agent with numerical tools, shared evaluation history, and method-specific continuation state; the agent selects operations, parameters, and evaluation allowances.Source: §1 · p. 2
• On low-dimensional S2MPJ problems, NextEval has higher aggregate evaluation-efficiency scores than the tested DFO baselines.Source: §1 · p. 2
• The agent receives only permitted evaluation history, derived summaries, and generic metadata—not the objective formula, benchmark identifier, derivatives, or reference optimum.Only objective calls consume the budget; analyzing existing observations is free, while a search tool may make several calls before returning control.Source: §1 · p. 2

2 NEXTEVAL SOLVER

• NextEval combines an LLM agent that selects tools and parameters with a numerical environment that executes requests, tracks evaluations, and returns feedback.The agent reasons over accumulated observations, while numerical methods generate candidate points; the overview also shows the environment maintaining evaluation history and method-specific state.Source: §2 · pp. 2–3, Figure 2 · p. 3

2.1 AGENT–ENVIRONMENT INTERACTION

• The agent can continue, reconfigure, or switch methods using feedback, while the environment manages the shared history, budget, incumbent, and each method’s state.Tool calls use the conversation and environment state; multi-step calls can generate further points without another agent decision after each evaluation, while analysis calls make no evaluations.Source: §2.1 · p. 3

2.2 ANALYSIS AND SEARCH TOOLS

• Analysis tools inspect progress and local-model reliability without evaluations, while search tools generate and evaluate candidates or obtain samples for later model-based search.The agent can use situation summaries and unevaluated model previews to decide whether more samples are needed; single-step calls return after one operation, while multi-step calls run up to an agent-specified evaluation limit capped by the remaining budget.Source: §2.2 · p. 4

2.3 STATE MANAGEMENT AND CONTINUATION

• NextEval shares evaluation history but retains each method’s parameters and numerical state separately, allowing continuation while accounting for changes made by other tools.When another method finds a better point, the environment checks whether a saved trust-region interpolation set needs updating or rebuilding; numerical state is kept separate from the agent’s conversation so methods can continue after compression.Source: §2.3 · p. 4

Benchmark protocol

• NextEval Bench evaluates agent configurations on matched tasks through a shared numerical environment, with fixed tools, budgets, and permitted information.Run records are verified before performance is compared using OptiProfiler.Source: Figure 3 · p. 5, §3 · p. 5

Tasks and problem features

• Tasks use OptiProfiler test problems and extend standard settings with perturbed starting points, observation noise, and rotated coordinates.Feature seeds are shared across agents; noise is freshly sampled on every evaluation, including repeated evaluations at the same point.Source: §3 · p. 5

Agent configurations and execution

• Each complete optimization session specifies the LLM, harness, prompts, inference settings, and resource limits, while the environment preserves history and numerical state.Objective evaluations, tool calls, LLM calls, and wall time are recorded separately; problem identities, formulas, and scoring references are hidden from the agent.Source: §3 · p. 5

Performance evaluation

• Performance is measured by evaluations to reach relative reduction targets, with verified run records used to construct OptiProfiler performance and data profiles.Noisy tasks are scored using clean objective values; when retries occur, profiles use the final authorized attempt and do not sum evaluation costs across attempts.Source: §3 · p. 5

4.1 EXPERIMENTAL SETUP

• The suite comprised 92 unconstrained S2MPJ problems with n ≤5, and each run had a budget of 50n objective evaluations.Performance and data profiles summarize target-reaching evaluations, normalized respectively by the smallest count per problem and by dimension plus one; higher curves indicate more targets reached. Figures show τ = 10^-6, while Table 2 also reports τ = 10^-4 scores.Source: §4.1 · p. 6

4.3 AUDITED CASE STUDY

• On SNAIL, GPT-5.6 Terra switched after stalling and reached a best recorded value of 1.157 × 10^-3 at evaluation 98, within the 100-evaluation budget.The clean objective value was 2.101×10^-4, but clean values and contours were evaluator-only and unavailable to the agent.Source: §4.3 · p. 7, Figure 5 · p. 7

4.2 PRIMARY PERFORMANCE COMPARISON WITH CLASSICAL DFO

• NextEval had the largest aggregate score in the rotated comparison; finite-difference BFGS was competitive without noise but degraded under noise.The result measures agent decisions together with numerical tools; the comparison used the built-in runner with DeepSeek V4 Flash, thinking disabled, and ten repetitions.Source: §4.2 · pp. 6–7, Figure 4 · p. 7, Table 2 · p. 8

4.4 CONTROLLED AGENT-CONFIGURATION STUDIES

• GPT-5.6 Sol, Terra, and Luna scored 0.981–0.997 versus 0.571 for GPT-5.5, but GPT-5.5 used 13% of the budget at the median versus 100% for GPT-5.6.With Codex and medium reasoning fixed, GPT-5.6 reached more targets; the difference reflects both search and stopping behavior, not reasoning ability alone.Source: §4.4 · p. 8, Figure 6 · p. 8
• With Qwen 3.8 Flash and Codex fixed, medium thinking raised the aggregate score from 0.924 to 1.000 and lowered the median final clean value on 53 problems, versus 39 with thinking off.Additional reasoning helped in this setting but did not improve every problem.Source: §4.4 · p. 8, Figure 6 · p. 8
• With DeepSeek V4 Flash and thinking off, Claude Code scored 0.999 versus 0.947 for the built-in runner under the same instructions, tools, decoding limits, and evaluation budgets.This comparison measures the complete harness, including context management and session recovery; partial failure histories were retained.Source: §4.4 · p. 8, Figure 6 · p. 8

Numerical optimization without derivatives

• Direct search, interpolation-based trust regions, evolution strategies, and BO are established DFO approaches, while TuRBO's adaptive allocation is part of a specified optimization algorithm.NextEval exposes numerical operations as tools rather than replacing their calculations with language-model arithmetic (Hansen, 2016; Eriksson et al., 2019).Source: §5 · p. 8

Language models and learned optimizers

• Unlike learned or prompted methods that mainly generate candidates, configurations, or surrogate quantities, NextEval lets an agent inspect history, select and configure methods, continue or switch them, and allocate bounded evaluations.Prior approaches include learned recurrent DFO (Chen et al., 2017), OPRO and EvoLLM (Yang et al., 2024; Lange et al., 2024), LLAMBO (Liu et al., 2024), and SemanticOpt (Meindl et al., 2025).Source: §5 · p. 9

Optimization agents

• Related optimization agents study strategy discovery, HPO coordination, multi-agent BO, physics-grounded simulations, reusable tool graphs, or computational-mathematics workflows, but not the same oracle-only DFO setting and evaluation budget.Code-to-Harness evolves and freezes portable strategies (Wu et al., 2026); related examples include AgentHPO, multi-agent BO, OptAgent (Jiang et al., 2026), GATE (Luo et al., 2025), optimal-control interaction (Luo et al., 2023), and Iteris (Chen et al., 2026).Source: §5 · p. 9

Evaluation infrastructure

• NextEval Bench combines oracle-controlled numerical tasks, agent-level records, and suite-level profiles, with manifests fixing evaluation conditions and the complete LLM–harness configuration.Its current scope is a protocol and local execution machinery, not hosted benchmarking; related infrastructure includes COCO and OptiProfiler, MetaBox-v2 (Ma et al., 2025), Harbor (Harbor contributors, 2026a;b), and Terminal-Bench-Science.Source: §5 · p. 9

6 LIMITATIONS AND BENCHMARK ROADMAP

• Development feedback shaped the tools, manuals, and prompts, so generalization to unseen problem families remains untested; perturbations and structure-preserving rotations do not rule out problem recognition.The rotation preserves the objective’s value structure and is not full anonymization.Source: §6 · p. 9
• The study measures evaluation counts rather than total runtime or LLM cost, and whole-system comparisons do not isolate gains attributable to the LLM.Appendix E’s controller-replacement and numerical-support ablations are planned tests, not completed results.Source: §6 · p. 9
• Higher-dimensional problems and additional problem families remain important tests of the solver and benchmark.The primary comparison covers public problems with at most five variables.Source: §6 · p. 9
• Training a parameterized controller, including through reinforcement learning, is a separate extension rather than a completed component of this study.Source: §6 · p. 9

7 CONCLUSION

• On the low-dimensional S2MPJ suite with perturbed starts and rotated coordinates, NextEval’s primary configuration has a higher aggregate score than the tested DFO baselines.The persistent agent selects history summaries and numerical methods, while the environment executes evaluations and retains method state; NextEval Bench compares LLM–harness configurations under a common evaluation budget.Source: §7 · p. 9

REPRODUCIBILITY STATEMENT

• OptiProfiler generates all displayed profiles from saved histories; Appendices A and B specify experimental settings and trial accounting, respectively.Historical experiments are reported under the settings used when they were conducted.Source: REPRODUCIBILITY STATEMENT · p. 10

AI USE STATEMENT

• AI assistants supported coding, literature discovery, experimental design, trajectory analysis, interpretation, drafting, language editing, and creation of Figures 1–3.The authors are responsible for verifying the final code, citations, and empirical claims.Source: AI USE STATEMENT · p. 10

B TRIAL ACCOUNTING AND REFERENCE-SET SENSITIVITY

• Completed trials, including normal early stops and verified session-limit stops, are retained; the selected result is the latest authorized terminal attempt, not the best-objective attempt.Failed attempts may be retried under the recorded recovery policy, while completed parents are not retried and all attempts remain in the audit trail.Source: A EXPERIMENTAL PROTOCOL · p. 14

A EXPERIMENTAL PROTOCOL

• Start directions and applicable rotations are shared across solvers within each problem–repetition pair, while observation noise is resampled on every evaluation.Only start perturbations and noise use level 10^-3; rotation is an orthogonal coordinate change without anisotropic scaling or observation noise.Source: A EXPERIMENTAL PROTOCOL · p. 14
• The primary rotated comparison uses DeepSeek V4 Flash across ten repetitions, with five solver arms sharing transformed starts and 50n objective-call budgets.Earlier perturbed-start cohorts use a different runner and three repetitions, so the cohorts are analyzed separately rather than pooled.Source: A EXPERIMENTAL PROTOCOL · p. 14

A.1 AGENT INFORMATION AND PROBLEM RECOGNITION

• The task payload withholds benchmark identifiers, formulas, reference optima, and evaluator paths, but these restrictions do not guarantee against problem recognition or training-data contamination.Observed points and values may reveal the function family, and the payload boundary alone does not establish filesystem isolation for an external coding harness.Source: §A.1 · p. 15

A.2 EXACT PROFILE SCORE AND NONFINITE CONVENTIONS

• Profile targets use the best clean value in the declared comparison set, including the initial value, and compare relative or dimension-scaled attainment costs across solvers.Higher performance and data-profile curves indicate more targets reached within the corresponding relative or dimension-scaled budget.Source: §A.2 · p. 16
• Aggregate scores average repetitions before integration, normalize within each tolerance, then average ten normalized performance scores; the displayed 10^-6 panels are not selected by model family.Perf. and Data report normalized performance- and data-profile areas, respectively.Source: §A.2 · p. 16
• The archived scorer maps NaN merits to +∞; with an infinite initial merit it declares all solvers to pass an infinite target, a degenerate case excluded from finite-start sensitivity analysis.That convention is not evidence of successful optimization.Source: §A.2 · p. 16

B.1 PRIMARY ROTATED SUITE

• The primary rotated suite contains 92 problems, ten repetitions, five solver arms, and 4,600 trials, with a shared 50n objective budget per arm.Each arm uses the same perturbed start followed by an orthogonal coordinate rotation, without observation noise; profiles are generated by reloading recorded histories in OptiProfiler.Source: §B.1 · p. 16
• Of 920 primary-suite agent trials, 914 completed and six failed; all six histories remain in the comparison, with none rerun or excluded.Five failures occurred on MGH17LS and one on MISRA1ALS; the failures followed repeated invalid evaluations or an eigensolver error.Source: §B.1 · pp. 16–17
• The feature replay audit verified rotated values against f(Qz) and checked stored transformations’ orthogonality, supporting feature identity but not unrecognizability of public test functions.Orthogonal rotations preserve objective values and can preserve recognizable structure.Source: §B.1 · p. 17

B.2 EARLIER PERTURBED-START STUDIES

• NextEval’s aggregate-score advantage over the strongest reference had 95% bootstrap intervals of [0.010, 0.217] for perturbed starts and [0.178, 0.477] with noise.Intervals describe variation over the problem suite, not arbitrary objectives; excluding the single problem with nonfinite perturbed starts in each earlier cohort preserved NextEval’s highest aggregate score.Source: §B.2 · p. 17
• For the earlier five-arm sets, 5,000 problem-bootstrap resamples retained all solvers and repetitions, with first-hit counts and profile normalization recomputed for each draw.The earlier four-arm completion-conditioned scores and intervals were not reused because changing participants changes empirical targets and profile normalization.Source: §B.2 · p. 17

B.3 HISTORICAL THREE-REPEAT GPT SNAPSHOT

• At 10^-6, historical four-GPT history-target counts were 171, 159, 160, and 48 of 273 for Sol, Terra, Luna, and GPT-5.5, respectively.Terra and Luna do not strictly follow their aggregate-score order at every tolerance; labels denote recorded configurations, not an external intelligence test.Source: §B.3 · p. 17
• GPT-5.5’s median budget use was 11% for the complete historical configuration, whose runner finalized after two consecutive replies without a tool call.The prompt asked agents not to finalize while budget remained, and terminal status alone cannot distinguish model-authored from runner-mediated stopping.Source: §B.3 · pp. 17–18
• In the historical four-agent comparison, Sol exceeded GPT-5.5 by 0.634 aggregate-score points, with a problem-bootstrap interval of [0.537, 0.698].Sol–Terra and Sol–Luna differences were 0.054 [−0.064, 0.161] and 0.072 [−0.038, 0.167], respectively, so the three GPT-5.6 scores do not establish a total order.Source: §B.3 · p. 18

C SELECTED OPTIMIZATION HISTORIES

• The historical screen covered ten repetitions each of four interpretability-selected problems, and its flagged events neither identify new model decisions nor establish escape from a local minimum.All 40 traces passed integrity checks, with warnings for legacy fields; ten trust-region traces had flagged events and 30 had none. The screening thresholds were selection conventions, not statistical tests.Source: C SELECTED OPTIMIZATION HISTORIES · p. 18
• HAIRY repetition 5 improved from 49.57 to 24.03 after the agent resumed a numerical campaign, while an EGGCRATE reset failed to improve materially before stopping at 39 evaluations.On HAIRY, the reduction occurred within the resumed numerical method; EGGCRATE’s best value remained near 9.488.Source: C SELECTED OPTIMIZATION HISTORIES · p. 18
• A flat best-so-far curve indicates observed stagnation, not a certified local minimum, and a tool call followed by improvement does not establish that the call caused it.Attributing choices to the model requires linking evaluations to campaigns and agent requests; comparative cases require the same instance, transformed start, budget, information policy, and controller configuration.Source: C SELECTED OPTIMIZATION HISTORIES · pp. 18–19
• On SNAIL repetition 9, a reset trust-region call followed evaluation 81 and the best recorded value reached 1.157 × 10−3 at evaluation 98 within the 100-evaluation budget.The clean evaluator check at that point was 2.101 × 10−4 and was not available to the agent; the sequence documents association, not causality or a certified local minimum.Source: C SELECTED OPTIMIZATION HISTORIES · p. 19
• In an earlier HIMMELBG trial, a Newton step worsened the objective after evaluation 33, but the environment retained the incumbent and the agent stopped with 66 evaluations remaining.Evaluation 33 reached 9.45 × 10−22; evaluation 34 had objective value 0.0544. The case illustrates incumbent preservation and session-level stopping, not a successful escape or benefit from the final tool switch.Source: C SELECTED OPTIMIZATION HISTORIES · p. 19
• On perturbed ROSENBR, NextEval’s median clean history-best value was 3.34 × 10−14, versus 3.37 × 10−3 for COBYQA and 5.92 × 10−2 for NEWUOA.The passage characterizes this as a large difference in final numerical precision.Source: C SELECTED OPTIMIZATION HISTORIES · p. 19
• On CLIFF, NextEval’s median clean history-best value was 0.2009, versus 0.1998 for both model-based references.The passage characterizes this as only a small difference in final attained values.Source: C SELECTED OPTIMIZATION HISTORIES · p. 19

C.1 STANDARDIZED TEN-REPEAT CASE RECORDS

• The standardized campaign records eight selected cases: two paired ROSENBRTU outcomes and six traces, presented descriptively with recorded actions, observations, budgets, and paired classical solver observations.The ROSENBRTU records use Qwen 3.8 Flash, paired starts, perturbation and noise strengths of 0.001, and 100-evaluation budgets; Table 5 and Figure 9 show their outcomes. Archived COBYQA and PDFO histories were not rerun; Powell used the frozen starts, feature seeds, noisy evaluators, and per-case budgets, and archived COBYQA observations were reproduced exactly before Powell runs were added.Source: §C.1 · pp. 19–20, Table 5 · p. 21, Figure 9 · p. 21

D.1 TEN-REPEAT MODEL AND REASONING STUDIES

• Scores were 0.997, 0.993, 0.981, and 0.571 for GPT-5.6 Sol, Terra, Luna, and GPT-5.5; at 10−6 they reached 544, 536, 540, and 271 targets out of 920.Codex and medium reasoning were fixed across the four models; the comparison covers 92 problems and ten repeats.Source: §D.1 · p. 21, Figure 11 · p. 23
• Median budget use was 100% for GPT-5.6 configurations and 13% for GPT-5.5, with stopping behavior reflecting the complete model–harness configuration.All configurations were instructed not to finalize with budget remaining, but repeated replies without tool calls could trigger a terminal status in the runner.Source: §D.1 · p. 21
• History-performance scores were 0.924 with thinking off and 1.000 with medium reasoning; medium had lower median final clean values on 53 problems versus 39.Across individual repetitions, medium recorded 484 lower final values, 426 higher values, and 10 ties; repetitions are not independent problem families. The final Qwen snapshot preserves authorized terminal attempts rather than choosing the best objective value, so profiles measure objective-evaluation efficiency rather than recovery overhead.Source: §D.1 · p. 21

D.2 DEEPSEEK EXECUTION-HARNESS COMPARISON

• Claude Code performed better than the built-in runner in the matched DeepSeek V4 Flash configuration, but the comparison does not identify which harness component caused the difference.The participants shared numerical implementation, model, instructions, tools, and matched repeats, but used different context management and recovery; the comparison represents complete harnesses.Source: §D.2 · p. 22, Figure 13 · p. 24
• In the separate historical three-repeat Qwen study, 273 complete pairs yielded 136/8/129 lower/equal/higher final values for Codex versus direct tool calling.Different instruction wrappers and API transports distinguish this study from the matched DeepSeek comparison, and its results were not pooled with that comparison.Source: §D.2 · p. 23

E PLANNED MECHANISM TESTS

• The planned mechanism tests replace either the controller while retaining the environment or the numerical support while retaining the model, and require matched controls and full reporting.The first can compare against continued use of the implemented trust-region method; the second can expose only budgeted evaluations and permitted history. These substitutions are not automatically a factorial experiment, and a plateau alone cannot establish a local minimum.Source: E PLANNED MECHANISM TESTS · p. 25

F NUMERICAL AND EXECUTION DETAILS

• The trust-region tool fits a quadratic model with geometry checks and regularization, while the agent controls method use and evaluation allowances; termination and operational interruptions are distinct.The section also specifies retaining final results and partial histories and requiring the strict benchmark executor to enforce oracle, filesystem, and network boundaries independently.Source: F NUMERICAL AND EXECUTION DETAILS · p. 25

G DESIGN RATIONALE AND RELATED APPROACHES

• NextEval uses adaptive agent decisions to control numerical methods, but this design rationale neither guarantees success on arbitrary objectives nor establishes that LLM decisions cause performance differences.The paper distinguishes this approach from one-shot predictors and complementary program-search and application-specific agents (Yao et al., 2023; Merrill et al., 2026; Ye et al., 2024; Ahmaditeshnizi et al., 2024; Jiang et al., 2026).Source: G DESIGN RATIONALE AND RELATED… · pp. 25–26

H NUMERICAL TOOL CATALOG

• Table 9 separates method state across families, while its final two analysis rows use existing observations without new objective calls.The catalog implements the numerical operations used by the agent in Section 2.2.Source: Table 9 · p. 27, H NUMERICAL TOOL CATALOG · p. 27