# ============================================================================
# T03 — Gradient-free optimization on non-convex benchmark functions
# ----------------------------------------------------------------------------
# Unlike paper_replication's P01-P07, the "synthesis" here frames a research
# QUESTION rather than a known paper's method. The model must design the
# experiment (conditions, metrics, datasets) — we only commit to what a
# competent study of this topic would include and what the rubric expects.
# ============================================================================

id: ML03
title: "CPU comparison of Nelder-Mead, Powell, and CMA-ES on non-convex functions"
arxiv_id: null
venue: "ARC-Bench 2026"
paper_asset: null

# The "synthesis" plays the role of the upstream briefing: research question,
# background, why the question matters, what "a reasonable experiment" looks
# like. It deliberately does NOT pre-specify a single method to reproduce.
synthesis: |
  Gradient-free optimization methods are widely used when derivatives are
  unavailable, unreliable, or expensive to compute. In low-to-medium
  dimensional non-convex landscapes, direct-search methods such as
  Nelder-Mead and Powell are frequently used because they are easy to call
  through scipy.optimize. Population-based approaches such as CMA-ES can be
  more robust to local minima and ill-conditioning, but they introduce extra
  hyperparameters and potentially higher per-iteration cost.

  A CPU-bounded benchmark can clarify practical trade-offs by evaluating these
  methods under a fixed function-evaluation budget on standard synthetic
  objective functions with known global minima. Rather than asking which method
  is universally best, the relevant question is whether one method reaches
  better objective values more reliably within the same budget and how much
  runtime overhead that reliability costs.

  A credible experiment should evaluate at least three optimization conditions
  (Nelder-Mead, Powell, CMA-ES) across multiple non-convex test functions,
  using repeated random starts and common stopping/budget rules. Reporting only
  final objective value is incomplete; runtime and success-rate-to-threshold
  should be included to capture both quality and efficiency. Statistical
  summaries over seeds are required because single runs are high variance.

  The resulting evidence should produce explicit pass/fail verdicts for each
  hypothesis: whether CMA-ES improves best-found objective value, whether
  Powell offers faster wall-clock convergence than CMA-ES at similar budgets,
  and whether Nelder-Mead underperforms on multimodal landscapes. This keeps
  the study measurable and feasible within a short single-core runtime window.

  *Under equal function-evaluation budgets on CPU, which gradient-free optimizer (Nelder-Mead, Powell, CMA-ES) gives the best trade-off between final objective quality, runtime, and success rate on non-convex benchmark functions?*

hypotheses:
  - id: H1
    statement: "CMA-ES achieves lower mean best objective value than both Nelder-Mead and Powell on at least 2 of 3 benchmark functions, averaged over >=10 random starts with the same evaluation budget."
    measurable: true
  - id: H2
    statement: "Powell has lower median wall-clock runtime than CMA-ES on at least 2 of 3 benchmark functions while finishing within the same function-evaluation cap."
    measurable: true
  - id: H3
    statement: "On the multimodal Rastrigin function, Nelder-Mead attains a lower success rate (fraction of runs reaching f(x) <= 1e-2) than CMA-ES by at least 0.20 absolute."
    measurable: true

experiment_design:
  research_question: "Under equal evaluation budgets, how do Nelder-Mead, Powell, and CMA-ES compare on objective quality, runtime, and success probability for non-convex synthetic objectives?"
  conditions:
    - name: "nelder_mead"
      description: "scipy.optimize.minimize with method='Nelder-Mead', random initial point per run, maxfev budget enforced."
    - name: "powell"
      description: "scipy.optimize.minimize with method='Powell', same initialization protocol and maxfev cap."
    - name: "cma_es"
      description: "CMA-ES implementation (lightweight numpy/scipy variant) with population updates under the same total function-evaluation budget."
    - name: "random_search_baseline"
      description: "Uniform random search within bounded domain using the same number of objective evaluations as other methods."
  baselines:
    - "random_search_baseline as a budget-matched non-adaptive baseline"
  metrics:
    - name: "primary_metric"
      direction: "minimize"
      description: "Mean best objective value found at termination (lower is better), aggregated over random starts."
    - name: "median_runtime_sec"
      direction: "minimize"
      description: "Median wall-clock runtime per run in seconds for each (method, function)."
    - name: "success_rate_eps"
      direction: "maximize"
      description: "Fraction of runs reaching objective value <= 1e-2 by termination."
  datasets:
    - name: "rastrigin_10d"
      source: "synthetic numpy implementation of Rastrigin function in 10 dimensions"
    - name: "rosenbrock_10d"
      source: "synthetic numpy implementation of Rosenbrock function in 10 dimensions"
    - name: "ackley_10d"
      source: "synthetic numpy implementation of Ackley function in 10 dimensions"
  compute_requirements:
    gpu_required: false
    estimated_wall_clock_sec: 480

rubric_path: "experiments/arc_bench/config/ml/rubrics/ML03.json"