CMA-ES (Covariance Matrix Adaptation Evolution Strategy)

CMA-ES is the theory-heavyweight of the founding four: a population-based evolution strategy developed by Nikolaus Hansen & Andreas Ostermeier (late 1990s), widely regarded as a top general-purpose continuous optimizer. Source: andrey-dik's MQL5 implementation (C_AO_CMAES).

How it works

Samples candidate solutions from a multivariate normal x_k ~ N(m, σ²C) (mean m, step-size σ, covariance C), then each iteration:

Ranks candidates by fitness, selects the best μ;
updates the mean by weighted recombination of top performers;
adapts the covariance matrix via rank-one and rank-μ updates along evolutionary paths;
adapts step-size by monitoring path length (stagnation detection).

Strengths & limits

Invariance to affine transforms of the search space: "equally efficiently solves f(x) and f(Ax + b)" — excellent on ill-conditioned, noisy, discontinuous, multi-modal landscapes.
Scalability wall: O(n²) memory, O(n³) operations — "for high-dimensional problems (n > 100), the resource intensity becomes disproportionate."

Benchmark (context-relative)

Ranked ~38/45 (48.33%) on the population-optimization-benchmark — fine at 5D/25D, failed at 500D on cost. A striking case for the no-free-lunch-theorem: high reputation, mid-low rank on this suite (see synthesis).

Caveat

Benchmark figure is from one MQL5 suite/implementation; CMA-ES rates far higher on many academic benchmarks — read the rank as suite-relative.

Why T2 (re-tiered 2026-08-25)

Held at T3 since ingest, which graded the venue — MQL5, a broker's community article site — rather than what the page rests on. This page's load-bearing claims are a mechanism description of an algorithm published by Hansen and Ostermeier (1996) and a comparative score, both from andrey-dik, who did not write the algorithm and has no stake in where it lands. That is somebody measuring somebody else's work, which QUALITY.md's ladder puts at T2.

The spoke's standing finding — that in this field the scorer is usually the author — is real and does not apply here; it applies to DOAm, which stays T3 because there Dik is scoring his own modification. The confound that does remain is implementation: Dik re-implements each optimizer in MQL5, so his figure grades his code as much as the method, and his scoring scheme changed mid-series (population-optimization-benchmark). Both stay flagged in prose, where a methodological limit belongs — not in the tier.

Note the floor is unmoved by this: T1 ÷ tiered counts T1 against the same denominator either way, so the change alters the tier mix and not the spoke's floor of 0.18.

metaheuristic-optimization · exploration-vs-exploitation · no-free-lunch-theorem · population-optimization-benchmark · andrey-dik · backtracking-search-algorithm