The heatmap is the negative log-likelihood surface over two parameters (μ₁, μ₂). Brighter/warmer = lower NLL = more likely. The gold diamond marks the true global optimum (found by dense grid search). Each colored path shows where an optimizer walks starting from the same point.

Switch to the hard surface (a 2-component mixture model, the textbook case where MLE is genuinely non-convex) and start both optimizers near the middle — several gradient-based and derivative-free methods collapse onto the degenerate point μ₁ = μ₂ = data mean, a well-known local trap in mixture-model likelihoods. Basin-hopping restarts from random perturbations and usually escapes it.

On the easy surface (two separate, independent Gaussian means — a genuinely convex NLL) every method lands in the same place almost immediately, because there's only one hill to descend.