Does the objective need to be differentiable?

Not always, but the chosen algorithm's assumptions determine which guarantees apply.

Is derivative-free optimization always random?

No. Both deterministic and randomized methods exist.

Can it use a surrogate model?

Yes. Models fitted to function values are a major approach; derivatives of the surrogate do not require derivatives from the original evaluator.

AI Glossary / Mathematics and optimization / Intermediate

Derivative-free optimization

Definition

Derivative-free optimization searches for good decision-variable values without requiring derivatives of the objective or constraints to be supplied. It uses information such as function evaluations, comparisons, and models built from sampled values. This is useful when an objective is available only through an experiment, simulation, or other evaluator that does not expose reliable gradients.

What it means

An optimizer needs a way to decide where to try next. A gradient-based method uses local rates of change, whereas a derivative-free method can work from observed outputs alone. For example, a simulation may return a design's error without explaining how that error changes with each input.

“Derivative-free” describes the information requirement, not the absence of mathematical structure. A method may fit a local model and use derivatives of that model internally. Nor does the label promise global optimality: many derivative-free algorithms seek a local solution under specified assumptions.

How it works

Define the decision variables, feasible region, objective, and evaluation budget. Evaluate one or more starting points. A direct-search method compares candidate points generated by a geometric rule; a model-based method builds an approximation from available values and chooses informative or promising new points.

After each evaluation, update the current candidate, search scale, or model. Constraints, failed evaluations, and noise need explicit handling. A simulation crash is not automatically a meaningful objective value, and one unusually favorable noisy result should not automatically replace the best verified candidate.

Finite-difference gradients are a related alternative, but can consume many evaluations and be sensitive to noise and step size. Terminology varies around derivative estimation; the central distinction here is that analytical or automatically differentiated derivatives need not be available from the original evaluator.

Three practical examples

1. Simulation calibration: adjust a small set of model parameters to reduce disagreement with measurements when the simulator returns only a fitting score.

2. Experimental design tuning: compare permitted machine settings using measured performance, with replication and feasibility rules appropriate to the experiment.

3. Machine-learning configuration: tune continuous hyperparameters using validation scores, keeping the final test data separate from the repeated tuning process.

Think of it this way

Think of improving a recipe by tasting carefully chosen trials without having a formula for how flavor changes with each ingredient. The search learns from the results and chooses the next trial.

The analogy has limits: mathematical algorithms specify rules for selecting and comparing trials, and some have convergence guarantees under assumptions. Taste-test intuition alone does not establish those guarantees, particularly with noisy observations or many interacting variables.

Worked example

Consider an original one-dimensional teaching objective f(x)=(x−2)²+1 on [0,4]. Pretend the optimizer receives only evaluated values. Start at x=0 with trial step 1 and use a simple coordinate-polling rule.

The starting value is f(0)=5. The feasible trial x=1 gives f(1)=2, so move there. Its next improving trial x=2 gives f(2)=1. From x=2, the neighbors x=1 and x=3 both give 2, so neither improves the incumbent.

Reduce the step to 0.5. The trials x=1.5 and x=2.5 both give 1.25. Again there is no improvement. This demonstrates value-based comparison and search-scale reduction without a gradient query.

Because the formula is visible to us as readers, we can independently prove the global minimum is 1 at x=2: the squared term is nonnegative. The optimizer's small collection of observations alone does not provide that proof for an arbitrary unknown function. The trace is hand-calculated, not a software execution or a general convergence demonstration.

Illustrative case study

Fictional case: Quarry Analytics calibrates three coefficients in a legacy simulation. Each run returns an error score, but the software offers no gradient interface. The team compares a derivative-free search with its existing manual tuning approach under the same evaluation budget.

It records failed runs, repeats apparently strong configurations when outputs vary, and checks the final parameters on separate scenarios. The decision depends on verified improvement per evaluation and robustness, not merely on how few optimizer iterations appear in a dashboard. The case describes an evaluation plan rather than fabricated performance results.

Common misconceptions and mistakes

Derivative-free does not mean computation-free. Function evaluations can dominate cost, and additional variables often make sample efficiency more difficult.

Do not assume all methods handle discontinuities, noise, or constraints equally well. A convergence result for smooth deterministic objectives may not apply to a stochastic simulator.

Avoid calling every local result the global optimum. Also, check whether useful derivative information can actually be obtained: automatic differentiation or known structural derivatives may be preferable when accessible and reliable.

Business use

The method is attractive when access to an evaluator is easier than access to its internal mathematics. Establish the value of one evaluation, acceptable constraints, and the budget before choosing an algorithm.

Compare methods using total evaluations, wall-clock time, feasibility, and validated objective quality. Warm starts, scaling, cached evaluations, and parallel trials can influence practical results. Retain a reproducible record of inputs and outputs so an apparently good candidate can be rechecked independently rather than trusted solely because the search selected it.

Check your understanding

Why does failure to improve at x=1.5 and x=2.5 not generally certify that x=2 is globally best?

Answer: Those evaluations inspect only two alternatives. Another part of an arbitrary objective could contain a better point. In this special example, global optimality follows from the known nonnegative-square formula, not merely from unsuccessful polling.

Related terms

Trust region
Surrogate model
Bayesian optimization
Black-box optimization
direct search
finite differences
noisy objective
local optimum
evaluation budget

Learn next

Implement a small value-only search on a function with a known minimum and record every evaluation. Then add controlled noise and compare single observations with replicated averages. Study direct-search and model-based trust-region methods before choosing a method for an expensive evaluator, paying particular attention to their assumptions and stopping criteria.

Trust region

Sources

Primary source: Larson, Menickelly and Wild, “Derivative-free optimization methods”, https://optimization-online.org/wp-content/uploads/2019/04/7153.pdf . The introduction, scope, and method classification were consulted. This entry explains the information-access concept without assigning one convergence guarantee to the entire field. The polling trace and fictional case are original.

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