LipschitzOptimizer

class LipschitzOptimizer(search_space: dict[str, list], initialize: dict[Literal['grid', 'vertices', 'random', 'warm_start'], int | list[dict]] = None, constraints: list[callable] = None, random_state: int = None, rand_rest_p: float = 0, nth_process: int = None, warm_start_smbo=None, max_sample_size: int = 10000000, sampling: dict[Literal['random'], int] = None, replacement: bool = True)[source]

Global optimizer using Lipschitz continuity bounds for deterministic search.

Lipschitz Optimization exploits the Lipschitz continuity property of objective functions to provide deterministic bounds on the optimum. If a function is Lipschitz continuous with constant L, the function value at any point cannot differ from nearby points by more than L times the distance. This property allows the algorithm to rule out regions that provably cannot contain the global optimum.

The algorithm estimates the Lipschitz constant from observed data and uses it to compute upper bounds on the objective function across the search space. Points with the highest potential (according to these bounds) are selected for evaluation, gradually tightening the bounds until convergence.

The algorithm is well-suited for:

  • Lipschitz continuous objective functions

  • Problems requiring global optimality guarantees

  • Low-dimensional optimization (bounds become loose in high dimensions)

  • Deterministic optimization without relying on random sampling

This method provides theoretical guarantees for finding the global optimum but requires the objective function to satisfy Lipschitz continuity. The estimated Lipschitz constant adapts as more observations are collected.

Parameters:
search_spacedict[str, list]

The search space to explore, defined as a dictionary mapping parameter names to arrays of possible values.

Each key is a parameter name (string), and each value is a numpy array or list of discrete values that the parameter can take. The optimizer will only evaluate positions that are on this discrete grid.

Example: A 2D search space with 100 points per dimension:

search_space = {
    "x": np.linspace(-10, 10, 100),
    "y": np.linspace(-10, 10, 100),
}

The resolution of each dimension (number of points in the array) directly affects optimization quality and speed. More points give finer resolution but increase the search space size exponentially.

initializedict[str, int], default={“vertices”: 4, “random”: 2}

Strategy for generating initial positions before the main optimization loop begins. Initialization samples are evaluated first, and the best one becomes the starting point for the optimizer.

Supported keys:

  • "grid": int – Number of positions on a regular grid.

  • "vertices": int – Number of corner/edge positions of the search space.

  • "random": int – Number of uniformly random positions.

  • "warm_start": list[dict] – Specific positions to evaluate, each as a dict mapping parameter names to values.

Multiple strategies can be combined:

initialize = {"vertices": 4, "random": 10}
initialize = {"warm_start": [{"x": 0.5, "y": 1.0}], "random": 5}

More initialization samples improve the starting point but consume iterations from n_iter. For expensive objectives, a few targeted warm-start points are often more efficient than many random samples.

constraintslist[callable], default=[]

A list of constraint functions that restrict the search space. Each constraint is a callable that receives a parameter dictionary and returns True if the position is valid, False if it should be rejected.

Rejected positions are discarded and regenerated: the optimizer resamples a new candidate position (up to 100 retries per step). During initialization, positions that violate constraints are filtered out entirely.

Example: Constrain the search to a circular region:

def circular_constraint(para):
    return para["x"]**2 + para["y"]**2 <= 25

constraints = [circular_constraint]

Multiple constraints are combined with AND logic (all must return True).

random_stateint or None, default=None

Seed for the random number generator to ensure reproducible results.

  • None: Use a new random state each run (non-deterministic).

  • int: Seed the random number generator for reproducibility.

Setting a fixed seed is recommended for debugging and benchmarking. Different seeds may lead to different optimization trajectories, especially for stochastic optimizers.

rand_rest_pfloat, default=0

Probability of performing a random restart instead of the normal algorithm step. At each iteration, a uniform random number is drawn; if it falls below rand_rest_p, the optimizer jumps to a random position instead of following its strategy.

  • 0.0: No random restarts (pure algorithm behavior).

  • 0.01-0.05: Light diversification, helps escape shallow local optima.

  • 0.1-0.3: Aggressive restarts, useful for highly multi-modal landscapes.

  • 1.0: Equivalent to random search.

This is especially useful for local search optimizers (Hill Climbing, Simulated Annealing) that can get trapped. For population-based optimizers, the effect is less pronounced since they already maintain diversity through multiple agents.

warm_start_smboobject or None, default=None

Previous SMBO state for warm-starting the surrogate model. Allows continuing optimization from a previous run by reusing the fitted model state, avoiding the cost of refitting from scratch.

Pass None to start fresh without warm-starting.

max_sample_sizeint, default=10000000

Maximum number of candidate points to consider when optimizing the acquisition function. The surrogate model predicts scores for these candidates, and the best one according to the acquisition function is selected for evaluation.

Larger values improve acquisition optimization quality but increase memory usage and computation time. For most problems, the default is more than sufficient. Reduce this if memory is a concern.

samplingdict, default={“random”: 1000000}

Configuration for how candidate points are generated for acquisition function optimization. The key specifies the sampling strategy and the value specifies the number of samples.

Currently supported: {"random": N} for uniform random sampling of N candidate points from the search space.

Example:

sampling = {"random": 500000}  # Fewer candidates, faster
replacementbool, default=True

Whether to sample candidate points with replacement when generating candidates for acquisition function optimization. When True, the same point can appear multiple times. When False, each candidate is unique, ensuring diversity but potentially slower for very large sample sizes.

Attributes:
best_para

Return the best parameters found as a dictionary.

best_value

Return the best values found (raw parameter values).

search_data

Lazily construct and return the search results DataFrame.

Methods

eval_time

init_stats

iter_time

search

See also

DirectAlgorithm

Deterministic global search without Lipschitz constant.

BayesianOptimizer

Surrogate-model-based approach using Gaussian Processes.

RandomSearchOptimizer

Simple alternative without Lipschitz assumptions.

Notes

The algorithm exploits the Lipschitz continuity property:

\[\begin{split}|f(x) - f(y)| \\leq L \\cdot \\|x - y\\|\end{split}\]

where L is the Lipschitz constant. This provides upper bounds on the function value at unobserved points:

\[\begin{split}f(x) \\leq f(x_i) + \\hat{L} \\cdot \\|x - x_i\\| \\quad \\forall i\end{split}\]

The algorithm estimates \(\\hat{L}\) from observed data and selects the point with the highest upper bound for the next evaluation. This gradually tightens the bounds and provably converges to the global optimum for Lipschitz continuous functions.

For visual explanations and tuning guides, see the Lipschitz Optimizer user guide.

Examples

>>> import numpy as np
>>> from gradient_free_optimizers import LipschitzOptimizer

>>> def smooth_function(para):
...     x, y = para["x"], para["y"]
...     return -(x**2 + y**2)  # Lipschitz continuous

>>> search_space = {
...     "x": np.linspace(-5, 5, 100),
...     "y": np.linspace(-5, 5, 100),
... }

>>> opt = LipschitzOptimizer(search_space)
>>> opt.search(smooth_function, n_iter=100)
property best_para[source]

Return the best parameters found as a dictionary.

Uses the Converter to transform the best position into user-friendly parameter names and values.

Returns:
dict or None

Dictionary mapping parameter names to their best values, or None if no evaluation has been performed yet.

property best_value[source]

Return the best values found (raw parameter values).

Returns:
list or None

List of best values in parameter order, or None if no evaluation has been performed yet.

property search_data: pd.DataFrame[source]

Lazily construct and return the search results DataFrame.

The DataFrame is only built when this property is accessed, avoiding a large memory spike at the end of high-dimensional optimizations. The result is cached so subsequent accesses don’t rebuild it.