RandomRestartHillClimbingOptimizer

class RandomRestartHillClimbingOptimizer(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, epsilon: float = 0.03, distribution: Literal['normal', 'laplace', 'gumbel', 'logistic'] = 'normal', n_neighbours: int = 3, n_iter_restart: int = 10)

Hill climbing variant that periodically restarts from random positions.

Random Restart Hill Climbing addresses the local optima problem by periodically resetting the search to a new random position after a fixed number of iterations. This simple yet effective strategy allows the algorithm to explore multiple regions of the search space, increasing the probability of finding the global optimum. The best solution found across all restarts is retained.

The algorithm is well-suited for:

• Multimodal optimization problems with many local optima

• Problems where the location of the global optimum is unknown

• Scenarios where multiple independent searches are beneficial

• Situations requiring a simple, parallelizable approach

The n_iter_restart parameter controls the frequency of restarts. Shorter intervals lead to more exploration but less exploitation of each local region, while longer intervals allow more thorough local search before restarting. The optimal value depends on the problem’s landscape and the expected basin of attraction size.

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.

epsilonfloat, default=0.03

Step size for generating neighbor positions, expressed as a fraction of each dimension’s range. Controls how far the optimizer looks from the current position when sampling neighbors.

• 0.01-0.02: Fine-grained local search, slow convergence.

• 0.03-0.05: Moderate step size, good default range.

• 0.1-0.3: Large steps, broader exploration.

• 0.5-1.0: Very large steps, nearly global jumps.

Example: For a dimension with np.linspace(0, 100, 1000) (range = 100):

• epsilon=0.03 leads to neighbors within ~3 units

• epsilon=0.1 leads to neighbors within ~10 units

Smaller values are better for fine-tuning near a known good solution. Larger values help escape local optima but may overshoot narrow peaks.

distribution{“normal”, “laplace”, “gumbel”, “logistic”}, default=”normal”

Probability distribution used to sample neighbor offsets. Each distribution produces different exploration patterns:

• "normal": Gaussian distribution. Most neighbors are close to the current position with rare far jumps. Best general-purpose choice.

• "laplace": Sharper peak than normal with heavier tails. Good for landscapes where occasional large jumps help.

• "gumbel": Asymmetric, skewed distribution. Can bias exploration in one direction.

• "logistic": Similar to normal but with slightly heavier tails. A middle ground between normal and Laplace.

The distribution interacts with epsilon: heavy-tailed distributions (Laplace) effectively increase the chance of large steps beyond what epsilon alone suggests.

n_neighboursint, default=3

Number of neighbor positions to sample and evaluate per iteration. The optimizer moves to the best among these neighbors (if it improves on the current position).

• 1: Minimal sampling, fast iterations but may miss good directions. Good for very cheap objective functions.

• 3-5: Moderate sampling, good balance of quality and speed.

• 10-20: Thorough neighborhood evaluation, better directional choices but more function evaluations per step.

Higher values act like a local beam search, giving the optimizer more information about the local landscape at each iteration.

n_iter_restartint, default=10

Number of iterations between random restarts. After this many iterations of hill climbing, the optimizer jumps to a new random position and begins climbing again. The best solution across all restarts is retained.

• 5-10: Frequent restarts, many short climbs. Good for landscapes with many shallow local optima.

• 20-50: Moderate restart frequency. Good balance for most problems.

• 100+: Infrequent restarts, thorough local search per restart. Good for landscapes with wide basins of attraction.

The optimal value depends on how many iterations the hill climber typically needs to reach a local optimum. With a total budget of n_iter iterations, you get approximately n_iter / n_iter_restart independent restarts.

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

HillClimbingOptimizer

Base hill climbing without restarts.

RepulsingHillClimbingOptimizer

Escapes local optima via step size increase.

RandomAnnealingOptimizer

Temperature-based step size reduction.

Notes

The algorithm alternates between hill climbing phases and random restarts:

1. Perform standard hill climbing for n_iter_restart iterations.

2. Jump to a uniformly random position in the search space.

3. Repeat until the iteration budget is exhausted.

The best solution found across all restart cycles is retained. With k restarts on a landscape with m local optima, the probability of finding the global optimum is approximately \(1 - (1 - 1/m)^k\), assuming roughly equal basin sizes.

For visual explanations and tuning guides, see the Random Restart Hill Climbing user guide.

Examples

>>> import numpy as np
>>> from gradient_free_optimizers import RandomRestartHillClimbingOptimizer

>>> def schwefel(para):
...     x, y = para["x"], para["y"]
...     return -(418.9829 * 2 - x * np.sin(np.sqrt(abs(x)))
...              - y * np.sin(np.sqrt(abs(y))))

>>> search_space = {
...     "x": np.linspace(-500, 500, 1000),
...     "y": np.linspace(-500, 500, 1000),
... }

>>> opt = RandomRestartHillClimbingOptimizer(search_space, n_iter_restart=20)
>>> opt.search(schwefel, n_iter=500)

property best_para

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

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

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.