RepulsingHillClimbingOptimizer

class RepulsingHillClimbingOptimizer(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, repulsion_factor: float = 5)

Hill climbing variant that increases step size when stuck to escape local optima.

Repulsing Hill Climbing is an adaptive variant of hill climbing that dynamically increases the search radius (epsilon) when no improvement is found. This “repulsion” mechanism helps the optimizer escape local optima by taking progressively larger steps when stuck, effectively being “pushed away” from the current region. Once a better solution is found, the step size resets to its original value for fine-grained local search.

The algorithm is well-suited for:

• Optimization landscapes with isolated local optima

• Problems where the distance between optima is unknown

• Scenarios requiring automatic adaptation of search radius

• Balancing local exploitation and global exploration without manual tuning

The repulsion_factor parameter controls how aggressively the step size increases when stuck. A factor of 5 means epsilon is multiplied by 5 each time no improvement is found. Higher values lead to faster escape from local optima but may overshoot good regions.

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.

repulsion_factorfloat, default=5

Multiplicative factor applied to epsilon each time no improvement is found. The step size grows geometrically while stuck, enabling escape from local optima basins of increasing size.

• 1.0-2.0: Gentle increase, slow escape from local optima.

• 3.0-5.0: Moderate increase, good default range.

• 10.0+: Aggressive increase, quickly jumps to distant regions but may overshoot good areas.

After an improvement is found, epsilon resets to its original value. The effective step size after k stuck iterations is epsilon * repulsion_factor^k.

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 adaptive step size.

RandomRestartHillClimbingOptimizer

Periodic random restarts.

SimulatedAnnealingOptimizer

Escapes local optima through probabilistic acceptance.

Notes

The algorithm extends standard Hill Climbing with an adaptive step size mechanism:

\[\begin{split}\\epsilon_{\\text{eff}} = \\epsilon \\cdot r^{k}\end{split}\]

where r is the repulsion_factor and k is the number of consecutive iterations without improvement. When an improvement is found, k resets to 0.

This creates an automatic exploration-exploitation cycle: the optimizer performs fine-grained local search when improving, then broadens its search when stuck, and returns to local search after finding a new promising region.

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

Examples

>>> import numpy as np
>>> from gradient_free_optimizers import RepulsingHillClimbingOptimizer

>>> def ackley(para):
...     x, y = para["x"], para["y"]
...     return -(-20 * np.exp(-0.2 * np.sqrt(0.5 * (x**2 + y**2)))
...              - np.exp(0.5 * (np.cos(2*np.pi*x) + np.cos(2*np.pi*y)))
...              + np.e + 20)

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

>>> opt = RepulsingHillClimbingOptimizer(search_space, repulsion_factor=3)
>>> opt.search(ackley, n_iter=200)

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.