class_docstr: Implement the acquisition function of Sparsity Exploring Bayesian
  Optimization (SEBO).
  
  The SEBO is a hyperparameter-free method to simultaneously maximize a target
  objective and sparsity. When L0 norm is used, SEBO uses a novel differentiable
  relaxation based on homotopy continuation to efficiently optimize for sparsity.
docstr: Optimize SEBO ACQF with L0 norm using homotopy.
docstr_tok: ["Optimize","SEBO","ACQF","with","L0","norm","using","homotopy","."]
code: def _optimize_with_homotopy(
      self,
      n: int,
      search_space_digest: SearchSpaceDigest,
      fixed_features: Optional[Dict[int, float]] = None,
      rounding_func: Optional[Callable[[Tensor], Tensor]] = None,
      optimizer_options: Optional[Dict[str, Any]] = None,
  ) -> Tuple[Tensor, Tensor]:
      """Optimize SEBO ACQF with L0 norm using homotopy."""
      # extend to fixed a no homotopy_schedule schedule
      _tensorize = partial(
          torch.tensor, dtype=self.dtype, device=self.device
      )
      ssd = search_space_digest
      bounds = _tensorize(ssd.bounds).t()
  
      homotopy_schedule = LogLinearHomotopySchedule(
          start=0.1, end=1e-3, num_steps=30
      )
  
      # Prepare arguments for optimizer
      optimizer_options_with_defaults = optimizer_argparse(
          self.acqf,
          bounds=bounds,
          q=n,
          optimizer_options=optimizer_options,
      )
  
      def callback():  # pyre-ignore
          if (
              self.acqf.cache_pending
          ):  # If true, pending points are concatenated with X_baseline
              if self.acqf._max_iep != 0:
                  raise ValueError(
                      "The maximum number of pending points (max_iep) must be 0"
                  )
              X_baseline = self.acqf._X_baseline_and_pending.clone()
              self.acqf.__init__(  # pyre-ignore
                  X_baseline=X_baseline,
                  model=self.surrogates["sebo"].model,
                  ref_point=self.acqf.ref_point,
                  objective=self.acqf.objective,
              )
          else:  # We can directly get the pending points here
              X_pending = self.acqf.X_pending
              self.acqf.__init__(  # pyre-ignore
                  X_baseline=self.X_observed,
                  model=self.surrogates["sebo"].model,
                  ref_point=self.acqf.ref_point,
                  objective=self.acqf.objective,
              )
              self.acqf.set_X_pending(X_pending)
  
      homotopy = Homotopy(
          homotopy_parameters=[
              HomotopyParameter(
                  parameter=self.deterministic_model._f.a,
                  schedule=homotopy_schedule,
              )
          ],
          callbacks=[callback],
      )
      # need to know sparse dimensions
      batch_initial_conditions = get_batch_initial_conditions(
          acq_function=self.acqf,
          raw_samples=optimizer_options_with_defaults["raw_samples"],
          X_pareto=self.X_pareto,
          target_point=self.target_point,
          num_restarts=optimizer_options_with_defaults["num_restarts"],
          **{"device": self.device, "dtype": self.dtype},
      )
      candidates, expected_acquisition_value = optimize_acqf_homotopy(
          q=n,
          acq_function=self.acqf,
          bounds=bounds,
          homotopy=homotopy,
          num_restarts=optimizer_options_with_defaults["num_restarts"],
          raw_samples=optimizer_options_with_defaults["raw_samples"],
          post_processing_func=rounding_func,
          fixed_features=fixed_features,
          batch_initial_conditions=batch_initial_conditions,
      )
  
      return candidates, expected_acquisition_value
  
imports: import functools
  from copy import deepcopy
  from functools import partial
  from typing import Any, Callable, Dict, List, Optional, Tuple, Type
  import torch
  from ax.core.search_space import SearchSpaceDigest
  from ax.models.torch.botorch_modular.acquisition import Acquisition
  from ax.models.torch.botorch_modular.optimizer_argparse import optimizer_argparse
  from ax.models.torch.botorch_modular.surrogate import Surrogate
  from ax.models.torch_base import TorchOptConfig
  from ax.utils.common.constants import Keys
  from ax.utils.common.typeutils import not_none
  from botorch.acquisition.acquisition import AcquisitionFunction
  from botorch.acquisition.multi_objective.monte_carlo import qExpectedHypervolumeImprovement
  from botorch.acquisition.penalized import L0Approximation
  from botorch.models.deterministic import GenericDeterministicModel
  from botorch.models.model import ModelList
  from botorch.optim import Homotopy, HomotopyParameter, LogLinearHomotopySchedule, optimize_acqf_homotopy
  from botorch.utils.datasets import SupervisedDataset
  from botorch.utils.multi_objective.pareto import is_non_dominated
  from torch import Tensor
  from torch.quasirandom import SobolEngine
label_desc: Use image node_id 5 for calling the SEBOAcquisition obj's underlying member method code with example usage: obj._optimize_with_homotopy(n, search_space_digest, fixed_features, rounding_func, optimizer_options) and returns: candidates,  expected_acquisition_value