code: def _log_improvement(
      self, obj_subsets: Tensor, view_shape: Union[Tuple, torch.Size]
  ) -> Tensor:
      # smooth out the clamp and take the log (previous step 3)
      # subtract cell lower bounds, clamp min at zero, but first
      # make obj_subsets broadcastable with cell bounds:
      # mc_samples x batch_shape x (num_cells = 1) x q_choose_i x i x m
      obj_subsets = obj_subsets.unsqueeze(-4)
      # making cell bounds broadcastable with obj_subsets:
      # (mc_samples = 1) x (batch_shape = 1) x num_cells x 1 x (i = 1) x m
      cell_lower_bounds = self.cell_lower_bounds.view(
          view_shape
      ).unsqueeze(-3)
      Z = obj_subsets - cell_lower_bounds
      log_Zi = self._log_smooth_relu(Z)
      return log_Zi
  
imports: from __future__ import annotations
  from typing import Callable, List, Optional, Tuple, Union
  import torch
  from botorch.acquisition.logei import TAU_MAX, TAU_RELU
  from botorch.acquisition.multi_objective import MultiObjectiveMCAcquisitionFunction
  from botorch.acquisition.multi_objective.objective import MCMultiOutputObjective
  from botorch.models.model import Model
  from botorch.sampling.base import MCSampler
  from botorch.utils.multi_objective.box_decompositions.non_dominated import NondominatedPartitioning
  from botorch.utils.multi_objective.hypervolume import NoisyExpectedHypervolumeMixin, SubsetIndexCachingMixin
  from botorch.utils.objective import compute_smoothed_feasibility_indicator
  from botorch.utils.safe_math import fatmin, log_fatplus, log_softplus, logdiffexp, logmeanexp, logplusexp, logsumexp, smooth_amin
  from botorch.utils.transforms import concatenate_pending_points, is_ensemble, match_batch_shape, t_batch_mode_transform
  from torch import Tensor
label_desc: Use image node_id 3 for calling the qLogExpectedHypervolumeImprovement obj's underlying member method code with example usage: obj._log_improvement(obj_subsets, view_shape) and returns: log_Zi