code: def test_optimize_w_posterior_transform(self):
      # This is mainly testing that we can optimize without errors.
      for dtype in (torch.float, torch.double):
          tkwargs = {"dtype": dtype, "device": self.device}
          mean = torch.tensor([1.0, 0.5], **tkwargs).expand(2, 1, 2)
          cov = torch.tensor(
              [[1.0, 0.1], [0.1, 0.5]], **tkwargs
          ).expand(2, 2, 2)
          posterior = GPyTorchPosterior(
              MultitaskMultivariateNormal(mean, cov)
          )
          model = MockModel(posterior)
          n_f = 4
          mean = torch.tensor([1.0, 0.5], **tkwargs).expand(
              n_f, 2, 1, 2
          )
          cov = torch.tensor(
              [[1.0, 0.1], [0.1, 0.5]], **tkwargs
          ).expand(n_f, 2, 2, 2)
          posterior = GPyTorchPosterior(
              MultitaskMultivariateNormal(mean, cov)
          )
          mfm = MockModel(posterior)
          bounds = torch.zeros(2, 2, **tkwargs)
          bounds[1] = 1
          options = {"num_inner_restarts": 2, "raw_inner_samples": 2}
          with mock.patch.object(
              MockModel, "fantasize", return_value=mfm
          ):
              kg = qMultiFidelityKnowledgeGradient(
                  model=model,
                  num_fantasies=n_f,
                  posterior_transform=ScalarizedPosteriorTransform(
                      weights=torch.rand(2, **tkwargs)
                  ),
              )
              # Mocking this to get around grad issues.
              with mock.patch(
                  f"{optimize_acqf.__module__}.gen_candidates_scipy",
                  return_value=(
                      torch.zeros(2, n_f + 1, 2, **tkwargs),
                      torch.zeros(2, **tkwargs),
                  ),
              ), mock.patch(
                  f"{optimize_acqf.__module__}._filter_kwargs",
                  wraps=lambda f, **kwargs: _filter_kwargs(
                      function=gen_candidates_scipy, **kwargs
                  ),
              ):
                  candidate, value = optimize_acqf(
                      acq_function=kg,
                      bounds=bounds,
                      q=1,
                      num_restarts=2,
                      raw_samples=2,
                      options=options,
                  )
          self.assertTrue(
              torch.equal(candidate, torch.zeros(1, 2, **tkwargs))
          )
  
imports: from contextlib import ExitStack
  from unittest import mock
  import torch
  from botorch.acquisition.analytic import PosteriorMean, ScalarizedPosteriorMean
  from botorch.acquisition.cost_aware import GenericCostAwareUtility
  from botorch.acquisition.knowledge_gradient import _get_value_function, _split_fantasy_points, ProjectedAcquisitionFunction, qKnowledgeGradient, qMultiFidelityKnowledgeGradient
  from botorch.acquisition.monte_carlo import qExpectedImprovement, qSimpleRegret
  from botorch.acquisition.objective import GenericMCObjective, ScalarizedPosteriorTransform
  from botorch.acquisition.utils import project_to_sample_points
  from botorch.exceptions.errors import UnsupportedError
  from botorch.generation.gen import gen_candidates_scipy
  from botorch.models import SingleTaskGP
  from botorch.optim.optimize import optimize_acqf
  from botorch.optim.utils import _filter_kwargs
  from botorch.posteriors.gpytorch import GPyTorchPosterior
  from botorch.sampling.normal import IIDNormalSampler, SobolQMCNormalSampler
  from botorch.utils.test_helpers import DummyNonScalarizingPosteriorTransform
  from botorch.utils.testing import BotorchTestCase, MockModel, MockPosterior
  from gpytorch.distributions import MultitaskMultivariateNormal
label_desc: Use image node_id 4 for calling the TestQMultiFidelityKnowledgeGradient obj's underlying member method code with example usage: obj.test_optimize_w_posterior_transform() without return types