diff --git a/bayes_opt/acquisition.py b/bayes_opt/acquisition.py
index 4cd6828a..7765330f 100644
--- a/bayes_opt/acquisition.py
+++ b/bayes_opt/acquisition.py
@@ -565,6 +565,7 @@ def get_acquisition_params(self) -> dict[str, Any]:
             "kappa": self.kappa,
             "exploration_decay": self.exploration_decay,
             "exploration_decay_delay": self.exploration_decay_delay,
+            "i": self.i,
         }
 
     def set_acquisition_params(self, params: dict[str, Any]) -> None:
@@ -578,6 +579,7 @@ def set_acquisition_params(self, params: dict[str, Any]) -> None:
         self.kappa = params["kappa"]
         self.exploration_decay = params["exploration_decay"]
         self.exploration_decay_delay = params["exploration_decay_delay"]
+        self.i = params.get("i", self.i)
 
 
 class ProbabilityOfImprovement(AcquisitionFunction):
@@ -745,6 +747,7 @@ def get_acquisition_params(self) -> dict[str, Any]:
             "xi": self.xi,
             "exploration_decay": self.exploration_decay,
             "exploration_decay_delay": self.exploration_decay_delay,
+            "i": self.i,
         }
 
     def set_acquisition_params(self, params: dict[str, Any]) -> None:
@@ -758,6 +761,7 @@ def set_acquisition_params(self, params: dict[str, Any]) -> None:
         self.xi = params["xi"]
         self.exploration_decay = params["exploration_decay"]
         self.exploration_decay_delay = params["exploration_decay_delay"]
+        self.i = params.get("i", self.i)
 
 
 class ExpectedImprovement(AcquisitionFunction):
@@ -934,6 +938,7 @@ def get_acquisition_params(self) -> dict[str, Any]:
             "xi": self.xi,
             "exploration_decay": self.exploration_decay,
             "exploration_decay_delay": self.exploration_decay_delay,
+            "i": self.i,
         }
 
     def set_acquisition_params(self, params: dict[str, Any]) -> None:
@@ -947,6 +952,7 @@ def set_acquisition_params(self, params: dict[str, Any]) -> None:
         self.xi = params["xi"]
         self.exploration_decay = params["exploration_decay"]
         self.exploration_decay_delay = params["exploration_decay_delay"]
+        self.i = params.get("i", self.i)
 
 
 class ConstantLiar(AcquisitionFunction):
diff --git a/tests/test_acquisition_state.py b/tests/test_acquisition_state.py
new file mode 100644
index 00000000..e9ad8d8d
--- /dev/null
+++ b/tests/test_acquisition_state.py
@@ -0,0 +1,87 @@
+from __future__ import annotations
+
+import json
+
+import pytest
+
+from bayes_opt import BayesianOptimization
+from bayes_opt.acquisition import ExpectedImprovement, ProbabilityOfImprovement, UpperConfidenceBound
+
+
+@pytest.mark.parametrize(
+    ("acquisition_type", "parameter"),
+    [(UpperConfidenceBound, "kappa"), (ProbabilityOfImprovement, "xi"), (ExpectedImprovement, "xi")],
+)
+@pytest.mark.parametrize(
+    ("delay", "suggestions_before_save"),
+    [(3, 2), (3, 4), (None, 2), (0, 2), (3, 0)],
+    ids=["before-delay", "after-delay", "no-delay", "zero-delay", "unused-acquisition"],
+)
+def test_save_load_preserves_exploration_schedule(
+    tmp_path, acquisition_type, parameter, delay, suggestions_before_save
+):
+    def make_optimizer():
+        acquisition = acquisition_type(
+            **{parameter: 1.0}, exploration_decay=0.5, exploration_decay_delay=delay
+        )
+        optimizer = BayesianOptimization(
+            f=lambda x: -((x - 0.3) ** 2),
+            pbounds={"x": (-1.0, 1.0)},
+            acquisition_function=acquisition,
+            random_state=7,
+            verbose=0,
+        )
+        optimizer.set_gp_params(n_restarts_optimizer=0)
+        return optimizer
+
+    original = make_optimizer()
+    for x in (-1.0, 0.0, 1.0):
+        original.probe({"x": x}, lazy=False)
+    for _ in range(suggestions_before_save):
+        original.suggest()
+
+    state_path = tmp_path / "optimizer.json"
+    original.save_state(state_path)
+    restored = make_optimizer()
+    restored.load_state(state_path)
+
+    assert restored.res == original.res
+    assert getattr(restored.acquisition_function, parameter) == getattr(
+        original.acquisition_function, parameter
+    )
+
+    for _ in range(2):
+        assert restored.suggest() == pytest.approx(original.suggest())
+        assert getattr(restored.acquisition_function, parameter) == getattr(
+            original.acquisition_function, parameter
+        )
+
+
+@pytest.mark.parametrize(
+    ("acquisition_type", "parameter"),
+    [(UpperConfidenceBound, "kappa"), (ProbabilityOfImprovement, "xi"), (ExpectedImprovement, "xi")],
+)
+def test_load_legacy_acquisition_state(tmp_path, acquisition_type, parameter):
+    acquisition = acquisition_type(**{parameter: 1.0}, exploration_decay=0.5, exploration_decay_delay=3)
+    original = BayesianOptimization(
+        f=lambda x: x, pbounds={"x": (-1.0, 1.0)}, acquisition_function=acquisition, random_state=7, verbose=0
+    )
+    state_path = tmp_path / "legacy.json"
+    original.save_state(state_path)
+    state = json.loads(state_path.read_text())
+    state["acquisition_params"].pop("i", None)
+    state_path.write_text(json.dumps(state))
+
+    restored_acquisition = acquisition_type(**{parameter: 2.0})
+    restored = BayesianOptimization(
+        f=lambda x: x,
+        pbounds={"x": (-1.0, 1.0)},
+        acquisition_function=restored_acquisition,
+        random_state=7,
+        verbose=0,
+    )
+    restored.load_state(state_path)
+    assert getattr(restored_acquisition, parameter) == 1.0
+    assert restored_acquisition.exploration_decay == 0.5
+    assert restored_acquisition.exploration_decay_delay == 3
+    assert restored.suggest() == original.suggest()