Warm start ADMM (#1202)

* warm start admm

* Update qiskit/optimization/algorithms/admm_optimizer.py

Co-authored-by: Steve Wood <40241007+woodsp-ibm@users.noreply.github.com>

* updated documentation

* moved reno to a new file

* Update releasenotes/notes/admm-warm-start-6f5b5c801bcdf78f.yaml

Co-authored-by: Manoel Marques <Manoel.Marques@ibm.com>
Co-authored-by: Steve Wood <40241007+woodsp-ibm@users.noreply.github.com>

qiskit/optimization/algorithms/admm_optimizer.py

@@ -53,6 +53,7 @@ def __init__(self,
                  tau_decr: float = 2,
                  mu_res: float = 10,
                  mu_merit: float = 1000,
+                 warm_start: bool = False,
                  max_iter: Optional[int] = None) -> None:
         """Defines parameters for ADMM optimizer and their default values.
 
@@ -76,6 +77,11 @@ def __init__(self,
             tau_decr: Parameter used in the rho update (UPDATE_RHO_BY_RESIDUALS).
             mu_res: Parameter used in the rho update (UPDATE_RHO_BY_RESIDUALS).
             mu_merit: Penalization for constraint residual. Used to compute the merit values.
+            warm_start: Start ADMM with pre-initialized values for binary and continuous variables
+                by solving a relaxed (all variables are continuous) problem first. This option does
+                not guarantee the solution will optimal or even feasible. The option should be
+                used when tuning other options does not help and should be considered as a hint
+                to the optimizer where to start its iterative process.
             max_iter: Deprecated, use maxiter.
         """
         super().__init__()
@@ -97,6 +103,7 @@ def __init__(self,
         self.factor_c = factor_c
         self.beta = beta
         self.rho_initial = rho_initial
+        self.warm_start = warm_start
 
     def __repr__(self) -> str:
         props = ", ".join(["{}={}".format(key, value) for (key, value) in vars(self).items()])
@@ -292,6 +299,9 @@ def solve(self, problem: QuadraticProgram) -> ADMMOptimizationResult:
         self._state.binary_indices = self._get_variable_indices(problem, Variable.Type.BINARY)
         self._state.continuous_indices = self._get_variable_indices(problem,
                                                                     Variable.Type.CONTINUOUS)
+        if self._params.warm_start:
+            # warm start injection for the initial values of the variables
+            self._warm_start(problem)
 
         # convert optimization problem to a set of matrices and vector that are used
         # at each iteration.
@@ -830,6 +840,26 @@ def _get_solution_residuals(self, iteration: int) -> Tuple[float, float]:
 
         return primal_residual, dual_residual
 
+    def _warm_start(self, problem: QuadraticProgram) -> None:
+        """Solves a relaxed (all variables are continuous) and initializes the optimizer state with
+            the found solution.
+
+        Args:
+            problem: a problem to solve.
+
+        Returns:
+            None
+        """
+        qp_copy = copy.deepcopy(problem)
+        for variable in qp_copy.variables:
+            variable.vartype = VarType.CONTINUOUS
+        cts_result = self._continuous_optimizer.solve(qp_copy)
+        logger.debug("Continuous relaxation: %s", cts_result.x)
+
+        self._state.x0 = cts_result.x[self._state.binary_indices]
+        self._state.u = cts_result.x[self._state.continuous_indices]
+        self._state.z = cts_result.x[self._state.binary_indices]
+
     @property
     def parameters(self) -> ADMMParameters:
         """Returns current parameters of the optimizer.

releasenotes/notes/admm-warm-start-6f5b5c801bcdf78f.yaml

@@ -0,0 +1,7 @@
+---
+features:
+  - |
+    Introduced an option `warm_start` that should be used when tuning other options does not help.
+    When this option is enabled, a relaxed problem (all variables are continuous) is solved first
+    and the solution is used to initialize the state of the optimizer before it starts the
+    iterative process in the `solve` method.

test/optimization/test_admm.py

@@ -163,6 +163,40 @@ def test_admm_ex5(self):
         self.assertIsNotNone(solution.state)
         self.assertIsInstance(solution.state, ADMMState)
 
+    def test_admm_ex5_warm_start(self):
+        """Example 5 but with a warm start"""
+        mdl = Model('ex5')
+
+        # pylint:disable=invalid-name
+        v = mdl.binary_var(name='v')
+        w = mdl.binary_var(name='w')
+        t = mdl.binary_var(name='t')
+
+        mdl.minimize(v + w + t)
+        mdl.add_constraint(2 * v + 2 * w + t <= 3, "cons1")
+        mdl.add_constraint(v + w + t >= 1, "cons2")
+        mdl.add_constraint(v + w == 1, "cons3")
+
+        op = QuadraticProgram()
+        op.from_docplex(mdl)
+
+        admm_params = ADMMParameters(
+            rho_initial=1001, beta=1000, factor_c=900,
+            maxiter=100, three_block=False, warm_start=True
+        )
+
+        solver = ADMMOptimizer(params=admm_params)
+        solution = solver.solve(op)
+
+        self.assertIsNotNone(solution)
+        self.assertIsInstance(solution, ADMMOptimizationResult)
+        self.assertIsNotNone(solution.x)
+        np.testing.assert_almost_equal([0., 1., 0.], solution.x, 3)
+        self.assertIsNotNone(solution.fval)
+        np.testing.assert_almost_equal(1., solution.fval, 3)
+        self.assertIsNotNone(solution.state)
+        self.assertIsInstance(solution.state, ADMMState)
+
     def test_admm_ex6(self):
         """Example 6 as a unit test. Example 6 is reported in:
         Gambella, C., & Simonetto, A. (2020).