Improve docstring and naming of test of kkt conditions.

tests/unit/aggregation/test_dual_cone_utils.py

@@ -6,24 +6,48 @@
 
 
 @mark.parametrize("shape", [(5, 7), (9, 37), (2, 14), (32, 114), (50, 100)])
-def test_lagrangian_satisfies_kkt_conditions(shape: tuple[int, int]):
+def test_solution_weights(shape: tuple[int, int]):
+    r"""
+    Tests that `_get_projection_weights` returns valid weights corresponding to the projection onto
+    the dual cone of a matrix with the specified shape.
+
+    Validation is performed by verifying that the solution satisfies the `KKT conditions
+    <https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions>`_ for the
+    quadratic program that projects vectors onto the dual cone of a matrix.
+    Specifically, the solution should satisfy the equivalent set of conditions described in Lemma 4
+    of [1].
+
+    Let:
+    - `u` be a vector of weights,
+    - `G` a positive semi-definite matrix,
+    - Consider the quadratic problem of minimizing `v^\top G v` subject to `u \preceq v`.
+
+    Then `w` is a solution if and only if it satisfies the following three conditions:
+    1. **Dual feasibility:** `u \preceq w`
+    2. **Primal feasibility:** `0 \preceq G w`
+    3. **Complementary slackness:** `u^\top G w = w^\top G w`
+
+    Reference:
+    [1] `Jacobian Descent For Multi-Objective Optimization <https://arxiv.org/pdf/2406.16232>`_.
+    """
     matrix = torch.randn(shape)
     weights = torch.rand(shape[0])
 
     gramian = matrix @ matrix.T
 
     projection_weights = _get_projection_weights(gramian, weights, "quadprog")
-    lagrange_multiplier = projection_weights - weights
+    dual_gap = projection_weights - weights
 
-    positive_lagrange_multiplier = lagrange_multiplier[lagrange_multiplier >= 0.0]
-    assert_close(
-        positive_lagrange_multiplier.norm(), lagrange_multiplier.norm(), atol=1e-05, rtol=0
-    )
+    # Dual feasibility
+    dual_gap_positive_part = dual_gap[dual_gap >= 0.0]
+    assert_close(dual_gap_positive_part.norm(), dual_gap.norm(), atol=1e-05, rtol=0)
 
-    constraint = gramian @ projection_weights
+    primal_gap = gramian @ projection_weights
 
-    positive_constraint = constraint[constraint >= 0]
-    assert_close(positive_constraint.norm(), constraint.norm(), atol=1e-04, rtol=0)
+    # Primal feasibility
+    primal_gap_positive_part = primal_gap[primal_gap >= 0]
+    assert_close(primal_gap_positive_part.norm(), primal_gap.norm(), atol=1e-04, rtol=0)
 
-    slackness = lagrange_multiplier @ constraint
+    # Complementary slackness
+    slackness = dual_gap @ primal_gap
     assert_close(slackness, torch.zeros_like(slackness), atol=3e-03, rtol=0)