Constraint Handling

Equality-manifold optimization maintained near-exact budget satisfaction in synthetic experiments. The slack-variable method recovered the optimum when equality methods wasted budget on under-budget optima. The manifold projection removes the constraint-normal component of biased straight-through gradients but does not eliminate tangential bias. The inequality-constrained extension adds a slack variable so unused budget can be absorbed. The slack-variable retraction is not globally smooth at the boundary where slack is zero.