Constrained optimization has two classical approaches. Projected gradient descent: take an unconstrained step, then project back onto the feasible set. Sequential Quadratic Programming (SQP): solve a quadratic approximation of the problem subject to linearized constraints, then step. These look different — one projects, the other linearizes. Goyens and Feppon (arXiv: 2603.24309) show they're both geodesic flows on the same manifold with different metrics.
The unifying framework: treat the constraint set as a Riemannian manifold. The metric — the definition of distance on the manifold — determines the geometry. Different metrics produce different geodesics (shortest paths). Projected gradient descent corresponds to one choice of metric. SQP corresponds to another. The augmented Lagrangian method corresponds to a third.
The “landing” method decomposes each step into tangent and normal components. The tangent component moves along the constraint surface (making progress). The normal component pushes toward the constraint surface (becoming feasible). Asymptotically, the iterates “land” on the feasible set — approaching it from the ambient space without expensive exact projections.
The key insight: the choice of metric determines the oblique projector that splits tangent from normal. By parameterizing the metric in terms of oblique projectors, the authors disentangle the roles of direction (which way to go) and feasibility (how to stay on the surface). This produces systematic guidelines for designing new metrics with closed-form update rules.
The through-claim: algorithms are coordinate systems on the same geometry. Projected gradient descent and SQP solve the same optimization problem on the same constraint manifold. They differ not in what they optimize but in how they measure distance. The algorithms are not alternatives — they're the same algorithm in different coordinates.
Goyens & Feppon, 2603.24309. Optimization / Riemannian geometry / constrained optimization / SQP / projected gradient descent.