Find your next learning step
Projecting onto one constraint and then another creates a simple iteration, but its limit depends on convexity, intersection, ambient geometry and the precise meaning of convergence.
Open the chapter contents · Topology root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Define the iteration
Route 2 · Work the projection ledger
Route 3 · Interpret limits and fixed points
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE ITERATION
1. Alternate two nearest-point maps
Let A and B be nonempty closed convex subsets of a Hilbert space, or of a metric setting where the needed projections are single-valued. Starting from x_0, form y_n=P_B(x_n) and x_{n+1}=P_A(y_n). Each half-step enforces one constraint, perhaps breaking the other.
The composition T=P_A P_B is a self-map of A after the first cycle. A limit x satisfying T(x)=x is a fixed point of the cycle. Whether that fixed point lies in A∩B, gives a best-approximation pair, or fails to exist depends on the hypotheses.
CHAPTER 2 OF 12 · DEFINE THE ITERATION
2. Convexity makes each step canonical
For a nonempty closed convex set in a Hilbert space, nearest-point projection exists, is unique and is nonexpansive. Complete CAT(0) spaces have an analogous nearest-point theorem for closed convex targets. This is the same geometric foundation used in Local Convexity, Nearest-Point Projections and Equivariant Retractions.
If a target is nonconvex, a nearest point may be nonunique and the iteration becomes set-valued unless a tie rule is declared. A drawn “projection” is not a theorem.
CHAPTER 3 OF 12 · WORK THE PROJECTION LEDGER
3. Work a line-and-strip ledger
In the Euclidean plane let A be the x-axis and B be the closed half-plane y≥1. Begin at x_0=(3,4). Project to B: the point already lies in B, so y_0=(3,4). Project to A: x_1=(3,0). The next B-projection is y_1=(3,1), and the next A-projection returns x_2=(3,0).
| Step | Point | Set enforced | Distance to the other set |
|---|---|---|---|
| start | (3,4) | neither required | 4 to A |
| P_B | (3,4) | B | 4 to A |
| P_A | (3,0) | A | 1 to B |
| P_B | (3,1) | B | 1 to A |
| P_A | (3,0) | A | 1 to B |
The cycle is a fixed pair, not a point in an empty intersection.
CHAPTER 4 OF 12 · WORK THE PROJECTION LEDGER
4. Intersecting lines reveal the contraction
Now take A and B to be two lines through the origin meeting at angle θ. One full cycle multiplies the component away from the intersection by a factor related to cos²θ. When θ is small, progress is slow; when the lines are orthogonal, one cycle reaches the intersection.
This worked geometry explains why convergence rate depends on how sets meet. The existence of an intersection does not guarantee useful speed. Near-tangent constraints can produce a long tail of tiny improvements.
CHAPTER 5 OF 12 · WORK THE PROJECTION LEDGER
5. Fixed points need careful interpretation
If A∩B is nonempty, every intersection point is fixed by both projections. Under standard convex Hilbert-space conditions, the alternating sequence converges in the appropriate sense to a feasible point. For two closed subspaces, von Neumann’s theorem gives strong convergence to the orthogonal projection onto their intersection.
For general closed convex sets in infinite-dimensional spaces, weak and strong convergence must not be conflated, and the limit need not equal the metric projection of x_0 onto A∩B. If that particular best feasible point is required, use an algorithm whose theorem provides it.
CHAPTER 6 OF 12 · INTERPRET LIMITS AND FIXED POINTS
6. Changed context: constraints become inconsistent
Move B from a line crossing A to the parallel line y=1. The intersection disappears, but alternating projections still settle into the pair (3,0) and (3,1). A stopping rule based only on repeated iterates may call this “converged” without noticing infeasibility.
Track both step size and feasibility residual. A positive limiting gap can indicate inconsistent constraints or an approximation pair. In more general settings the minimum distance may not be attained, so even the best-pair story requires closedness and geometry.
CHAPTER 7 OF 12 · INTERPRET LIMITS AND FIXED POINTS
7. Extend the method to CAT(0) spaces carefully
In a complete CAT(0) space, convex targets have unique projections and geodesic convexity replaces linear combinations. Alternating projection results use metric notions of weak convergence, often called Δ-convergence, and additional regularity or compactness may be needed for strong convergence.
Do not import a Hilbert-space inner-product proof line by line. State the ambient space, convexity definition, projection theorem and convergence mode. A tree provides an accessible CAT(0) example: projecting alternately onto two subtrees follows unique gates and can visibly reach their intersection.
CHAPTER 8 OF 12 · INTERPRET LIMITS AND FIXED POINTS
8. Compare nearby projection algorithms
Alternating projections seeks feasibility, but it is not identical to Dykstra’s algorithm, averaged projections or Douglas–Rachford reflection. Correction vectors in Dykstra’s method target the actual projection onto an intersection in settings where plain alternation may reach another feasible point.
Use Dykstra’s Algorithm, Correction Vectors and Projections onto Intersections and Douglas–Rachford Reflections, Convex Feasibility and Operator Splitting as adjacent methods, not interchangeable names.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Convergence of residuals does not always imply strong convergence of iterates. A fixed point of P_A P_B is not automatically a point of A∩B when the sets are disjoint. Nonconvex projections can branch, cycle or converge to locally meaningful points without global guarantees.
Finite numerical precision can create apparent stationarity. Changing the order of more than two projections can change behaviour. A topological deformation retraction and an iterative metric projection answer different questions even when both use the word projection.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Build an evidence check
State the space, metric, target sets and proof of closedness and convexity. Record projection formulas, starting point, order, residual d(x_n,A∩B) when computable, inter-set gap, step size and objective. Name weak, strong or Δ-convergence precisely.
Test an intersecting case, a nearly tangent case, a disjoint case and a nonconvex counterexample. Verify the same code against a problem with an analytic solution. Stop only when the declared feasibility and iteration criteria are both satisfied.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Questions readers should ask
Does alternating projection always return the nearest point in the intersection?
No. Special cases do; general feasibility convergence does not automatically identify that particular projection.
What does a two-point cycle mean?
For suitable disjoint convex sets, it can represent a best-approximation pair rather than algorithmic failure.
Why state the convergence mode?
Weak, strong and Δ-convergence carry different guarantees, especially in infinite-dimensional or nonlinear metric spaces.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue through verified routes
Begin with the Topology root, Gold Standard of Education and How X Works. Continue through Augmented-Core Flow, CAT(0) Links and Contractibility, Core Slices, Convex Fibres and One-Dimensional Compatibility and Compatibility Fans, Irreducible Trees and Length-Function Sums.
Primary reading includes Bačák, Searston and Sims’ alternating projections in CAT(0) spaces, Bauschke and Borwein’s projection algorithms and convex feasibility, and Lewis, Luke and Malick’s local convergence for nonconvex projections. The line-and-strip and intersecting-line ledgers are original teaching examples.
Top · Contents · Previous · How X Works