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A nonsmooth objective may behave smoothly after an algorithm identifies the correct active manifold. Before that moment, the iterates can cross strata with different local geometry, so one rate certificate may hide the real transition.
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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 · Frame the problem
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · FRAME THE PROBLEM
1. A critical neighbourhood can be stratified
Stratification partitions a set into smooth pieces, or strata, that fit together under regularity conditions. In nonsmooth optimisation, different support patterns, active constraints or ranks can define different local pieces.
The Kurdyka–Łojasiewicz, or KL, property controls how objective gaps relate to slopes near critical points. When the neighbourhood contains several strata, the relevant local constants or desingularising behaviour may change as the iterate moves between them.
CHAPTER 2 OF 12 · FRAME THE PROBLEM
2. Partial smoothness makes an active manifold visible
A function is partly smooth relative to a manifold when it is smooth along that manifold and its nonsmooth variation is organised normally to it. Under nondegeneracy and algorithm-specific conditions, iterates can identify the manifold in finite time.
After identification, a sparse problem may keep the same zero pattern or a constrained problem may keep the same active face. The algorithm can then exploit a lower-dimensional smooth model. Identification is a theorem with hypotheses, not a pattern inferred from a short plot.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A support-stratum ledger
Consider a proximal method for an objective with an absolute-value regulariser. The iterates approach a solution whose second coordinate is zero.
| Iteration range | Observed support | Candidate stratum | Interpretation |
|---|---|---|---|
| 1–8 | x and y nonzero | Two-dimensional smooth piece | No active zero identified |
| 9–13 | y alternates near zero | Boundary transitions | Identification not certified |
| 14–30 | y exactly zero | Active manifold y=0 | Finite identification candidate |
| Later perturbation test | y remains zero | Same manifold | Supports, but does not prove, containment |
Rate analysis before iteration 14 must allow boundary crossings. After verified identification, the proof may restrict to the manifold y=0 and use sharper local curvature.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked geometry for a simple composite objective
Take F(x,y)=½(x−3)²+½y²+λ|y| with λ=1. The minimiser is (3,0). Away from y=0 the objective is smooth on each sign stratum; at y=0 it is nonsmooth in the ambient plane but smooth along the active manifold.
| Point | Active description | Subgradient in y | Local reading |
|---|---|---|---|
| (2,0.6) | y>0 | y+1=1.6 | Positive-sign stratum |
| (2,−0.6) | y<0 | y−1=−1.6 | Negative-sign stratum |
| (2,0) | y=0 | [−1,1] | Active manifold boundary |
| (3,0) | Critical | 0 belongs to [−1,1] | Identified solution |
Once y=0 is identified, the remaining objective along the manifold is ½(x−3)², giving a familiar quadratic local model. Before identification, a proof must include the nonsmooth normal direction.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Piecewise KL certificates need transition rules
A piecewise certificate can assign one valid KL modulus to each stratum and a safe envelope on overlaps. It also needs a rule proving which modulus applies to each tail segment and how the estimates telescope across a transition.
Choosing the fastest local exponent after observing convergence is circular. The certificate must establish entry, containment or a finite number of controlled transitions independently of the rate it claims.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Identification can unlock local acceleration
After active-manifold identification, Newton or quasi-Newton steps may be applied to the smooth reduced problem. The speedup comes from using intrinsic geometry, not from declaring the ambient nonsmoothness gone.
Degenerate intersections can delay or prevent stable identification. If strict complementarity or transversality fails, iterates may skim several strata and the local fast rate may not start.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: low-rank recovery
Transfer the logic to a matrix problem whose strata are fixed-rank manifolds. A proximal algorithm may eventually identify the rank of the limit, after which optimisation can proceed on a smooth fixed-rank geometry.
The transfer check must distinguish numerical small singular values from exact rank identification. Repeated singular values and nontransverse constraints can change the active geometry and invalidate a naive reduced Hessian.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect stratification to the KL branch
The variable-KL guide tracks changing local moduli along a critical set. The uniformised-KL guide gives one safe inequality around a compact cluster set.
Use the metric-subregularity and error-bound guide when a local residual bound supports a linear certificate after the active geometry is fixed.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
Finite identification is not automatic for every nonsmooth method. Approximate subproblem solves, noise, degeneracy and changing step rules can prevent exact manifold entry. A support that looks stable for ten iterations may change later.
A KL rate does not prove convergence to a global minimum. Active-manifold acceleration can converge quickly to a saddle or nonglobal critical point unless other structure excludes it.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
State the stratification, active manifold, tangent and normal spaces, subdifferential, nondegeneracy condition and algorithmic identification theorem. Record when the iterate enters the manifold and why it cannot leave.
For each rate segment, give the objective gap, residual, KL modulus or exponent, descent constant and transition rule. Compare ambient and reduced models without fitting a theorem from the same tail it is meant to certify.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is an active set the same as an active manifold?
An active set is often a combinatorial label. Partial smoothness turns the identified structure into a local manifold with tangent and normal geometry.
Does identification happen when a value is merely small?
No. Numerical thresholds are diagnostics. Finite identification is exact in the mathematical algorithm under stated assumptions.
Why use a piecewise rate?
It can describe a slow transition across strata followed by a faster rate on the identified manifold, while keeping each claim tied to the geometry where it holds.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Hare and Lewis develop active-constraint identification through partial smoothness and prox-regularity. Lewis and Wylie connect partial smoothness to active-set Newton methods. Lewis and Tian study identifiability, KL geometry and subgradient curves. The geometric ledgers above are original teaching examples.
Continue through the Topology root, variable local KL geometry and KL exponents and finite-length trajectories. The Gold Standard of Education and How X Works provide the wider route.
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