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A budgeted weighted rational Remez exchange terminates within its finite cap

Statement

For a finite integer budget B∈N0, the procedure Finite-budget weighted rational Remez exchange with defect status makes at most B+1 alternation solver calls and at most B+1 extrema-oracle calls (at most 2(B+1) calls total) and returns an explicit status, provided each input procedure terminates. This does not assert that the nonlinear solver succeeds or that the exchange iterates converge.

Facts & Assumptions

Given: The input and status contract of Finite-budget weighted rational Remez exchange with defect status, with a finite integer budget B∈N0 and terminating solver/oracle calls.

[F1]

Finite-budget weighted rational Remez exchange with defect status: each iteration makes one solver call and, only after a normal candidate, at most one extrema-oracle call; it either returns a status or exchanges one reference node.

Proof

1.1givenF1

Every entered iteration has exactly one solver call; if the solver returns a status, the procedure terminates. If it returns a defective candidate, the explicit defect branch terminates. Otherwise a normal candidate receives one oracle call. An inconclusive response returns oracle-inconclusive; every other response either returns a certificate or zero-level status, or searches the finite reference set for an exchange and then returns no-valid-exchange or updates one node. Thus each iteration terminates under the given termination hypothesis.

2.1givenF1step 1.1algebra∎

The loop has only the indices 0,…,B, so it has at most B+1 iterations. Each iteration makes one solver call and at most one oracle call, giving at most B+1 calls to each procedure. If no earlier branch returns, the final iteration is followed by budget-exhausted. Therefore the procedure terminates within the stated total call bound.

Depends on

Used by

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