TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Finite merit-globalization budgets return a defined status

Statement

For finite caps $R, J_{\max} \geq 1$ and finite-valued merit evaluations, the procedure of L1 merit globalization with finite penalty and backtracking budgets terminates with exactly one of its specified accepted, QP, nonpositive-prediction, domain-exit, or line-search-budget statuses. Each accepted record contains the trial point and QP multiplier vectors; an ordinary accepted record also contains its fraction. In particular, the theorem asserts no unconditional accepted step.

Facts & Assumptions

Given: The globalization algorithm, finite positive caps, and well-defined finite arithmetic evaluations at every tested in-domain point.

[F1]

L1 merit globalization with finite penalty and backtracking budgets performs at most $R$ penalty attempts and at most $J_{\max}$ ordinary trial tests per attempt, plus at most one optional in-domain SOC test per attempt, and records whether any positive prediction and any in-domain trial test occurred.

[F2]

The exhaustive active-set QP solve terminates with the unique minimizer exactly when feasible proves that the exact exhaustive QP solve terminates with a subproblem status.

[F3]

Minimum-norm second-order correction for identified active constraints specifies a single active-residual evaluation and a finite-dimensional rank test and linear solve, or its zero-row or failure return.

[A1]

The caps $R, J_{\max}$ are positive finite integers and every in-domain merit evaluation terminates with a finite value.

Proof

1.1A1F1F2F3algebra

By [F2], the initial exact QP call returns after finitely many tests. An infeasible or failure status terminates the globalization routine immediately. Otherwise each penalty attempt computes one finite prediction and either advances or executes a finite list of tests. Each optional SOC call consists of the finite operations in [F3], including its zero-row and failure branches, so it also terminates.

2.1A1F1step 1.1algebra∎

There are at most $R (1 + J_{\max})$ globalization tests. A passing test returns an accepted record. If none passes, the stored flags select exactly one of the nonpositive-prediction, domain-exit, or line-search-budget-exhausted statuses according to whether any positive prediction and any in-domain test occurred. Thus the routine terminates with a defined status, without implying that any test must pass.

Depends on

Used by

Nothing in the library uses this result yet.

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Sources

Jorge Nocedal and Stephen J. Wright, Numerical Optimization, 2nd ed. (standard reference, not scraped)